{"id":15689,"date":"2019-09-05T12:08:19","date_gmt":"2019-09-05T16:08:19","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/sequences\/"},"modified":"2019-09-05T12:08:19","modified_gmt":"2019-09-05T16:08:19","slug":"sequences","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/sequences\/","title":{"raw":"Sequences","rendered":"Sequences"},"content":{"raw":"[latexpage]<div class=\"textbox textbox--learning-objectives\"><h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>By the end of this section, you will be able to: <ul><li>Write the first few terms of a sequence<\/li><li>Find a formula for the general term (nth term) of a sequence<\/li><li>Use factorial notation<\/li><li>Find the partial sum<\/li><li>Use summation notation to write a sum<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167836335254\" class=\"be-prepared\"><p id=\"fs-id1167836613299\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167836334429\" type=\"1\"><li>Evaluate \\(2n+3\\) for the integers 1, 2, 3, and 4.<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>Evaluate \\({\\left(-1\\right)}^{n}\\) for the integers 1, 2, 3, and 4.<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536158\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>If \\(f\\left(n\\right)={n}^{2}+2,\\) find \\(f\\left(1\\right)+f\\left(2\\right)+f\\left(3\\right).\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/5e548626-8f0f-496d-ab87-4f0358ca2fd3#fs-id1167829859398\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><\/ol><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836362095\"><h3 data-type=\"title\">Write the First Few Terms of a Sequence<\/h3><p id=\"fs-id1167833025720\">Let\u2019s look at the function \\(f\\left(x\\right)=2x\\) and evaluate it for just the counting numbers.<\/p><table id=\"fs-id1167836595941\" summary=\"Table has 2 columns and 8 rows. The first row lists the function f of x equals 2 x. The first column lists values of x 1 through 5 with an ellipsis in the last row. The second column lists values of 2 x with values 2, 4, 6, 8, and 10 with an ellipsis in the last row.\" class=\"unnumbered\"><tbody><tr valign=\"top\"><td colspan=\"2\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(f\\left(x\\right)=2x\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">\\(x\\)<\/strong><\/td><td data-valign=\"bottom\" data-align=\"center\"><strong data-effect=\"bold\">\\(2x\\)<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">1<\/td><td data-valign=\"middle\" data-align=\"center\">2<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">4<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">3<\/td><td data-valign=\"middle\" data-align=\"center\">6<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">8<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">5<\/td><td data-valign=\"middle\" data-align=\"center\">10<\/td><\/tr><tr valign=\"top\"><td data-valign=\"middle\" data-align=\"center\">\u2026<\/td><td data-valign=\"middle\" data-align=\"center\">\u2026<\/td><\/tr><\/tbody><\/table><p id=\"fs-id1167836410426\">If we list the function values in order as 2, 4, 6, 8, and 10, \u2026 we have a sequence. A <span data-type=\"term\">sequence<\/span> is a function whose domain is the counting numbers.<\/p><div data-type=\"note\" id=\"fs-id1167836352765\"><div data-type=\"title\">Sequences<\/div><p id=\"fs-id1167832982337\">A <strong data-effect=\"bold\">sequence<\/strong> is a function whose domain is the counting numbers.<\/p><\/div><p id=\"fs-id1167829691634\">A sequence can also be seen as an ordered list of numbers and each number in the list is a <em data-effect=\"italics\">term<\/em>. A sequence may have an infinite number of terms or a finite number of terms. Our sequence has three dots (ellipsis) at the end which indicates the list never ends. If the domain is the set of all counting numbers, then the sequence is an <span data-type=\"term\">infinite sequence<\/span>. Its domain is all counting numbers and there is an infinite number of counting numbers.<\/p><div data-type=\"equation\" id=\"fs-id1167836507257\" class=\"unnumbered\" data-label=\"\">\\(2,4,6,8,10,\\text{\u2026},\\)<\/div><p id=\"fs-id1167836410266\">If we limit the domain to a finite number of counting numbers, then the sequence is a <span data-type=\"term\">finite sequence<\/span>. If we use only the first four counting numbers, 1, 2, 3, 4 our sequence would be the finite sequence,<\/p><div data-type=\"equation\" id=\"fs-id1167836693849\" class=\"unnumbered\" data-label=\"\">\\(2,4,6,8\\)<\/div><p id=\"fs-id1167833024298\">Often when working with sequences we do not want to write out all the terms. We want more compact way to show how each term is defined. When we worked with functions, we wrote \\(f\\left(x\\right)=2x\\) and we said the expression \\(2x\\) was the rule that defined values in the range. While a sequence is a function, we do not use the usual function notation. Instead of writing the function as \\(f\\left(x\\right)=2x,\\) we would write it as \\({a}_{n}=2n.\\) The \\({a}_{n}\\) is the <em data-effect=\"italics\">n<\/em>th term of the sequence, the term in the <em data-effect=\"italics\">n<\/em>th position where <em data-effect=\"italics\">n<\/em> is a value in the domain. The formula for writing the <em data-effect=\"italics\">n<\/em>th term of the sequence is called the <span data-type=\"term\">general term<\/span> or formula of the sequence.<\/p><div data-type=\"note\" id=\"fs-id1167836484969\"><div data-type=\"title\">General Term of a Sequence<\/div><p id=\"fs-id1167836571092\">The <strong data-effect=\"bold\">general term<\/strong> of the sequence is found from the formula for writing the <em data-effect=\"italics\">n<\/em>th term of the sequence. The <em data-effect=\"italics\">n<\/em>th term of the sequence, <em data-effect=\"italics\">a<sub>n<\/sub><\/em>, is the term in the <em data-effect=\"italics\">n<\/em>th position where <em data-effect=\"italics\">n<\/em> is a value in the domain.<\/p><\/div><p id=\"fs-id1167836367818\">When we are given the general term of the sequence, we can find the terms by replacing <em data-effect=\"italics\">n<\/em> with the counting numbers in order. For \\({a}_{n}=2n,\\)<\/p><table id=\"fs-id1167836626974\" summary=\"Table has 2 rows and 7 columns. The first row lists values of n from 1 through 5 with the last value as a sub n. The second row lists values of a sub n with mutiples of 2, from 2 through 10 with the last value as 2n. Below the table are values a sub 1, a sub 2, a sub 3, a sub 4, a sub 5, and a sub n followed by the values 2, 4, 6, 8, and 10.\" class=\"unnumbered\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\">\\(n\\)<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">1<\/td><td data-valign=\"middle\" data-align=\"center\">2<\/td><td data-valign=\"middle\" data-align=\"center\">3<\/td><td data-valign=\"middle\" data-align=\"center\">4<\/td><td data-valign=\"middle\" data-align=\"center\">5<\/td><td data-valign=\"top\" data-align=\"center\">\\({a}_{n}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\">\\({a}_{n}\\)<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}2\u00b71\\hfill \\\\ \\hfill 2\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}2\u00b72\\hfill \\\\ \\hfill 4\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}2\u00b73\\hfill \\\\ \\hfill 6\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}2\u00b74\\hfill \\\\ \\hfill 8\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}2\u00b75\\hfill \\\\ \\hfill 10\\hfill \\end{array}\\)<\/td><td data-valign=\"top\" data-align=\"center\">2<em data-effect=\"italics\">n<\/em><\/td><\/tr><\/tbody><\/table><div data-type=\"equation\" id=\"fs-id1167836512648\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\hfill {a}_{1},&amp; \\hfill {a}_{2},&amp; \\hfill {a}_{3},&amp; \\hfill {a}_{4},&amp; {a}_{5}\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\text{,}\\hfill &amp; {a}_{n}\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\hfill \\\\ \\hfill 2,&amp; \\hfill 4,&amp; \\hfill 6,&amp; \\hfill 8,&amp; 10\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\hfill &amp; \\end{array}\\)<\/div><p id=\"fs-id1167836535441\">To find the values of a sequence, we substitute in the counting numbers in order into the general term of the sequence.<\/p><div data-type=\"example\" id=\"fs-id1167836424231\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836408426\"><div data-type=\"problem\" id=\"fs-id1167836477802\"><p id=\"fs-id1167836547838\">Write the first five terms of the sequence whose general term is \\({a}_{n}=4n-3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836415705\"><p id=\"fs-id1167836545291\">We substitute the values 1, 2, 3, 4, and 5 into the formula, \\({a}_{n}=4n-3,\\) in order.<\/p><span data-type=\"media\" id=\"fs-id1167836551928\" data-alt=\"This figure shows three rows and five columns. The first row reads nth term equals 4 times n minus 3 written five times. The second row reads a sub 1 equals 4 times g times 1 minus 3, a sub 2 equals 4 times g times 2 minus 3, a sub 3 equals 4 times g times 3 minus 3, a sub 4 equals 4 times g times 4 minus 3, a sub 5 equals 4 times g times 5 minus 3. The third row reads, a sub 1 equals 1, a sub 2 equals 5, a sub 3 equals 9, a sub 4 equals 13, a sub 5 equals 17.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows three rows and five columns. The first row reads nth term equals 4 times n minus 3 written five times. The second row reads a sub 1 equals 4 times g times 1 minus 3, a sub 2 equals 4 times g times 2 minus 3, a sub 3 equals 4 times g times 3 minus 3, a sub 4 equals 4 times g times 4 minus 3, a sub 5 equals 4 times g times 5 minus 3. The third row reads, a sub 1 equals 1, a sub 2 equals 5, a sub 3 equals 9, a sub 4 equals 13, a sub 5 equals 17.\" \/><\/span><p id=\"fs-id1167836362021\">The first five terms of the sequence are 1, 5, 9, 13, and 17.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836512330\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836526397\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836327796\"><p id=\"fs-id1167836376014\">Write the first five terms of the sequence whose general term is \\({a}_{n}=3n-4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836622808\"><p id=\"fs-id1167836523502\">\\(-1,2,5,8,11\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829598214\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836327003\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836662447\"><p id=\"fs-id1167836309374\">Write the first five terms of the sequence whose general term is \\({a}_{n}=2n-5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836414969\"><p id=\"fs-id1167836418956\">\\(-3,-1,1,3,5\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836418844\">For some sequences, the variable is an exponent.<\/p><div data-type=\"example\" id=\"fs-id1167836410316\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836356303\"><div data-type=\"problem\" id=\"fs-id1167836516667\"><p id=\"fs-id1167829694727\">Write the first five terms of the sequence whose general term is \\({a}_{n}={2}^{n}+1.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836542257\"><p id=\"fs-id1167836311916\">We substitute the values 1, 2, 3, 4, and 5 into the formula, \\({a}_{n}={2}^{n}+1,\\) in order.<\/p><span data-type=\"media\" id=\"fs-id1167836557890\" data-alt=\"This figure shows three rows and five columns. The first row reads &#x201c;nth term equals 2 to the nth power plus 1&#x201d; written five times. The second row reads, &#x201c;a sub 1 equals 2 times 1 plus 1, a sub 2 equals 2 to the power of 2 plus 1, a sub 3 equals 2 to the power 3 plus 1, a sub 4 equals 2 to the power of 4 plus 1, a sub 5 equals 2 to the power 5 plus 1&#x201d;. The last row reads &#x201c;a sub 1 equals 3, a sub 2 equals 5, a sub 3 equals 9, a sub 4 equals 17, a sub 5 equals 33&#x201d;.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows three rows and five columns. The first row reads &#x201c;nth term equals 2 to the nth power plus 1&#x201d; written five times. The second row reads, &#x201c;a sub 1 equals 2 times 1 plus 1, a sub 2 equals 2 to the power of 2 plus 1, a sub 3 equals 2 to the power 3 plus 1, a sub 4 equals 2 to the power of 4 plus 1, a sub 5 equals 2 to the power 5 plus 1&#x201d;. The last row reads &#x201c;a sub 1 equals 3, a sub 2 equals 5, a sub 3 equals 9, a sub 4 equals 17, a sub 5 equals 33&#x201d;.\" \/><\/span><p id=\"fs-id1167836691944\">The first five terms of the sequence are 3, 5, 9, 17, and 33.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836700380\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836521836\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836367277\"><p id=\"fs-id1167836327173\">Write the first five terms of the sequence whose general term is \\({a}_{n}={3}^{n}+4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836320223\"><p id=\"fs-id1167829740836\">\\(7,13,31,85,247\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836691832\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836688426\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836510398\"><p id=\"fs-id1167829695706\">Write the first five terms of the sequence whose general term is \\({a}_{n}={2}^{n}-5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836362572\"><p id=\"fs-id1167836608138\">\\(-3,-1,3,11,27\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167836287578\">It is not uncommon to see the expressions \\({\\left(-1\\right)}^{n}\\) or \\({\\left(-1\\right)}^{n+1}\\) in the general term for a sequence. If we evaluate each of these expressions for a few values, we see that this expression alternates the sign for the terms.<\/p><table id=\"fs-id1167836521976\" summary=\"Table shows 6 columns and 3 rows. The first row lists values of n from 1 to 5. The second row lists values of negative 1 to the power of n as negative 1, 1, negative 1, 1, and negative 1. The third row lists values of negative 1 to the power of n plus 1 as 1 negative 1, 1, negative 1, and 1. Below the table are values a sub 1, a sub 2, a sub 3, a sub 4, a sub 5, and a sub n followed by ellipsis. Below are the values negative 1, 1, negative 1, 1, and negative 1. Below that are the numbers 1 negative 1, 1, negative 1, and 1.\" class=\"unnumbered\"><tbody><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\">\\(n\\)<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">1<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">2<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">3<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">4<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">5<\/strong><\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\">\\({\\left(-1\\right)}^{n}\\)<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{1}\\hfill \\\\ \\\\ \\hfill -1\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{2}\\hfill \\\\ \\\\ \\hfill 1\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{3}\\hfill \\\\ \\\\ \\hfill -1\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{4}\\hfill \\\\ \\\\ \\hfill 1\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{5}\\hfill \\\\ \\\\ \\hfill -1\\hfill \\end{array}\\)<\/td><\/tr><tr valign=\"top\"><td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\">\\({\\left(-1\\right)}^{n+1}\\)<\/strong><\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{1+1}\\hfill \\\\ \\\\ \\hfill 1\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{2+1}\\hfill \\\\ \\\\ \\hfill -1\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{3+1}\\hfill \\\\ \\\\ \\hfill 1\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{4+1}\\hfill \\\\ \\\\ \\hfill -1\\hfill \\end{array}\\)<\/td><td data-valign=\"middle\" data-align=\"center\">\\(\\begin{array}{c}\\hfill {\\left(-1\\right)}^{5+1}\\hfill \\\\ \\\\ \\hfill 1\\hfill \\end{array}\\)<\/td><\/tr><\/tbody><\/table><div data-type=\"equation\" id=\"fs-id1167836550517\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{cccccc}\\hfill {a}_{1},&amp; \\hfill {a}_{2},&amp; \\hfill {a}_{3},&amp; \\hfill {a}_{4},&amp; \\hfill {a}_{5}\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\text{,}&amp; \\hfill {a}_{n}\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\\\ \\hfill -1,&amp; \\hfill 1,&amp; \\hfill -1,&amp; \\hfill 1,&amp; -1\\text{\u2026}\\hfill &amp; \\\\ \\hfill 1,&amp; \\hfill -1,&amp; \\hfill 1,&amp; \\hfill -1,&amp; \\phantom{\\rule{0.6em}{0ex}}1\\text{\u2026}\\hfill &amp; \\end{array}\\)<\/div><p id=\"fs-id1167833059788\">The terms in the next example will alternate signs as a result of the powers of \\(-1.\\)<\/p><div data-type=\"example\" id=\"fs-id1167836340793\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836540214\"><div data-type=\"problem\" id=\"fs-id1167836512624\"><p id=\"fs-id1167836534763\">Write the first five terms of the sequence whose general term is \\({a}_{n}={\\left(-1\\right)}^{n}{n}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833051065\"><p id=\"fs-id1167836707050\">We substitute the values 1, 2, 3, 4, and 5 into the formula, \\({a}_{n}={\\left(-1\\right)}^{n}{n}^{3},\\) in order.<\/p><span data-type=\"media\" id=\"fs-id1167836622463\" data-alt=\"This figure shows three rows and five columns. The first row reads &#x201c;nth term equals negative 1 to the nth power times n cubed&#x201d; written five times. The second row reads a sub 1 equals negative 1 to the power of 1 times g times 1 cubed, a sub 2 equals negative 1 squared time g times 2 cubed, a sub 3 equals negative 1 cubed times g times 23 cubed, a sub 4 equals negative 1 to the power of 4 times g times 4 cubed, a sub 5 equals negative 1 to the power of 5 times g times 5 cubed. The last row reads, &#x201c;a sub 1 equals negative 1, a sub 2 equals 8, a sub 3 equals negative 27, a sub 4 equals 64, and a sub 5 equals negative 125.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows three rows and five columns. The first row reads &#x201c;nth term equals negative 1 to the nth power times n cubed&#x201d; written five times. The second row reads a sub 1 equals negative 1 to the power of 1 times g times 1 cubed, a sub 2 equals negative 1 squared time g times 2 cubed, a sub 3 equals negative 1 cubed times g times 23 cubed, a sub 4 equals negative 1 to the power of 4 times g times 4 cubed, a sub 5 equals negative 1 to the power of 5 times g times 5 cubed. The last row reads, &#x201c;a sub 1 equals negative 1, a sub 2 equals 8, a sub 3 equals negative 27, a sub 4 equals 64, and a sub 5 equals negative 125.\" \/><\/span><p id=\"fs-id1167836627870\">The first five terms of the sequence are \\(-1,8,-27,64,\\) and \\(-125.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836503186\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833019254\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836597185\"><p id=\"fs-id1167829720825\">Write the first five terms of the sequence whose general term is \\({a}_{n}={\\left(-1\\right)}^{n}{n}^{2}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829720139\"><p id=\"fs-id1167836520591\">\\(-1,4,-9,16,-25\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836367160\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829627648\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836553112\"><p id=\"fs-id1167829695390\">Write the first five terms of the sequence whose general term is \\({a}_{n}={\\left(-1\\right)}^{n+1}{n}^{3}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836352298\"><p id=\"fs-id1167836529276\">\\(1,-8,27,-64,125\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836526589\"><h3 data-type=\"title\">Find a Formula for the General Term (<em data-effect=\"italics\">n<\/em>th Term) of a Sequence<\/h3><p id=\"fs-id1167829740541\">Sometimes we have a few terms of a sequence and it would be helpful to know the general term or <em data-effect=\"italics\">n<\/em>th term. To find the general term, we look for patterns in the terms. Often the patterns involve multiples or powers. We also look for a pattern in the signs of the terms.<\/p><div data-type=\"example\" id=\"fs-id1167836330323\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836688056\"><div data-type=\"problem\" id=\"fs-id1167836299692\"><p id=\"fs-id1167836700504\">Find a general term for the sequence whose first five terms are shown.<\/p><div data-type=\"equation\" id=\"fs-id1167836621355\" class=\"unnumbered\" data-label=\"\">\\(4,8,12,16,20\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\dots \\)<\/div><\/div><div data-type=\"solution\" id=\"fs-id1167836392172\"><table id=\"fs-id1167836548526\" summary=\"This figure shows five rows. The first row reads, &#x201c;4&#x201d;, &#x201c;8&#x201d;, &#x201c;12&#x201d;, &#x201c;16&#x201d;, &#x201c;20&#x201d;, and an &#x201c;ellipsis&#x201d;. The second row reads &#x201c;n&#x201d;, &#x201c;1&#x201d;, &#x201c;2&#x201d;, &#x201c;3&#x201d;, &#x201c;4&#x201d;, &#x201c;5&#x201d;, and an &#x201c;ellipsis&#x201d;. The third row reads &#x201c;We look for a pattern in terms&#x201d;, &#x201c;Terms&#x201d;, &#x201c;4&#x201d;, &#x201c;8&#x201d;, &#x201c;12&#x201d;, &#x201c;16&#x201d;, &#x201c;20&#x201d;, and an &#x201c;ellipsis&#x201d;. The four row reads, &#x201c;The numbers are all multiples of 4&#x201d;, &#x201c;Pattern&#x201d;, &#x201c;4 times g times 1&#x201d;, &#x201c;4 times g times 3&#x201d;, 4 times g times 4&#x201d;, &#x201c;4 times g times 5&#x201d;, and an &#x201c;ellipsis&#x201d;, &#x201c;4 times g times n&#x201d;. The last row reads &#x201c;The general term of the sequence is a nth term equals 4 times n&#x201d;.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836509695\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836492778\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">We look for a pattern in the terms.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836700375\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">The numbers are all multiples of 4.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836330596\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\">The general term of the sequence is \\({a}_{n}=4n.\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829594848\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836356048\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829691674\"><p id=\"fs-id1167829579943\">Find a general term for the sequence whose first five terms are shown.<\/p><p id=\"fs-id1167836310267\">\\(3,6,9,12,15,\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836540322\"><p id=\"fs-id1167836511209\">\\({a}_{n}=3n\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836312781\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836688166\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833021343\"><p id=\"fs-id1167833158233\">Find a general term for the sequence whose first five terms are shown.<\/p><p id=\"fs-id1167829645005\">\\(5,10,15,20,25,\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833060799\"><p id=\"fs-id1167829853134\">\\({a}_{n}=5n\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167829628072\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836538101\"><div data-type=\"problem\" id=\"fs-id1167836504080\"><p id=\"fs-id1167836520449\">Find a general term for the sequence whose first five terms are shown.<\/p><div data-type=\"equation\" id=\"fs-id1167836292789\" class=\"unnumbered\" data-label=\"\">\\(2,-4,8,-16,32,\\text{\u2026}\\)<\/div><\/div><div data-type=\"solution\" id=\"fs-id1167836310482\"><table id=\"fs-id1167829756446\" summary=\"This figure shows six rows. The first row reads, &#x201c;2&#x201d;, &#x201c;negative 4&#x201d;, &#x201c;8&#x201d;, &#x201c;negative 16&#x201d;, &#x201c;32&#x201d;, and an &#x201c;ellipsis&#x201d;. The second row reads &#x201c;n&#x201d;, &#x201c;1&#x201d;, &#x201c;2&#x201d;, &#x201c;3&#x201d;, &#x201c;4&#x201d;, &#x201c;5&#x201d;, &#x201c;ellipsis&#x201d;, &#x201c;n&#x201d;. The third row reads &#x201c;We look for a pattern in the terms&#x201d;, &#x201c;Terms 2,&#x201d; &#x201c;negative 4&#x201d;, &#x201c;8&#x201d;, &#x201c;negative 16&#x201d;, &#x201c;32&#x201d;, and an &#x201c;ellipsis&#x201d;. The fourth row reads &#x201c;The numbers are powers of 2&#x201d;, and &#x201c;Pattern&#x201d;. The fifth row reads &#x201c;the signs are alternating with even n negative&#x201d;, &#x201c;negative 1 to the power of 1 plus 1 times 2 to the power of 1, negative 1 to the power of 2 plus 1 times 2 to the power of 2, negative 1 to the power of 3 plus 1 times 2 to the power of 3, negative 1 to the power of 4 plus 1 times 2 to the power of 1, negative 1 to the power of 5 plus 1 times 2 to the power of 5, ellipsis, negative 1 to the power of n plus 1 times 2 to then power of n. The last row reads, &#x201c;The general term of the sequence is a sub n equals negative 1 to the power of n plus 1 times 2 to the power of n.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833054918\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836432055\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">We look for a pattern in the terms.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836554839\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">The numbers are powers of 2. The signs are<span data-type=\"newline\"><br \/><\/span>alternating, with even \\(n\\) negative.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833369157\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\">The general term of the sequence is \\({a}_{n}={\\left(-1\\right)}^{n+1}{2}^{n}.\\) <\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829790449\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829694947\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829694949\"><p id=\"fs-id1167836558252\">Find a general term for the sequence whose first five terms are shown.<\/p><p id=\"fs-id1167836558255\">\\(-3,9,-27,81,-243\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836398987\"><p id=\"fs-id1167836398989\">\\({a}_{n}={\\left(-1\\right)}^{n}{3}^{n}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836417349\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836506804\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836506806\"><p id=\"fs-id1167836506808\">Find a general term for the sequence whose first five terms are shown<\/p><p id=\"fs-id1167829809843\">\\(1,-4,9,-16,25\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836536576\"><p id=\"fs-id1167829596213\">\\({a}_{n}={\\left(-1\\right)}^{n+1}{n}^{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"example\" id=\"fs-id1167836417304\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836417306\"><div data-type=\"problem\" id=\"fs-id1167836417309\"><p id=\"fs-id1167829628377\">Find a general term for the sequence whose first five terms are shown.<\/p><div data-type=\"equation\" id=\"fs-id1167829628380\" class=\"unnumbered\" data-label=\"\">\\(\\frac{1}{3},\\frac{1}{9},\\frac{1}{27},\\frac{1}{81},\\frac{1}{243},\\text{\u2026}\\)<\/div><\/div><div data-type=\"solution\" id=\"fs-id1167836448361\"><table id=\"fs-id1167836635242\" summary=\"This figure shows five rows. The first row reads, &#x201c;one-third&#x201d;, &#x201c;one-ninth&#x201d;, &#x201c;one-twenty-seventh&#x201d;, &#x201c;1 divided by 81&#x201d;, &#x201c;1 divided by 243&#x201d;, and an ellipsis. The second row reads, &#x201c;n&#x201d;, &#x201c;1&#x201d;, &#x201c;2&#x201d;, &#x201c;3&#x201d;, &#x201c;4&#x201d;, &#x201c;5&#x201d; and an ellipsis, &#x201c;n&#x201d;. The third row reads &#x201c;We look for a pattern in the terms&#x201d;, &#x201c;Terms&#x201d;, &#x201c;one-third&#x201d;, &#x201c;one-ninth&#x201d;, &#x201c;one-twenty-seventh&#x201d;, &#x201c;1 divided by 81&#x201d;, &#x201c;1 divided by 243&#x201d;, and an ellipsis. The fourth row reads, &#x201c;The numerators are all at 1&#x201d;, &#x201c;Pattern&#x201d;, &#x201c;one-third to the power of 1 times 9, one-third to the power of 2 times 9, one-third to the power of 3 times 4 times 9, one-third to the power of 3 times 5 times 9, ellipsis, one-third to the power of n&#x201d;. The fifth row reads, &#x201c;The denominators are powers of 3. The general term of the sequence is a sub n equals one-third to the power of n&#x201d;.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829580229\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829716495\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">We look for a pattern in the terms.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836691952\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">The numerators are all 1.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836609920\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">The denominators are powers of 3.<\/td><td data-align=\"left\">The general term of the sequence is \\({a}_{n}=\\frac{1}{{3}^{n}}.\\) <\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836318650\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836318653\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836525725\"><p id=\"fs-id1167836525727\">Find a general term for the sequence whose first five terms are shown.<\/p><p id=\"fs-id1167836525730\">\\(\\frac{1}{2},\\frac{1}{4},\\frac{1}{8},\\frac{1}{16},\\frac{1}{32},\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836559949\"><p id=\"fs-id1167836559951\">\\({a}_{n}=\\frac{1}{{2}^{n}}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167832989121\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836360706\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836360708\"><p id=\"fs-id1167836360710\">Find a general term for the sequence whose first five terms are shown.<\/p><p id=\"fs-id1167836375920\">\\(\\frac{1}{1},\\frac{1}{4},\\frac{1}{9},\\frac{1}{16},\\frac{1}{25},\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833061756\"><p id=\"fs-id1167833061758\">\\({a}_{n}=\\frac{1}{{n}^{2}}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836547277\"><h3 data-type=\"title\">Use Factorial Notation<\/h3><p id=\"fs-id1167836506757\">Sequences often have terms that are products of consecutive integers. We indicate these products with a special notation called <em data-effect=\"italics\">factorial notation<\/em>. For example,\\(5!\\), read 5 factorial, means \\(5\u00b74\u00b73\u00b72\u00b71.\\) The exclamation point is not punctuation here; it indicates the <span data-type=\"term\">factorial notation<\/span>.<\/p><div data-type=\"note\" id=\"fs-id1167836492202\"><div data-type=\"title\">Factorial Notation<\/div><p id=\"fs-id1167836492208\">If <em data-effect=\"italics\">n<\/em> is a positive integer, then \\(n!\\) is<\/p><div data-type=\"equation\" id=\"fs-id1167829754626\" class=\"unnumbered\" data-label=\"\">\\(n!=n\\left(n-1\\right)\\left(n-2\\right)\\dots \\)<\/div><p id=\"fs-id1167829791300\">We define \\(0!\\) as 1, so \\(0!=1.\\)<\/p><\/div><p id=\"fs-id1167833048324\">The values of \\(n!\\) for the first 5 positive integers are shown.<\/p><div data-type=\"equation\" id=\"fs-id1167836552570\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccccccccccccc}\\hfill 1!\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}2!\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}3!\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}4!\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}5!\\hfill \\\\ \\hfill 1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}2\\cdot 1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}3\\cdot 2\\cdot 1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}4\\cdot 3\\cdot 2\\cdot 1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}5\\cdot 4\\cdot 3\\cdot 2\\cdot 1\\hfill \\\\ \\hfill 1\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}2\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}6\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}24\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2em}{0ex}}120\\hfill \\end{array}\\)<\/div><div data-type=\"example\" id=\"fs-id1167836704331\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836704333\"><div data-type=\"problem\" id=\"fs-id1167836704335\"><p id=\"fs-id1167829754906\">Write the first five terms of the sequence whose general term is \\({a}_{n}=\\frac{1}{n!}\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836611218\"><p id=\"fs-id1167829809728\">We substitute the values 1, 2, 3, 4, 5 into the formula, \\({a}_{n}=\\frac{1}{n!},\\) in order.<\/p><span data-type=\"media\" id=\"fs-id1167836433458\" data-alt=\"This figure shows four rows and five columns. The first row reads, &#x201c;nth term equals one divided by n factorial&#x201d; written five times. The second row reads &#x201c;a sub 1 equals one divided by 1 factorial, a sub 2 equals 1 divided by 2 factorial, a sub 3 equals 1 divided by 3 factorial, a sub 4 equals 1 divided by 4 factorial, a sub 5 equals 1 divided by 5 factorial&#x201d;. The third row reads &#x201c;a sub 1 equals 1 divided 1&#x201d;, &#x201c;a sub 2 equals 1 divided by 2 times g times 1&#x201d;, &#x201c;a sub 3 equals 1 divided by 3 times g times 2 g times 1&#x201d;, &#x201c;a sub 4 equals 1 divided 4 times g times 3 times g times 2 times g times 1&#x201d;, &#x201c;a sub 5 equals 1 divided by 5 g times 4 times g times 3 times g times 2 times g times 1&#x201d;, &#x201c;a sub 1 equals 1, a sub 2 equals one-half&#x201d;, &#x201c;a sub 3 equals one-sixth&#x201d;, &#x201c;a sub 4 equals 1 divided by 24&#x201d;, &#x201c;a sub 5 equals 1 divided by 120&#x201d;.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows four rows and five columns. The first row reads, &#x201c;nth term equals one divided by n factorial&#x201d; written five times. The second row reads &#x201c;a sub 1 equals one divided by 1 factorial, a sub 2 equals 1 divided by 2 factorial, a sub 3 equals 1 divided by 3 factorial, a sub 4 equals 1 divided by 4 factorial, a sub 5 equals 1 divided by 5 factorial&#x201d;. The third row reads &#x201c;a sub 1 equals 1 divided 1&#x201d;, &#x201c;a sub 2 equals 1 divided by 2 times g times 1&#x201d;, &#x201c;a sub 3 equals 1 divided by 3 times g times 2 g times 1&#x201d;, &#x201c;a sub 4 equals 1 divided 4 times g times 3 times g times 2 times g times 1&#x201d;, &#x201c;a sub 5 equals 1 divided by 5 g times 4 times g times 3 times g times 2 times g times 1&#x201d;, &#x201c;a sub 1 equals 1, a sub 2 equals one-half&#x201d;, &#x201c;a sub 3 equals one-sixth&#x201d;, &#x201c;a sub 4 equals 1 divided by 24&#x201d;, &#x201c;a sub 5 equals 1 divided by 120&#x201d;.\" \/><\/span><p id=\"fs-id1167829747222\">The first five terms of the sequence are \\(1,\\frac{1}{2},\\frac{1}{6},\\frac{1}{24},\\frac{1}{120}.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836698688\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829811091\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829811093\"><p id=\"fs-id1167829811095\">Write the first five terms of the sequence whose general term is \\({a}_{n}=\\frac{2}{n!}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829755838\"><p id=\"fs-id1167829755841\">\\(2,1,\\frac{1}{3},\\frac{1}{12},\\frac{1}{60}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833047217\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836622285\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836622287\"><p id=\"fs-id1167836300267\">Write the first five terms of the sequence whose general term is \\({a}_{n}=\\frac{3}{n!}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833023017\"><p id=\"fs-id1167829683662\">\\(3,\\frac{3}{2},\\frac{1}{2},\\frac{1}{8},\\frac{1}{40}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165927785098\">When there is a fraction with factorials in the numerator and denominator, we line up the factors vertically to make our calculations easier.<\/p><div data-type=\"example\" id=\"fs-id1167836548831\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836548833\"><div data-type=\"problem\" id=\"fs-id1167829720760\"><p id=\"fs-id1167829720762\">Write the first five terms of the sequence whose general term is \\({a}_{n}=\\frac{\\left(n+1\\right)!}{\\left(n-1\\right)!}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833370051\"><p id=\"fs-id1167833370053\">We substitute the values 1, 2, 3, 4, 5 into the formula, \\({a}_{n}=\\frac{\\left(n+1\\right)!}{\\left(n-1\\right)!},\\) in order.<\/p><span data-type=\"media\" id=\"fs-id1167833240164\" data-alt=\"This figure shows five columns and five rows. The first row shows the sequence &#x201c;nth term equals n plus 1 times factorial divided by n minus 1 times factorial&#x201d; written five times. The second row is &#x201c;a sub 1 equals 1 plus 1 times factorial divided by 1 minus 1 times factorial&#x201d;, &#x201c;a sub 2 equals 2 plus 1 times factorial divided by 2 minus 1 times factorial&#x201d;, &#x201c;a sub 3 equals 3 plus 1 times factorial divided by 3 minus 1 times factorial&#x201d;, &#x201c;a sub 4 equals 4 plus 1 times factorial divided by 4 minus 1 times factorial&#x201d;, &#x201c;a sub 5 equals 5 plus 1 times factorial divided by 5 minus 1 times factorial&#x201d;. The third row reads &#x201c;a sub 1 equals 2 times factorial divided by 0 times factorial&#x201d;, &#x201c;a sub 2 equals 3 times factorial divided by 1 times factorial&#x201d;, &#x201c;a sub 3 equals 4 times factorial divided by 2 times factorial&#x201d;, &#x201c;a sub 3 equals 4 times factorial divided by 2 times factorial&#x201d;, &#x201c;a sub 4 equals 5 times factorial divided by 3 times factorial&#x201d;, &#x201c;a sub 5 equals 6 times factorial divided by 4 times factorial&#x201d;. The fourth row reads, &#x201c;a sub 1 equals 2 times g time 1 divided by 1&#x201d;, &#x201c;a sub 2 equals 3 times g times 2 times g times 1 divided by 1&#x201d;, &#x201c;a sub 3 equals 4 times g times 3 times g times 2 times g times 1 divided by 2 times g times 1&#x201d;, &#x201c;a sub 4 equals 5 times g times 4 times g times 3 times g times 2 times g times 1 divided by 3 g times 2 times g times 1&#x201d;, and &#x201c;a sub 5 equals 6 times g times 5 times g times 4 times g times 3 times g times 2 times g times 1 divided by 4 times g times 3 times g times 2 times g times 1&#x201d;. The fifth row reads &#x201c;a sub 1 equals 2&#x201d;, &#x201c;a sub 2 equals 6&#x201d;, &#x201c;a sub 3 equals 12&#x201d;, &#x201c;a sub 4 equals 20&#x201d;, &#x201c;a sub 5 equals 30&#x201d;.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_005_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows five columns and five rows. The first row shows the sequence &#x201c;nth term equals n plus 1 times factorial divided by n minus 1 times factorial&#x201d; written five times. The second row is &#x201c;a sub 1 equals 1 plus 1 times factorial divided by 1 minus 1 times factorial&#x201d;, &#x201c;a sub 2 equals 2 plus 1 times factorial divided by 2 minus 1 times factorial&#x201d;, &#x201c;a sub 3 equals 3 plus 1 times factorial divided by 3 minus 1 times factorial&#x201d;, &#x201c;a sub 4 equals 4 plus 1 times factorial divided by 4 minus 1 times factorial&#x201d;, &#x201c;a sub 5 equals 5 plus 1 times factorial divided by 5 minus 1 times factorial&#x201d;. The third row reads &#x201c;a sub 1 equals 2 times factorial divided by 0 times factorial&#x201d;, &#x201c;a sub 2 equals 3 times factorial divided by 1 times factorial&#x201d;, &#x201c;a sub 3 equals 4 times factorial divided by 2 times factorial&#x201d;, &#x201c;a sub 3 equals 4 times factorial divided by 2 times factorial&#x201d;, &#x201c;a sub 4 equals 5 times factorial divided by 3 times factorial&#x201d;, &#x201c;a sub 5 equals 6 times factorial divided by 4 times factorial&#x201d;. The fourth row reads, &#x201c;a sub 1 equals 2 times g time 1 divided by 1&#x201d;, &#x201c;a sub 2 equals 3 times g times 2 times g times 1 divided by 1&#x201d;, &#x201c;a sub 3 equals 4 times g times 3 times g times 2 times g times 1 divided by 2 times g times 1&#x201d;, &#x201c;a sub 4 equals 5 times g times 4 times g times 3 times g times 2 times g times 1 divided by 3 g times 2 times g times 1&#x201d;, and &#x201c;a sub 5 equals 6 times g times 5 times g times 4 times g times 3 times g times 2 times g times 1 divided by 4 times g times 3 times g times 2 times g times 1&#x201d;. The fifth row reads &#x201c;a sub 1 equals 2&#x201d;, &#x201c;a sub 2 equals 6&#x201d;, &#x201c;a sub 3 equals 12&#x201d;, &#x201c;a sub 4 equals 20&#x201d;, &#x201c;a sub 5 equals 30&#x201d;.\" \/><\/span><p id=\"fs-id1167836621846\">The first five terms of the sequence are 2, 6, 12, 20, and 30.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829789958\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833023070\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833023072\"><p id=\"fs-id1167833023074\">Write the first five terms of the sequence whose general term is \\({a}_{n}=\\frac{\\left(n-1\\right)!}{\\left(n+1\\right)!}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836398976\"><p id=\"fs-id1167836398978\">\\(\\frac{1}{2},\\frac{1}{6},\\frac{1}{12},\\frac{1}{20},\\frac{1}{30}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836613306\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836613309\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836613312\"><p id=\"fs-id1167836622307\">Write the first five terms of the sequence whose general term is \\({a}_{n}=\\frac{n!}{\\left(n+1\\right)!}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833056115\"><p id=\"fs-id1167833056118\">\\(\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5},\\frac{1}{6}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829719000\"><h3 data-type=\"title\">Find the Partial Sum<\/h3><p id=\"fs-id1167829596997\">Sometimes in applications, rather than just list the terms, it is important for us to add the terms of a sequence. Rather than just connect the terms with plus signs, we can use <span data-type=\"term\">summation notation<\/span>.<\/p><p id=\"fs-id1167836732726\">For example, \\({a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}+{a}_{5}\\) can be written as \\(\\sum _{i=1}^{5}{a}_{i}.\\) We read this as \u201cthe sum of <em data-effect=\"italics\">a<\/em> sub <em data-effect=\"italics\">i<\/em> from <em data-effect=\"italics\">i<\/em> equals one to five.\u201d The symbol \\(\\sum \\) means to add and the <em data-effect=\"italics\">i<\/em> is the index of summation. The 1 tells us where to start (initial value) and the 5 tells us where to end (terminal value).<\/p><div data-type=\"note\" id=\"fs-id1167832982280\"><div data-type=\"title\">Summation Notation<\/div><p id=\"fs-id1167836391189\">The sum of the first <em data-effect=\"italics\">n<\/em> terms of a sequence whose <em data-effect=\"italics\">n<\/em>th term is \\({a}_{n}\\) is written in summation notation as:<\/p><div data-type=\"equation\" id=\"fs-id1167836626906\" class=\"unnumbered\" data-label=\"\">\\(\\sum _{i=1}^{n}{a}_{i}={a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}+{a}_{5}+\\dots +{a}_{n}\\)<\/div><p id=\"fs-id1167829719007\">The <em data-effect=\"italics\">i<\/em> is the index of summation and the 1 tells us where to start and the <em data-effect=\"italics\">n<\/em> tells us where to end.<\/p><\/div><p id=\"fs-id1167833019032\">When we add a finite number of terms, we call the sum a <span data-type=\"term\">partial sum<\/span>.<\/p><div data-type=\"example\" id=\"fs-id1167833019038\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836561241\"><div data-type=\"problem\" id=\"fs-id1167836561243\"><p id=\"fs-id1167836561245\">Expand the partial sum and find its value: \\(\\sum _{i=1}^{5}2i.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836684163\"><p id=\"fs-id1167836389436\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\sum _{i=1}^{5}2i\\hfill \\\\ \\\\ \\\\ \\text{We substitute the values 1, 2, 3, 4, 5 in order.}\\hfill &amp; &amp; &amp; \\hfill 2\u00b71+2\u00b72+2\u00b73+2\u00b74+2\u00b75\\hfill \\\\ \\\\ \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill 2+4+6+8+10\\hfill \\\\ \\\\ \\\\ \\text{Add.}\\hfill &amp; &amp; &amp; \\hfill 30\\hfill \\\\ &amp; &amp; &amp; \\hfill \\sum _{i=1}^{5}2i=30\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829747135\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167833396844\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833396846\"><p id=\"fs-id1167833396848\">Expand the partial sum and find its value: \\(\\sum _{i=1}^{5}3i.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829719048\"><p id=\"fs-id1167829719050\">45<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836415595\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836287270\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836287272\"><p id=\"fs-id1167836287274\">Expand the partial sum and find its value: \\(\\sum _{i=1}^{5}4i.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832984037\"><p id=\"fs-id1167832984039\">60<\/p><\/div><\/div><\/div><p id=\"fs-id1167836450406\">The index does not always have to be <em data-effect=\"italics\">i<\/em> we can use any letter, but <em data-effect=\"italics\">i<\/em> and <em data-effect=\"italics\">k<\/em> are commonly used. The index does not have to start with 1 either\u2014it can start and end with any positive integer.<\/p><div data-type=\"example\" id=\"fs-id1167829747210\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167829747212\"><div data-type=\"problem\" id=\"fs-id1167829747214\"><p id=\"fs-id1167829747216\">Expand the partial sum and find its value: \\(\\sum _{k=0}^{3}\\frac{1}{k!}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829747192\"><p id=\"fs-id1167829747195\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\sum _{k=0}^{3}\\frac{1}{k!}\\hfill \\\\ \\text{We substitute the values 0, 1, 2, 3, in order.}\\hfill &amp; &amp; &amp; \\hfill \\frac{1}{0!}+\\frac{1}{1!}+\\frac{1}{2!}+\\frac{1}{3!}\\hfill \\\\ \\text{Evaluate the factorials.}\\hfill &amp; &amp; &amp; \\hfill \\frac{1}{1}+\\frac{1}{1}+\\frac{1}{2}+\\frac{1}{6}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill 1+1+\\frac{3}{6}+\\frac{1}{6}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\frac{16}{6}\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\frac{8}{3}\\hfill \\\\ &amp; &amp; &amp; \\hfill \\sum _{k=0}^{3}\\frac{1}{k!}=\\frac{8}{3}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167833158242\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836610913\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836610915\"><p id=\"fs-id1167836610917\">Expand the partial sum and find its value: \\(\\sum _{k=0}^{3}\\frac{2}{k!}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836534087\"><p id=\"fs-id1167836534089\">\\(\\frac{16}{3}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829717879\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829717882\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836540528\"><p id=\"fs-id1167836540530\">Expand the partial sum and find its value: \\(\\sum _{k=0}^{3}\\frac{3}{k!}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829747803\"><p id=\"fs-id1167829747805\">8<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836294722\"><h3 data-type=\"title\">Use Summation Notation to Write a Sum<\/h3><p id=\"fs-id1167836294727\">In the last two examples, we went from summation notation to writing out the sum. Now we will start with a sum and change it to summation notation. This is very similar to finding the general term of a sequence. We will need to look at the terms and find a pattern. Often the patterns involve multiples or powers.<\/p><div data-type=\"example\" id=\"fs-id1167836688410\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836688412\"><div data-type=\"problem\" id=\"fs-id1167829720741\"><p id=\"fs-id1167829720743\">Write the sum using summation notation: \\(1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829809741\"><p id=\"fs-id1167829809743\">\\(\\begin{array}{cccc}&amp; &amp; &amp; \\hfill \\phantom{\\rule{5.5em}{0ex}}1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}\\hfill \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{1.5em}{0ex}}n\\text{:}\\phantom{\\rule{1em}{0ex}}1,2,3,4,5\\hfill \\\\ \\text{We look for a pattern in the terms.}\\hfill &amp; &amp; &amp; \\hfill \\text{Terms:}\\phantom{\\rule{1em}{0ex}}1,\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5}\\hfill \\\\ \\text{The numerators are all one.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{2.5em}{0ex}}\\text{Pattern:}\\phantom{\\rule{1em}{0ex}}\\frac{1}{1},\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5},\\dots \\frac{1}{n}\\hfill \\\\ \\text{The denominators are the counting numbers}\\hfill &amp; &amp; &amp; \\\\ \\text{from one to five.}\\hfill &amp; &amp; &amp; \\hfill \\text{The sum written in summation notation is}\\hfill \\\\ &amp; &amp; &amp; \\hfill 1+\\frac{1}{2}+\\frac{1}{3}+\\frac{1}{4}+\\frac{1}{5}=\\sum _{n=1}^{5}\\frac{1}{n}.\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829627520\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836605384\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836605386\"><p id=\"fs-id1167836605388\">Write the sum using summation notation: \\(\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}+\\frac{1}{32}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833061159\"><p id=\"fs-id1167833061162\">\\(\\sum _{n=1}^{5}\\frac{1}{{2}^{n}}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836531526\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836531529\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836531531\"><p id=\"fs-id1167833014902\">Write the sum using summation notation: \\(1+\\frac{1}{4}+\\frac{1}{9}+\\frac{1}{16}+\\frac{1}{25}.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829908624\"><p id=\"fs-id1167829908626\">\\(\\sum _{n=1}^{5}\\frac{1}{{n}^{2}}\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165923635499\">When the terms of a sum have negative coefficients, we must carefully analyze the pattern of the signs.<\/p><div data-type=\"example\" id=\"fs-id1167836507518\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167836507520\"><div data-type=\"problem\" id=\"fs-id1167836507522\"><p id=\"fs-id1167836507524\">Write the sum using summation notation: \\(-1+8-27+64-125.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836530770\"><table id=\"fs-id1167836530772\" summary=\"This figure shows a table with seven columns and six rows. The first row reads the summation notation as &#x201c;negative 1 plus 8 minus 27 plus 64 minus 125&#x201d;. The second row reads &#x201c;n&#x201d;, then &#x201c;1&#x201d;, &#x201c;2&#x201d;, &#x201c;3&#x201d;, &#x201c;4&#x201d; and &#x201c;5&#x201d;. The third row reads, &#x201c;We look for a pattern in the terms&#x201d;, &#x201c;Terms&#x201d;, &#x201c;-1&#x201d;, &#x201c;negative 8&#x201d;, &#x201c;negative 27&#x201d;, &#x201c;64&#x201d;, and &#x201c;negative 125&#x201d;. The fourth row reads, &#x201c;The signs of the terms alternate, and the odd terms are negative&#x201d;, &#x201c;Pattern&#x201d;, &#x201c;negative 1 times the power of 1 times g times 1, negative 1 squared times g times 8, negative 1 cubed times g times 27, negative 1 to the power of 4 times g times 64, negative 1 to the power of 5 times g times 125. The fifth row reads, &#x201c;The numbers are the cubes of the counting numbers from 1 to 5&#x201d;, &#x201c;Pattern&#x201d;, &#x201c;negative 1 to the power of 1 times g times l cubed, negative 1 to the power of 2, times g times 2 cubed, negative 1 to the power of 3 times g times 3 cubed, negative 1 to the power of 4 times g times 4 cubed, negative 1 to the power of 5 times g times 5 cubed&#x201d;. The sixth row reads &#x201c;Pattern is negative 1 to the power of n times g times n to the power of 3. The seventh row is &#x201c;The sum written as summation notation is negative 1 plus 8 minus 27 plus 64 minus 124 equals the sum of a sub i from i equals 1 to 5 times negative 1 to the power of n times g times n to the power of 3.&#x201d;\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836433359\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr valign=\"top\"><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836685110\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">We look for a pattern in the terms.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836728372\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">The signs of the terms alternate,<span data-type=\"newline\"><br \/><\/span>and the odd terms are negative.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829741797\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">The numbers are the cubes of the<span data-type=\"newline\"><br \/><\/span>counting numbers from one to five.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836444781\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836646292\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td><\/td><td data-align=\"left\">The sum written in summation notation is<\/td><\/tr><tr><td><\/td><td data-align=\"left\">\\(-1+8-27+64-125=\\sum _{n=1}^{5}{\\left(-1\\right)}^{n}\\cdot {n}^{3}\\)<\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167836542582\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167836542586\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836542588\"><p id=\"fs-id1167836492236\">Write each sum using summation notation: \\(1-4+9-16+25.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836363682\"><p id=\"fs-id1167836363219\">\\(\\sum _{n=1}^{5}{\\left(-1\\right)}^{n+1}{n}^{2}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829627922\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167829627926\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829627928\"><p id=\"fs-id1167829627930\">Write each sum using summation notation: \\(-2+4-6+8-10.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833048339\"><p id=\"fs-id1167833048341\">\\(\\sum _{n=1}^{5}{\\left(-1\\right)}^{n}2n\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167829597663\" class=\"media-2\"><p id=\"fs-id1167829597666\">Access this online resource for additional instruction and practice with sequences.<\/p><ul id=\"fs-id1167836321416\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37serseqfindpat\">Series and Sequences-Finding Patterns<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836363337\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167836363344\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">Factorial Notation<\/strong><span data-type=\"newline\"><br \/><\/span> If <em data-effect=\"italics\">n<\/em> is a positive integer, then \\(n!\\) is<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1167833017967\" class=\"unnumbered\" data-label=\"\">\\(n!=n\\left(n-1\\right)\\left(n-2\\right)\\dots \\left(3\\right)\\left(2\\right)\\left(1\\right).\\)<\/div><span data-type=\"newline\"><br \/><\/span> We define \\(0!\\) as 1, so \\(0!=1\\)<\/li><li><strong data-effect=\"bold\">Summation Notation<\/strong><span data-type=\"newline\"><br \/><\/span> The sum of the first <em data-effect=\"italics\">n<\/em> terms of a sequence whose <em data-effect=\"italics\">n<\/em>th term \\({a}_{n}\\) is written in summation notation as:<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1167836692030\" class=\"unnumbered\" data-label=\"\">\\(\\sum _{i=1}^{n}{a}_{i}={a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}+{a}_{5}+\\dots +{a}_{n}\\)<\/div><span data-type=\"newline\"><br \/><\/span> The <em data-effect=\"italics\">i<\/em> is the index of summation and the 1 tells us where to start and the <em data-effect=\"italics\">n<\/em> tells us where to end.<\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836692470\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836692473\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167836692479\"><strong data-effect=\"bold\">Write the First Few Terms of a Sequence<\/strong><\/p><p id=\"fs-id1167836542434\">In the following exercises, write the first five terms of the sequence whose general term is given.<\/p><div data-type=\"exercise\" id=\"fs-id1167836542437\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836542439\"><p id=\"fs-id1167836732091\">\\({a}_{n}=2n-7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833047542\"><p id=\"fs-id1167833047545\">\\(-5,-3,-1,1,3\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836697712\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836697714\"><p id=\"fs-id1167836662577\">\\({a}_{n}=5n-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833030960\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833030962\"><p id=\"fs-id1167829593872\">\\({a}_{n}=3n+1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829586574\"><p id=\"fs-id1167829586576\">\\(4,7,10,13,16\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836433028\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836433030\"><p id=\"fs-id1167836622408\">\\({a}_{n}=4n+2\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836554862\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836554864\"><p id=\"fs-id1167836621806\">\\({a}_{n}={2}^{n}+3\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836626353\"><p id=\"fs-id1167836626355\">\\(5,7,11,19,35\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829747310\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836433289\"><p id=\"fs-id1167836433291\">\\({a}_{n}={3}^{n}-1\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829745911\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836663089\"><p id=\"fs-id1167836663091\">\\({a}_{n}={3}^{n}-2n\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833025530\"><p id=\"fs-id1167833025532\">\\(1,5,21,73,233\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833008151\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833008153\"><p id=\"fs-id1167833008155\">\\({a}_{n}={2}^{n}-3n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836518570\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836518572\"><p id=\"fs-id1167836518574\">\\({a}_{n}=\\frac{{2}^{n}}{{n}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829745278\"><p id=\"fs-id1167829745280\">\\(2,1,\\frac{8}{9},1,\\frac{32}{25}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836697736\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836697738\"><p id=\"fs-id1167836697740\">\\({a}_{n}=\\frac{{3}^{n}}{{n}^{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836728891\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836728893\"><p id=\"fs-id1167836728895\">\\({a}_{n}=\\frac{4n-2}{{2}^{n}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829747157\"><p id=\"fs-id1167829747160\">\\(1,\\frac{3}{2},\\frac{5}{4},\\frac{7}{8},\\frac{9}{16}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836610291\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836610293\"><p id=\"fs-id1167829788284\">\\({a}_{n}=\\frac{3n+3}{{3}^{n}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836731549\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836731552\"><p id=\"fs-id1167836731554\">\\({a}_{n}={\\left(-1\\right)}^{n}\u00b72n\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836663177\"><p id=\"fs-id1167833018382\">\\(-2,4,-6,8,-10\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836409447\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836409449\"><p id=\"fs-id1167836409451\">\\({a}_{n}={\\left(-1\\right)}^{n}\u00b73n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836444704\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836444706\"><p id=\"fs-id1167836444708\">\\({a}_{n}={\\left(-1\\right)}^{n+1}{n}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836542558\"><p id=\"fs-id1167836542560\">\\(1,-4,9,-16,25\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833056685\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833056687\"><p id=\"fs-id1167833056689\">\\({a}_{n}={\\left(-1\\right)}^{n+1}{n}^{4}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833071792\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833071795\"><p id=\"fs-id1167836613274\">\\({a}_{n}=\\frac{{\\left(-1\\right)}^{n+1}}{{n}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836630557\"><p id=\"fs-id1167836630559\">\\(1,-\\frac{1}{4},\\frac{1}{9},-\\frac{1}{16},\\frac{1}{25}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836688954\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836688957\"><p id=\"fs-id1167829719948\">\\({a}_{n}=\\frac{{\\left(-1\\right)}^{n+1}}{2n}\\)<\/p><\/div><\/div><p id=\"fs-id1167833018343\"><strong data-effect=\"bold\">Find a Formula for the General Term (<em data-effect=\"italics\">n<\/em>th Term) of a Sequence<\/strong><\/p><p id=\"fs-id1167836560597\">In the following exercises, find a general term for the sequence whose first five terms are shown.<\/p><div data-type=\"exercise\" id=\"fs-id1167836560600\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829599598\"><p id=\"fs-id1167829599600\">\\(8,16,24,32,40,\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836530056\"><p id=\"fs-id1167836530058\">\\({a}_{n}=8n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836663456\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836663458\"><p id=\"fs-id1167836663460\">\\(7,14,21,28,35,\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833369184\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833369187\"><p id=\"fs-id1167833369189\">\\(6,7,8,9,10,\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829746838\"><p id=\"fs-id1167829746840\">\\({a}_{n}=n+5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836433330\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836433332\"><p id=\"fs-id1167836433335\">\\(-3,-2,-1,0,1,\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836612792\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836612794\"><p id=\"fs-id1167836612796\">\\({e}^{3},{e}^{4},{e}^{5},{e}^{6},{e}^{7},\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836728288\"><p id=\"fs-id1167836728290\">\\({a}_{n}={e}^{n+2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829811395\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829811398\"><p id=\"fs-id1167829811400\">\\(\\frac{1}{{e}^{2}},\\frac{1}{e},1,e,{e}^{2},\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742040\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836363311\"><p id=\"fs-id1167836363313\">\\(-5,10,-15,20,-25,\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836731605\"><p id=\"fs-id1167836731607\">\\({a}_{n}={\\left(-1\\right)}^{n}5n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836560420\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836560422\"><p id=\"fs-id1167836560424\">\\(-6,11,-16,21,-26,\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836532531\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829752946\"><p id=\"fs-id1167829752948\">\\(-1,8,-27,64,-125,\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833019697\"><p id=\"fs-id1167833019699\">\\({a}_{n}={\\left(-1\\right)}^{n}{n}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832945880\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832945882\"><p id=\"fs-id1167832945885\">\\(2,-5,10,-17,26,\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836477546\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836477548\"><p id=\"fs-id1167836477550\">\\(-2,4,-6,8,-10,\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836559964\"><p id=\"fs-id1167836559966\">\\({a}_{n}={\\left(-1\\right)}^{n}2n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829906696\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836293168\"><p id=\"fs-id1167836293170\">\\(1,-3,5,-7,9,\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833380689\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833380691\"><p id=\"fs-id1167833380693\">\\(\\frac{1}{4},\\frac{1}{16},\\frac{1}{64},\\frac{1}{256},\\frac{1}{1,024},\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836415616\"><p id=\"fs-id1167836635475\">\\({a}_{n}=\\frac{1}{{4}^{n}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836386988\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836386990\"><p id=\"fs-id1167836386992\">\\(\\frac{1}{1},\\frac{1}{8},\\frac{1}{27},\\frac{1}{64},\\frac{1}{125},\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829747828\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829747830\"><p id=\"fs-id1167829747832\">\\(-\\frac{1}{2},-\\frac{2}{3},-\\frac{3}{4},-\\frac{4}{5},-\\frac{5}{6},\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836728478\"><p id=\"fs-id1167836728480\">\\({a}_{n}=-\\frac{n}{n+1}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829753358\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829753360\"><p id=\"fs-id1167829753362\">\\(-2,-\\frac{3}{2},-\\frac{4}{3},-\\frac{5}{4},-\\frac{6}{5},\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836684137\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836684139\"><p id=\"fs-id1167833021620\">\\(-\\frac{5}{2},-\\frac{5}{4},-\\frac{5}{8},-\\frac{5}{16},-\\frac{5}{32},\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829711771\"><p id=\"fs-id1167829711773\">\\(-\\frac{5}{{2}^{n}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829751436\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829751438\"><p id=\"fs-id1167829751440\">\\(4,\\frac{1}{2},\\frac{4}{27},\\frac{4}{64},\\frac{4}{125},\\text{\u2026}\\)<\/p><\/div><\/div><p id=\"fs-id1167829720967\"><strong data-effect=\"bold\">Use Factorial Notation<\/strong><\/p><p id=\"fs-id1167829720972\">In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.<\/p><div data-type=\"exercise\" id=\"fs-id1167829720976\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829720978\"><p id=\"fs-id1167836398761\">\\({a}_{n}=\\frac{4}{n!}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829906646\"><p id=\"fs-id1167829906648\">\\(4,2,\\frac{2}{3},\\frac{1}{6},\\frac{1}{30}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836362732\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836362734\"><p id=\"fs-id1167836362736\">\\({a}_{n}=\\frac{5}{n!}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829745763\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829745765\"><p id=\"fs-id1167829745767\">\\({a}_{n}=3n!\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829753610\"><p id=\"fs-id1167829753612\">\\(3,6,18,72,360\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833018403\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833018406\"><p id=\"fs-id1167833018408\">\\({a}_{n}=2n!\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833025452\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833025454\"><p id=\"fs-id1167833025456\">\\({a}_{n}=\\left(2n\\right)!\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829906197\"><p id=\"fs-id1167829906199\">\\(2,24,720,40320,3628800\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829742666\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829742668\"><p id=\"fs-id1167829742670\">\\({a}_{n}=\\left(3n\\right)!\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836289638\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836289640\"><p id=\"fs-id1167836289642\">\\({a}_{n}=\\frac{\\left(n-1\\right)!}{\\left(n\\right)!}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833379764\"><p id=\"fs-id1167829904857\">\\(1,\\frac{1}{2},\\frac{1}{3},\\frac{1}{4},\\frac{1}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836511165\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829850885\"><p id=\"fs-id1167829850887\">\\({a}_{n}=\\frac{n!}{\\left(n+1\\right)!}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829627454\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829627456\"><p id=\"fs-id1167836699088\">\\({a}_{n}=\\frac{n!}{{n}^{\\text{2}}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836621263\"><p id=\"fs-id1167836621266\">\\(1,\\frac{1}{2},\\frac{2}{3},\\frac{3}{2},\\frac{24}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836398785\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836398787\"><p id=\"fs-id1167836398789\">\\({a}_{n}=\\frac{{n}^{\\text{2}}}{n!}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829681252\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829681254\"><p id=\"fs-id1167829681257\">\\({a}_{n}=\\frac{\\left(n+1\\right)!}{{n}^{2}}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836607396\"><p id=\"fs-id1167829936533\">\\(2,\\frac{3}{2},\\frac{8}{3},\\frac{15}{2},\\frac{144}{5}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836662794\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836662797\"><p id=\"fs-id1167829754708\">\\({a}_{n}=\\frac{\\left(n+1\\right)!}{2n}\\)<\/p><\/div><\/div><p id=\"fs-id1167836433402\"><strong data-effect=\"bold\">Find the Partial Sum<\/strong><\/p><p id=\"fs-id1167836433408\">In the following exercises, expand the partial sum and find its value.<\/p><div data-type=\"exercise\" id=\"fs-id1167836433411\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836433413\"><p id=\"fs-id1167836433415\">\\(\\sum _{i=1}^{5}{i}^{2}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829752801\"><p id=\"fs-id1167829752803\">\\(1+4+9+16+25=55\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829808424\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829808427\"><p id=\"fs-id1167829808429\">\\(\\sum _{i=1}^{5}{i}^{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829741717\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829741719\"><p id=\"fs-id1167829741722\">\\(\\sum _{i=1}^{6}\\left(2i+3\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836635456\"><p id=\"fs-id1167836635459\">\\(5+7+9+11+13+15=60\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836528185\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836528187\"><p id=\"fs-id1167836528189\">\\(\\sum _{i=1}^{6}\\left(3i-2\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836548302\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836548304\"><p id=\"fs-id1167836732280\">\\(\\sum _{i=1}^{4}{2}^{i}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833021801\"><p id=\"fs-id1167833021803\">\\(2+4+8+16=30\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836510199\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836510201\"><p id=\"fs-id1167836510204\">\\(\\sum _{i=1}^{4}{3}^{i}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836386938\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833381315\"><p id=\"fs-id1167833381317\">\\(\\sum _{k=0}^{3}\\frac{4}{k!}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833202871\"><p id=\"fs-id1167833202873\">\\(\\frac{4}{1}+\\frac{4}{1}+\\frac{4}{2}+\\frac{4}{6}+\\frac{32}{3}=10\\frac{2}{3}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836444835\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836433208\"><p id=\"fs-id1167836433210\">\\(\\sum _{k=0}^{4}-\\frac{1}{k!}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829905677\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829905679\"><p id=\"fs-id1167829905681\">\\(\\sum _{k=1}^{5}k\\left(k+1\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832994357\"><p id=\"fs-id1167832994359\">\\(2+6+12+20+30=70\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833381075\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833381078\"><p id=\"fs-id1167833381080\">\\(\\sum _{k=1}^{5}k\\left(2k-3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836728850\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836728852\"><p id=\"fs-id1167836728854\">\\(\\sum _{n=1}^{5}\\frac{n}{n+1}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836363431\"><p id=\"fs-id1167836363433\">\\(\\frac{1}{2}+\\frac{2}{3}+\\frac{3}{4}+\\frac{4}{5}+\\frac{5}{6}=\\frac{71}{20}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829755864\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829755867\"><p id=\"fs-id1167829755869\">\\(\\sum _{n=1}^{4}\\frac{n}{n+2}\\)<\/p><\/div><\/div><p id=\"fs-id1167836732252\"><strong data-effect=\"bold\">Use Summation Notation to write a Sum<\/strong><\/p><p id=\"fs-id1167836732257\">In the following exercises, write each sum using summation notation.<\/p><div data-type=\"exercise\" id=\"fs-id1167836732260\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836732262\"><p id=\"fs-id1167836732265\">\\(\\frac{1}{3}+\\frac{1}{9}+\\frac{1}{27}+\\frac{1}{81}+\\frac{1}{243}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829627761\"><p id=\"fs-id1167829627763\">\\(\\sum _{n=1}^{5}\\frac{1}{{3}^{n}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833379856\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833379858\"><p id=\"fs-id1167833379860\">\\(\\frac{1}{4}+\\frac{1}{16}+\\frac{1}{64}+\\frac{1}{256}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829720089\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829720091\"><p id=\"fs-id1167829720093\">\\(1+\\frac{1}{8}+\\frac{1}{27}+\\frac{1}{64}+\\frac{1}{125}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836450380\"><p id=\"fs-id1167836450383\">\\(\\sum _{n=1}^{5}\\frac{1}{{n}^{3}}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836331126\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836331128\"><p id=\"fs-id1167829693463\">\\(\\frac{1}{5}+\\frac{1}{25}+\\frac{1}{125}+\\frac{1}{625}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833380085\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833380087\"><p id=\"fs-id1167833380089\">\\(2+1+\\frac{2}{3}+\\frac{1}{2}+\\frac{2}{5}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829627310\"><p id=\"fs-id1167829627312\">\\(\\sum _{n=1}^{5}\\frac{2}{n}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829811078\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829811080\"><p id=\"fs-id1167829811082\">\\(3+\\frac{3}{2}+1+\\frac{3}{4}+\\frac{3}{5}+\\frac{1}{2}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836728731\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836728733\"><p id=\"fs-id1167836728735\">\\(3-6+9-12+15\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836729430\"><p id=\"fs-id1167836729432\">\\(\\sum _{n=1}^{5}{\\left(-1\\right)}^{n+1}3n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836626890\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836626892\"><p id=\"fs-id1167836626894\">\\(-5+10-15+20-25\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829905564\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829905566\"><p id=\"fs-id1167829905569\">\\(-2+4-6+8-10+\\dots +20\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167829905835\"><p id=\"fs-id1167829905837\">\\(\\sum _{n=1}^{10}{\\left(-1\\right)}^{n}2n\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836728528\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836728530\"><p id=\"fs-id1167836728532\">\\(1-3+5-7+9+\\dots +21\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167829790474\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167829790476\"><p id=\"fs-id1167829790478\">\\(14+16+18+20+22+24+26\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836732228\"><p id=\"fs-id1167836732231\">\\(\\sum _{n=1}^{7}\\left(2n+12\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833379616\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833379618\"><p id=\"fs-id1167833379620\">\\(9+11+13+15+17+19+21\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836730456\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167836730097\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836730099\"><p id=\"fs-id1167836730101\">In your own words, explain how to write the terms of a sequence when you know the formula. Show an example to illustrate your explanation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167836730107\"><p id=\"fs-id1167836730109\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167836730114\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167836730116\"><p id=\"fs-id1167836730118\">Which terms of the sequence are negative when the <em data-effect=\"italics\">n<\/em><sup>th<\/sup> term of the sequence is \\({a}_{n}={\\left(-1\\right)}^{n}\\left(n+2\\right)?\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833381108\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833381110\"><p id=\"fs-id1167833381112\">In your own words, explain what is meant by \\(n!\\) Show some examples to illustrate your explanation.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167833381124\"><p id=\"fs-id1167833381126\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167833381132\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167833381134\"><p id=\"fs-id1167829904879\">Explain what each part of the notation \\(\\sum _{k=1}^{12}2k\\) means.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829904916\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167829809598\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p><span data-type=\"media\" id=\"fs-id1167829809613\" data-alt=\"This figure shows a table with four columns and six rows. The first row is the header row and labels each column, &#x201c;I can&#x201d;, &#x201c;Confidently&#x201d;, &#x201c;With some help&#x201d;, and &#x201c;No I don&#x2019;t get it!&#x201d;. The first row in the second column reads, &#x201c;Write the first few terms of a sequence&#x201d;, the third row, first column reads, &#x201c;Find a Formula for the nth Term of a Sequence&#x201d;, the fourth row first column reads &#x201c;Use Factorial Notation, the fifth row, first column reads, Find the partial sum&#x201d;, and the last row, first column reads, &#x201c;Use Summation Notation to write a Sum&#x201d;. The remaining three columns and rows are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a table with four columns and six rows. The first row is the header row and labels each column, &#x201c;I can&#x201d;, &#x201c;Confidently&#x201d;, &#x201c;With some help&#x201d;, and &#x201c;No I don&#x2019;t get it!&#x201d;. The first row in the second column reads, &#x201c;Write the first few terms of a sequence&#x201d;, the third row, first column reads, &#x201c;Find a Formula for the nth Term of a Sequence&#x201d;, the fourth row first column reads &#x201c;Use Factorial Notation, the fifth row, first column reads, Find the partial sum&#x201d;, and the last row, first column reads, &#x201c;Use Summation Notation to write a Sum&#x201d;. The remaining three columns and rows are blank.\" \/><\/span><p id=\"fs-id1167829809621\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p><p id=\"fs-id1167829809628\">\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p><p id=\"fs-id1167836513507\">\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p><p id=\"fs-id1167836513518\">\u2026no - I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167836513531\"><dt>finite sequence<\/dt><dd id=\"fs-id1167836513536\">A sequence with a domain that is limited to a finite number of counting numbers.<\/dd><\/dl><dl id=\"fs-id1167836513540\"><dt>general term of a sequence<\/dt><dd id=\"fs-id1167829809632\">The general term of the sequence is the formula for writing the <em data-effect=\"italics\">n<\/em>th term of the sequence. The <em data-effect=\"italics\">n<\/em>th term of the sequence, \\({a}_{n},\\) is the term in the <em data-effect=\"italics\">n<\/em>th position where <em data-effect=\"italics\">n<\/em> is a value in the domain.<\/dd><\/dl><dl id=\"fs-id1167829742733\"><dt>infinite sequence<\/dt><dd id=\"fs-id1167829742738\">A sequence whose domain is all counting numbers and there is an infinite number of counting numbers.<\/dd><\/dl><dl id=\"fs-id1167836731128\"><dt>partial sum<\/dt><dd id=\"fs-id1167836731133\">When we add a finite number of terms of a sequence, we call the sum a partial sum.<\/dd><\/dl><dl id=\"fs-id1167836731137\"><dt>sequence<\/dt><dd id=\"fs-id1167836731141\">A sequence is a function whose domain is the counting numbers.<\/dd><\/dl><\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<p>By the end of this section, you will be able to: <\/p>\n<ul>\n<li>Write the first few terms of a sequence<\/li>\n<li>Find a formula for the general term (nth term) of a sequence<\/li>\n<li>Use factorial notation<\/li>\n<li>Find the partial sum<\/li>\n<li>Use summation notation to write a sum<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836335254\" class=\"be-prepared\">\n<p id=\"fs-id1167836613299\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167836334429\" type=\"1\">\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27c2caf41fd59655a6e2c28929a4d34a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#110;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"50\" style=\"vertical-align: -2px;\" \/> for the integers 1, 2, 3, and 4.<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/d3697553-3900-453f-8c08-8f74d55711ba#fs-id1167836530265\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ced2ddb4631ac07149d157ee7d08348d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/> for the integers 1, 2, 3, and 4.<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/05eab039-6d1c-4d80-8c8c-94469164a52c#fs-id1167834536158\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<li>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-071bca41e4754fbce2b36667065c4831_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#43;&#50;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-47724d5bc49ef56fca804d85909dc5b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"159\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/5e548626-8f0f-496d-ab87-4f0358ca2fd3#fs-id1167829859398\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li>\n<\/ol>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836362095\">\n<h3 data-type=\"title\">Write the First Few Terms of a Sequence<\/h3>\n<p id=\"fs-id1167833025720\">Let\u2019s look at the function <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d268a37dadc23b636bc6955a3e233880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/> and evaluate it for just the counting numbers.<\/p>\n<table id=\"fs-id1167836595941\" summary=\"Table has 2 columns and 8 rows. The first row lists the function f of x equals 2 x. The first column lists values of x 1 through 5 with an ellipsis in the last row. The second column lists values of 2 x with values 2, 4, 6, 8, and 10 with an ellipsis in the last row.\" class=\"unnumbered\">\n<tbody>\n<tr valign=\"top\">\n<td colspan=\"2\" data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d268a37dadc23b636bc6955a3e233880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ede05c264bba0eda080918aaa09c4658_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\" \/><\/strong><\/td>\n<td data-valign=\"bottom\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da4556c0a02b580047678d308649edf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/><\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">1<\/td>\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">3<\/td>\n<td data-valign=\"middle\" data-align=\"center\">6<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\">8<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">5<\/td>\n<td data-valign=\"middle\" data-align=\"center\">10<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"middle\" data-align=\"center\">\u2026<\/td>\n<td data-valign=\"middle\" data-align=\"center\">\u2026<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p id=\"fs-id1167836410426\">If we list the function values in order as 2, 4, 6, 8, and 10, \u2026 we have a sequence. A <span data-type=\"term\">sequence<\/span> is a function whose domain is the counting numbers.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836352765\">\n<div data-type=\"title\">Sequences<\/div>\n<p id=\"fs-id1167832982337\">A <strong data-effect=\"bold\">sequence<\/strong> is a function whose domain is the counting numbers.<\/p>\n<\/div>\n<p id=\"fs-id1167829691634\">A sequence can also be seen as an ordered list of numbers and each number in the list is a <em data-effect=\"italics\">term<\/em>. A sequence may have an infinite number of terms or a finite number of terms. Our sequence has three dots (ellipsis) at the end which indicates the list never ends. If the domain is the set of all counting numbers, then the sequence is an <span data-type=\"term\">infinite sequence<\/span>. Its domain is all counting numbers and there is an infinite number of counting numbers.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836507257\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8f320b11df5cac4005b78272d0453412_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#52;&#44;&#54;&#44;&#56;&#44;&#49;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167836410266\">If we limit the domain to a finite number of counting numbers, then the sequence is a <span data-type=\"term\">finite sequence<\/span>. If we use only the first four counting numbers, 1, 2, 3, 4 our sequence would be the finite sequence,<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836693849\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99ca801e6ba8a2423a25c55a0a76484c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#52;&#44;&#54;&#44;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167833024298\">Often when working with sequences we do not want to write out all the terms. We want more compact way to show how each term is defined. When we worked with functions, we wrote <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d268a37dadc23b636bc6955a3e233880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"80\" style=\"vertical-align: -4px;\" \/> and we said the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da4556c0a02b580047678d308649edf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"19\" style=\"vertical-align: 0px;\" \/> was the rule that defined values in the range. While a sequence is a function, we do not use the usual function notation. Instead of writing the function as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3abaf258dcaf1dff75e7ce22d1ab4337_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#102;&#92;&#108;&#101;&#102;&#116;&#40;&#120;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#120;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"84\" style=\"vertical-align: -4px;\" \/> we would write it as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-04332b7a161fef17283c5134fa87eeae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#50;&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/> The <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1d5f11b142b357c0576e6792d1cef0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> is the <em data-effect=\"italics\">n<\/em>th term of the sequence, the term in the <em data-effect=\"italics\">n<\/em>th position where <em data-effect=\"italics\">n<\/em> is a value in the domain. The formula for writing the <em data-effect=\"italics\">n<\/em>th term of the sequence is called the <span data-type=\"term\">general term<\/span> or formula of the sequence.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836484969\">\n<div data-type=\"title\">General Term of a Sequence<\/div>\n<p id=\"fs-id1167836571092\">The <strong data-effect=\"bold\">general term<\/strong> of the sequence is found from the formula for writing the <em data-effect=\"italics\">n<\/em>th term of the sequence. The <em data-effect=\"italics\">n<\/em>th term of the sequence, <em data-effect=\"italics\">a<sub>n<\/sub><\/em>, is the term in the <em data-effect=\"italics\">n<\/em>th position where <em data-effect=\"italics\">n<\/em> is a value in the domain.<\/p>\n<\/div>\n<p id=\"fs-id1167836367818\">When we are given the general term of the sequence, we can find the terms by replacing <em data-effect=\"italics\">n<\/em> with the counting numbers in order. For <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a353c6f36edbb34d5d2df147319f80e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#50;&#110;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"66\" style=\"vertical-align: -4px;\" \/><\/p>\n<table id=\"fs-id1167836626974\" summary=\"Table has 2 rows and 7 columns. The first row lists values of n from 1 through 5 with the last value as a sub n. The second row lists values of a sub n with mutiples of 2, from 2 through 10 with the last value as 2n. Below the table are values a sub 1, a sub 2, a sub 3, a sub 4, a sub 5, and a sub n followed by the values 2, 4, 6, 8, and 10.\" class=\"unnumbered\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\">1<\/td>\n<td data-valign=\"middle\" data-align=\"center\">2<\/td>\n<td data-valign=\"middle\" data-align=\"center\">3<\/td>\n<td data-valign=\"middle\" data-align=\"center\">4<\/td>\n<td data-valign=\"middle\" data-align=\"center\">5<\/td>\n<td data-valign=\"top\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1d5f11b142b357c0576e6792d1cef0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1d5f11b142b357c0576e6792d1cef0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6e1103fb42c871815c17695d1b6a192d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&middot;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"17\" style=\"vertical-align: -11px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-594f4cf242568c531d5ef3ada3b0bad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&middot;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"35\" width=\"17\" style=\"vertical-align: -12px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c7d73997e666225ab8560fca2ee82ada_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&middot;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"18\" style=\"vertical-align: -11px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc8e95b2ec22bbcd15fb2b9a67fdbb45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&middot;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"18\" style=\"vertical-align: -11px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-25b32c0c9ec61f5b78326fccd6326107_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#50;&middot;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"18\" style=\"vertical-align: -12px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"center\">2<em data-effect=\"italics\">n<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"equation\" id=\"fs-id1167836512648\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-036c1bf770f2837a8b45db4d1db99904_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#51;&#125;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#52;&#125;&#44;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#54;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#56;&#44;&#38;&#32;&#49;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"34\" width=\"222\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"fs-id1167836535441\">To find the values of a sequence, we substitute in the counting numbers in order into the general term of the sequence.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836424231\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836408426\">\n<div data-type=\"problem\" id=\"fs-id1167836477802\">\n<p id=\"fs-id1167836547838\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0fcf577176238e88e07bd3d40ea0d49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#52;&#110;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836415705\">\n<p id=\"fs-id1167836545291\">We substitute the values 1, 2, 3, 4, and 5 into the formula, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-81f3d8da2e1183cbe6d6fb4ded6e93cc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#52;&#110;&#45;&#51;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -4px;\" \/> in order.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836551928\" data-alt=\"This figure shows three rows and five columns. The first row reads nth term equals 4 times n minus 3 written five times. The second row reads a sub 1 equals 4 times g times 1 minus 3, a sub 2 equals 4 times g times 2 minus 3, a sub 3 equals 4 times g times 3 minus 3, a sub 4 equals 4 times g times 4 minus 3, a sub 5 equals 4 times g times 5 minus 3. The third row reads, a sub 1 equals 1, a sub 2 equals 5, a sub 3 equals 9, a sub 4 equals 13, a sub 5 equals 17.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows three rows and five columns. The first row reads nth term equals 4 times n minus 3 written five times. The second row reads a sub 1 equals 4 times g times 1 minus 3, a sub 2 equals 4 times g times 2 minus 3, a sub 3 equals 4 times g times 3 minus 3, a sub 4 equals 4 times g times 4 minus 3, a sub 5 equals 4 times g times 5 minus 3. The third row reads, a sub 1 equals 1, a sub 2 equals 5, a sub 3 equals 9, a sub 4 equals 13, a sub 5 equals 17.\" \/><\/span><\/p>\n<p id=\"fs-id1167836362021\">The first five terms of the sequence are 1, 5, 9, 13, and 17.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836512330\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836526397\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836327796\">\n<p id=\"fs-id1167836376014\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3e5d7c8b8c961e4c5316f13d828d767_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#51;&#110;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836622808\">\n<p id=\"fs-id1167836523502\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d1357f05d47d986c9f05e143640ca22b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#50;&#44;&#53;&#44;&#56;&#44;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"97\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829598214\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836327003\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836662447\">\n<p id=\"fs-id1167836309374\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-746b1e7c9c996b091936069d40981aea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#50;&#110;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836414969\">\n<p id=\"fs-id1167836418956\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-51d6b08f7a4e3eccf68a92b5c54b1b52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#45;&#49;&#44;&#49;&#44;&#51;&#44;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836418844\">For some sequences, the variable is an exponent.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836410316\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836356303\">\n<div data-type=\"problem\" id=\"fs-id1167836516667\">\n<p id=\"fs-id1167829694727\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9837762b88d333d951508a6bf2e1ab6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836542257\">\n<p id=\"fs-id1167836311916\">We substitute the values 1, 2, 3, 4, and 5 into the formula, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e7fb89eb60273f3ac57096755af47a5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#43;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -4px;\" \/> in order.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836557890\" data-alt=\"This figure shows three rows and five columns. The first row reads &#x201c;nth term equals 2 to the nth power plus 1&#x201d; written five times. The second row reads, &#x201c;a sub 1 equals 2 times 1 plus 1, a sub 2 equals 2 to the power of 2 plus 1, a sub 3 equals 2 to the power 3 plus 1, a sub 4 equals 2 to the power of 4 plus 1, a sub 5 equals 2 to the power 5 plus 1&#x201d;. The last row reads &#x201c;a sub 1 equals 3, a sub 2 equals 5, a sub 3 equals 9, a sub 4 equals 17, a sub 5 equals 33&#x201d;.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows three rows and five columns. The first row reads &#x201c;nth term equals 2 to the nth power plus 1&#x201d; written five times. The second row reads, &#x201c;a sub 1 equals 2 times 1 plus 1, a sub 2 equals 2 to the power of 2 plus 1, a sub 3 equals 2 to the power 3 plus 1, a sub 4 equals 2 to the power of 4 plus 1, a sub 5 equals 2 to the power 5 plus 1&#x201d;. The last row reads &#x201c;a sub 1 equals 3, a sub 2 equals 5, a sub 3 equals 9, a sub 4 equals 17, a sub 5 equals 33&#x201d;.\" \/><\/span><\/p>\n<p id=\"fs-id1167836691944\">The first five terms of the sequence are 3, 5, 9, 17, and 33.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836700380\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836521836\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836367277\">\n<p id=\"fs-id1167836327173\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a51717f337428d41f4c730f364734610_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#51;&#125;&#94;&#123;&#110;&#125;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836320223\">\n<p id=\"fs-id1167829740836\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7676be28194eb75eae3834193ea5e079_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#44;&#49;&#51;&#44;&#51;&#49;&#44;&#56;&#53;&#44;&#50;&#52;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"121\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836691832\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836688426\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836510398\">\n<p id=\"fs-id1167829695706\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bfde651480c024980f3f53dd236302e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"95\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836362572\">\n<p id=\"fs-id1167836608138\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b92bf2f5e3e22b4d1d1f8ad606934cfd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#45;&#49;&#44;&#51;&#44;&#49;&#49;&#44;&#50;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836287578\">It is not uncommon to see the expressions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ced2ddb4631ac07149d157ee7d08348d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/> or <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1f662ecfe035fd90b5a408c43db8d28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/> in the general term for a sequence. If we evaluate each of these expressions for a few values, we see that this expression alternates the sign for the terms.<\/p>\n<table id=\"fs-id1167836521976\" summary=\"Table shows 6 columns and 3 rows. The first row lists values of n from 1 to 5. The second row lists values of negative 1 to the power of n as negative 1, 1, negative 1, 1, and negative 1. The third row lists values of negative 1 to the power of n plus 1 as 1 negative 1, 1, negative 1, and 1. Below the table are values a sub 1, a sub 2, a sub 3, a sub 4, a sub 5, and a sub n followed by ellipsis. Below are the values negative 1, 1, negative 1, 1, and negative 1. Below that are the numbers 1 negative 1, 1, negative 1, and 1.\" class=\"unnumbered\">\n<tbody>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">1<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">2<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">3<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">4<\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><strong data-effect=\"bold\">5<\/strong><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ced2ddb4631ac07149d157ee7d08348d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"43\" style=\"vertical-align: -4px;\" \/><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edc082c2635eaf3abe2c92050232cd80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"42\" style=\"vertical-align: -24px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5ba590f4ce1a59f64163e03c10f9f1fb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"43\" style=\"vertical-align: -24px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-766102477fe78856305bebd8bce8a571_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"43\" style=\"vertical-align: -24px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-800cfe7215d7e288b04f7282765fe0d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"43\" style=\"vertical-align: -24px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-623cb2ad5f722c83816dc4560356b3bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"43\" style=\"vertical-align: -24px;\" \/><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td data-valign=\"top\" data-align=\"center\"><strong data-effect=\"bold\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b1f662ecfe035fd90b5a408c43db8d28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/strong><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9fbeb3de471da59d3d54ab744985305a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#49;&#43;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"60\" style=\"vertical-align: -24px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f5180568185bbedadeb9178641f282e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#50;&#43;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"60\" style=\"vertical-align: -24px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-90b6879e132a04844bad486fa0952bdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#51;&#43;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"60\" style=\"vertical-align: -24px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ddb8ce8a62364fb0992dd3f8473038e4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#52;&#43;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"60\" style=\"vertical-align: -24px;\" \/><\/td>\n<td data-valign=\"middle\" data-align=\"center\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0cd8740817f77e8887b6645944890c69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#53;&#43;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"63\" width=\"60\" style=\"vertical-align: -24px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div data-type=\"equation\" id=\"fs-id1167836550517\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b37f1b3ad40f8e1cb86aca2803dcb1d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#51;&#125;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#52;&#125;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#44;&#38;&#32;&#45;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#44;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#44;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"56\" width=\"243\" style=\"vertical-align: -26px;\" \/><\/div>\n<p id=\"fs-id1167833059788\">The terms in the next example will alternate signs as a result of the powers of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc343c70c6ffe28a6cd57dfb53f250b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<div data-type=\"example\" id=\"fs-id1167836340793\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836540214\">\n<div data-type=\"problem\" id=\"fs-id1167836512624\">\n<p id=\"fs-id1167836534763\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56a3fc3019c0aac0b42fbe12fb4f1584_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833051065\">\n<p id=\"fs-id1167836707050\">We substitute the values 1, 2, 3, 4, and 5 into the formula, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-eab2839130ac73cfc75bd4a1af8e991f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/> in order.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836622463\" data-alt=\"This figure shows three rows and five columns. The first row reads &#x201c;nth term equals negative 1 to the nth power times n cubed&#x201d; written five times. The second row reads a sub 1 equals negative 1 to the power of 1 times g times 1 cubed, a sub 2 equals negative 1 squared time g times 2 cubed, a sub 3 equals negative 1 cubed times g times 23 cubed, a sub 4 equals negative 1 to the power of 4 times g times 4 cubed, a sub 5 equals negative 1 to the power of 5 times g times 5 cubed. The last row reads, &#x201c;a sub 1 equals negative 1, a sub 2 equals 8, a sub 3 equals negative 27, a sub 4 equals 64, and a sub 5 equals negative 125.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_003_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows three rows and five columns. The first row reads &#x201c;nth term equals negative 1 to the nth power times n cubed&#x201d; written five times. The second row reads a sub 1 equals negative 1 to the power of 1 times g times 1 cubed, a sub 2 equals negative 1 squared time g times 2 cubed, a sub 3 equals negative 1 cubed times g times 23 cubed, a sub 4 equals negative 1 to the power of 4 times g times 4 cubed, a sub 5 equals negative 1 to the power of 5 times g times 5 cubed. The last row reads, &#x201c;a sub 1 equals negative 1, a sub 2 equals 8, a sub 3 equals negative 27, a sub 4 equals 64, and a sub 5 equals negative 125.\" \/><\/span><\/p>\n<p id=\"fs-id1167836627870\">The first five terms of the sequence are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c13b30080194ad903a7045349fe53443_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#56;&#44;&#45;&#50;&#55;&#44;&#54;&#52;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"108\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5635b9acd283f40e02e8fa56b48a5c7d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"44\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836503186\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833019254\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836597185\">\n<p id=\"fs-id1167829720825\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7c178034b51e6f182764c4f519d36d04_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"110\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829720139\">\n<p id=\"fs-id1167836520591\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-94c6f23f4fc34c31aaf726d2e6f62b72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#52;&#44;&#45;&#57;&#44;&#49;&#54;&#44;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"133\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836367160\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829627648\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836553112\">\n<p id=\"fs-id1167829695390\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc4a166a6e32fa016dcc4a13e21f87c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"128\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836352298\">\n<p id=\"fs-id1167836529276\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f78914892bc2f95d0b861be313a33a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#45;&#56;&#44;&#50;&#55;&#44;&#45;&#54;&#52;&#44;&#49;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"137\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836526589\">\n<h3 data-type=\"title\">Find a Formula for the General Term (<em data-effect=\"italics\">n<\/em>th Term) of a Sequence<\/h3>\n<p id=\"fs-id1167829740541\">Sometimes we have a few terms of a sequence and it would be helpful to know the general term or <em data-effect=\"italics\">n<\/em>th term. To find the general term, we look for patterns in the terms. Often the patterns involve multiples or powers. We also look for a pattern in the signs of the terms.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836330323\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836688056\">\n<div data-type=\"problem\" id=\"fs-id1167836299692\">\n<p id=\"fs-id1167836700504\">Find a general term for the sequence whose first five terms are shown.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836621355\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab503cea337bc2d5078c9541b7702039_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#56;&#44;&#49;&#50;&#44;&#49;&#54;&#44;&#50;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#100;&#111;&#116;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836392172\">\n<table id=\"fs-id1167836548526\" summary=\"This figure shows five rows. The first row reads, &#x201c;4&#x201d;, &#x201c;8&#x201d;, &#x201c;12&#x201d;, &#x201c;16&#x201d;, &#x201c;20&#x201d;, and an &#x201c;ellipsis&#x201d;. The second row reads &#x201c;n&#x201d;, &#x201c;1&#x201d;, &#x201c;2&#x201d;, &#x201c;3&#x201d;, &#x201c;4&#x201d;, &#x201c;5&#x201d;, and an &#x201c;ellipsis&#x201d;. The third row reads &#x201c;We look for a pattern in terms&#x201d;, &#x201c;Terms&#x201d;, &#x201c;4&#x201d;, &#x201c;8&#x201d;, &#x201c;12&#x201d;, &#x201c;16&#x201d;, &#x201c;20&#x201d;, and an &#x201c;ellipsis&#x201d;. The four row reads, &#x201c;The numbers are all multiples of 4&#x201d;, &#x201c;Pattern&#x201d;, &#x201c;4 times g times 1&#x201d;, &#x201c;4 times g times 3&#x201d;, 4 times g times 4&#x201d;, &#x201c;4 times g times 5&#x201d;, and an &#x201c;ellipsis&#x201d;, &#x201c;4 times g times n&#x201d;. The last row reads &#x201c;The general term of the sequence is a nth term equals 4 times n&#x201d;.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836509695\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_006a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836492778\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_006b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We look for a pattern in the terms.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836700375\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_006c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The numbers are all multiples of 4.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836330596\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_006d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">The general term of the sequence is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2e81cec180ccbff36faf149cc81a0cf9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#52;&#110;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829594848\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836356048\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829691674\">\n<p id=\"fs-id1167829579943\">Find a general term for the sequence whose first five terms are shown.<\/p>\n<p id=\"fs-id1167836310267\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f290748fa789345e5a5b58fc6ef6b95_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#44;&#54;&#44;&#57;&#44;&#49;&#50;&#44;&#49;&#53;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"98\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836540322\">\n<p id=\"fs-id1167836511209\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-471a1d44dc0a73ed4078813e9e040e2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#51;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"62\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836312781\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836688166\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833021343\">\n<p id=\"fs-id1167833158233\">Find a general term for the sequence whose first five terms are shown.<\/p>\n<p id=\"fs-id1167829645005\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e14e21d4a70e086dd459e2afae874f60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#49;&#48;&#44;&#49;&#53;&#44;&#50;&#48;&#44;&#50;&#53;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833060799\">\n<p id=\"fs-id1167829853134\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e7b3401f72cea1d5efc340d022a0abd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#53;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"62\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167829628072\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836538101\">\n<div data-type=\"problem\" id=\"fs-id1167836504080\">\n<p id=\"fs-id1167836520449\">Find a general term for the sequence whose first five terms are shown.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836292789\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3681269dbdb2bd4460cddc032a2ecc93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#45;&#52;&#44;&#56;&#44;&#45;&#49;&#54;&#44;&#51;&#50;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"125\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836310482\">\n<table id=\"fs-id1167829756446\" summary=\"This figure shows six rows. The first row reads, &#x201c;2&#x201d;, &#x201c;negative 4&#x201d;, &#x201c;8&#x201d;, &#x201c;negative 16&#x201d;, &#x201c;32&#x201d;, and an &#x201c;ellipsis&#x201d;. The second row reads &#x201c;n&#x201d;, &#x201c;1&#x201d;, &#x201c;2&#x201d;, &#x201c;3&#x201d;, &#x201c;4&#x201d;, &#x201c;5&#x201d;, &#x201c;ellipsis&#x201d;, &#x201c;n&#x201d;. The third row reads &#x201c;We look for a pattern in the terms&#x201d;, &#x201c;Terms 2,&#x201d; &#x201c;negative 4&#x201d;, &#x201c;8&#x201d;, &#x201c;negative 16&#x201d;, &#x201c;32&#x201d;, and an &#x201c;ellipsis&#x201d;. The fourth row reads &#x201c;The numbers are powers of 2&#x201d;, and &#x201c;Pattern&#x201d;. The fifth row reads &#x201c;the signs are alternating with even n negative&#x201d;, &#x201c;negative 1 to the power of 1 plus 1 times 2 to the power of 1, negative 1 to the power of 2 plus 1 times 2 to the power of 2, negative 1 to the power of 3 plus 1 times 2 to the power of 3, negative 1 to the power of 4 plus 1 times 2 to the power of 1, negative 1 to the power of 5 plus 1 times 2 to the power of 5, ellipsis, negative 1 to the power of n plus 1 times 2 to then power of n. The last row reads, &#x201c;The general term of the sequence is a sub n equals negative 1 to the power of n plus 1 times 2 to the power of n.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833054918\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_007a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836432055\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_007b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We look for a pattern in the terms.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836554839\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_007c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The numbers are powers of 2. The signs are<span data-type=\"newline\"><br \/><\/span>alternating, with even <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> negative.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167833369157\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_007d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">The general term of the sequence is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b789e65f620bb01dbb3e04fb146813e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"127\" style=\"vertical-align: -4px;\" \/> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829790449\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829694947\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829694949\">\n<p id=\"fs-id1167836558252\">Find a general term for the sequence whose first five terms are shown.<\/p>\n<p id=\"fs-id1167836558255\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f3e025f8a442aa7c4fc38d153a0158c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#57;&#44;&#45;&#50;&#55;&#44;&#56;&#49;&#44;&#45;&#50;&#52;&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836398987\">\n<p id=\"fs-id1167836398989\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf0025132182374121b9df271512c96a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#123;&#51;&#125;&#94;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836417349\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836506804\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836506806\">\n<p id=\"fs-id1167836506808\">Find a general term for the sequence whose first five terms are shown<\/p>\n<p id=\"fs-id1167829809843\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b51c6d606dcdc5a358df5672eb492280_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#45;&#52;&#44;&#57;&#44;&#45;&#49;&#54;&#44;&#50;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#44;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836536576\">\n<p id=\"fs-id1167829596213\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c6e2475fdd795adf440aabd1a44d6f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" id=\"fs-id1167836417304\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836417306\">\n<div data-type=\"problem\" id=\"fs-id1167836417309\">\n<p id=\"fs-id1167829628377\">Find a general term for the sequence whose first five terms are shown.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829628380\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-91d24443dcefc89ebc93c18276808fa3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#49;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#52;&#51;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"115\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836448361\">\n<table id=\"fs-id1167836635242\" summary=\"This figure shows five rows. The first row reads, &#x201c;one-third&#x201d;, &#x201c;one-ninth&#x201d;, &#x201c;one-twenty-seventh&#x201d;, &#x201c;1 divided by 81&#x201d;, &#x201c;1 divided by 243&#x201d;, and an ellipsis. The second row reads, &#x201c;n&#x201d;, &#x201c;1&#x201d;, &#x201c;2&#x201d;, &#x201c;3&#x201d;, &#x201c;4&#x201d;, &#x201c;5&#x201d; and an ellipsis, &#x201c;n&#x201d;. The third row reads &#x201c;We look for a pattern in the terms&#x201d;, &#x201c;Terms&#x201d;, &#x201c;one-third&#x201d;, &#x201c;one-ninth&#x201d;, &#x201c;one-twenty-seventh&#x201d;, &#x201c;1 divided by 81&#x201d;, &#x201c;1 divided by 243&#x201d;, and an ellipsis. The fourth row reads, &#x201c;The numerators are all at 1&#x201d;, &#x201c;Pattern&#x201d;, &#x201c;one-third to the power of 1 times 9, one-third to the power of 2 times 9, one-third to the power of 3 times 4 times 9, one-third to the power of 3 times 5 times 9, ellipsis, one-third to the power of n&#x201d;. The fifth row reads, &#x201c;The denominators are powers of 3. The general term of the sequence is a sub n equals one-third to the power of n&#x201d;.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829580229\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_008a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829716495\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_008b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We look for a pattern in the terms.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836691952\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_008c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The numerators are all 1.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836609920\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_008d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The denominators are powers of 3.<\/td>\n<td data-align=\"left\">The general term of the sequence is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8fdcff658614537d5a8374338cd33e78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#51;&#125;&#94;&#123;&#110;&#125;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"65\" style=\"vertical-align: -6px;\" \/> <\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836318650\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836318653\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836525725\">\n<p id=\"fs-id1167836525727\">Find a general term for the sequence whose first five terms are shown.<\/p>\n<p id=\"fs-id1167836525730\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-19dbcb3e0087e8e593a1b96c82f3287f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#50;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"101\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836559949\">\n<p id=\"fs-id1167836559951\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5914fac140c0fe611f165b7c56b19269_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167832989121\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836360706\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836360708\">\n<p id=\"fs-id1167836360710\">Find a general term for the sequence whose first five terms are shown.<\/p>\n<p id=\"fs-id1167836375920\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-022f8921408dfe805613a0020fe0959e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"101\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833061756\">\n<p id=\"fs-id1167833061758\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b27f04999333431857f7204e0e723d28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"60\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836547277\">\n<h3 data-type=\"title\">Use Factorial Notation<\/h3>\n<p id=\"fs-id1167836506757\">Sequences often have terms that are products of consecutive integers. We indicate these products with a special notation called <em data-effect=\"italics\">factorial notation<\/em>. For example,<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1aa90678cd6bf384861b42b23e66404e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\" \/>, read 5 factorial, means <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-12f4b45e169b2a0f772c3310112b4b7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&middot;&#52;&middot;&#51;&middot;&#50;&middot;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"49\" style=\"vertical-align: -1px;\" \/> The exclamation point is not punctuation here; it indicates the <span data-type=\"term\">factorial notation<\/span>.<\/p>\n<div data-type=\"note\" id=\"fs-id1167836492202\">\n<div data-type=\"title\">Factorial Notation<\/div>\n<p id=\"fs-id1167836492208\">If <em data-effect=\"italics\">n<\/em> is a positive integer, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da0ef996f36e1b32a0f26f6e896e1771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/> is<\/p>\n<div data-type=\"equation\" id=\"fs-id1167829754626\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-293c13b696951bd07fdb3ca721c93d66_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#33;&#61;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#111;&#116;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"188\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167829791300\">We define <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b00e8fb86b6f4891f9d2fbed6a7670b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> as 1, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f4299fb0e2692b5751c097e082f036e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#33;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"50\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<p id=\"fs-id1167833048324\">The values of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da0ef996f36e1b32a0f26f6e896e1771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/> for the first 5 positive integers are shown.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836552570\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-11ee4b0ed0b46a6f092471262e739322_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#33;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#33;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#33;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#33;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#33;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#99;&#100;&#111;&#116;&#32;&#52;&#92;&#99;&#100;&#111;&#116;&#32;&#51;&#92;&#99;&#100;&#111;&#116;&#32;&#50;&#92;&#99;&#100;&#111;&#116;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"58\" width=\"600\" style=\"vertical-align: -23px;\" \/><\/div>\n<div data-type=\"example\" id=\"fs-id1167836704331\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836704333\">\n<div data-type=\"problem\" id=\"fs-id1167836704335\">\n<p id=\"fs-id1167829754906\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-942f41cf257670b470f22d8c8348ec01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#33;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836611218\">\n<p id=\"fs-id1167829809728\">We substitute the values 1, 2, 3, 4, 5 into the formula, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b9d340daef6dd771b74f3c4d57e75751_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#33;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/> in order.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167836433458\" data-alt=\"This figure shows four rows and five columns. The first row reads, &#x201c;nth term equals one divided by n factorial&#x201d; written five times. The second row reads &#x201c;a sub 1 equals one divided by 1 factorial, a sub 2 equals 1 divided by 2 factorial, a sub 3 equals 1 divided by 3 factorial, a sub 4 equals 1 divided by 4 factorial, a sub 5 equals 1 divided by 5 factorial&#x201d;. The third row reads &#x201c;a sub 1 equals 1 divided 1&#x201d;, &#x201c;a sub 2 equals 1 divided by 2 times g times 1&#x201d;, &#x201c;a sub 3 equals 1 divided by 3 times g times 2 g times 1&#x201d;, &#x201c;a sub 4 equals 1 divided 4 times g times 3 times g times 2 times g times 1&#x201d;, &#x201c;a sub 5 equals 1 divided by 5 g times 4 times g times 3 times g times 2 times g times 1&#x201d;, &#x201c;a sub 1 equals 1, a sub 2 equals one-half&#x201d;, &#x201c;a sub 3 equals one-sixth&#x201d;, &#x201c;a sub 4 equals 1 divided by 24&#x201d;, &#x201c;a sub 5 equals 1 divided by 120&#x201d;.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_004_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows four rows and five columns. The first row reads, &#x201c;nth term equals one divided by n factorial&#x201d; written five times. The second row reads &#x201c;a sub 1 equals one divided by 1 factorial, a sub 2 equals 1 divided by 2 factorial, a sub 3 equals 1 divided by 3 factorial, a sub 4 equals 1 divided by 4 factorial, a sub 5 equals 1 divided by 5 factorial&#x201d;. The third row reads &#x201c;a sub 1 equals 1 divided 1&#x201d;, &#x201c;a sub 2 equals 1 divided by 2 times g times 1&#x201d;, &#x201c;a sub 3 equals 1 divided by 3 times g times 2 g times 1&#x201d;, &#x201c;a sub 4 equals 1 divided 4 times g times 3 times g times 2 times g times 1&#x201d;, &#x201c;a sub 5 equals 1 divided by 5 g times 4 times g times 3 times g times 2 times g times 1&#x201d;, &#x201c;a sub 1 equals 1, a sub 2 equals one-half&#x201d;, &#x201c;a sub 3 equals one-sixth&#x201d;, &#x201c;a sub 4 equals 1 divided by 24&#x201d;, &#x201c;a sub 5 equals 1 divided by 120&#x201d;.\" \/><\/span><\/p>\n<p id=\"fs-id1167829747222\">The first five terms of the sequence are <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b3a46b3dd615c53a9ffd77a68b27c78e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#48;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"107\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836698688\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829811091\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829811093\">\n<p id=\"fs-id1167829811095\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-218b5ebed33c22db211a653a240f5a48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#110;&#33;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829755838\">\n<p id=\"fs-id1167829755841\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2926cd778f6dcbbbe7f4b3b4e203510f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833047217\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836622285\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836622287\">\n<p id=\"fs-id1167836300267\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36db46163b850d943918f311ce83f8c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#110;&#33;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833023017\">\n<p id=\"fs-id1167829683662\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de7d63a8eac8e43e68c19ca38c27c10c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"89\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165927785098\">When there is a fraction with factorials in the numerator and denominator, we line up the factors vertically to make our calculations easier.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836548831\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836548833\">\n<div data-type=\"problem\" id=\"fs-id1167829720760\">\n<p id=\"fs-id1167829720762\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-568cc6217c54b3ead031682411e81b8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"91\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833370051\">\n<p id=\"fs-id1167833370053\">We substitute the values 1, 2, 3, 4, 5 into the formula, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-acd0de5b9835fa0fb8ff5aedef5379b5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"91\" style=\"vertical-align: -9px;\" \/> in order.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167833240164\" data-alt=\"This figure shows five columns and five rows. The first row shows the sequence &#x201c;nth term equals n plus 1 times factorial divided by n minus 1 times factorial&#x201d; written five times. The second row is &#x201c;a sub 1 equals 1 plus 1 times factorial divided by 1 minus 1 times factorial&#x201d;, &#x201c;a sub 2 equals 2 plus 1 times factorial divided by 2 minus 1 times factorial&#x201d;, &#x201c;a sub 3 equals 3 plus 1 times factorial divided by 3 minus 1 times factorial&#x201d;, &#x201c;a sub 4 equals 4 plus 1 times factorial divided by 4 minus 1 times factorial&#x201d;, &#x201c;a sub 5 equals 5 plus 1 times factorial divided by 5 minus 1 times factorial&#x201d;. The third row reads &#x201c;a sub 1 equals 2 times factorial divided by 0 times factorial&#x201d;, &#x201c;a sub 2 equals 3 times factorial divided by 1 times factorial&#x201d;, &#x201c;a sub 3 equals 4 times factorial divided by 2 times factorial&#x201d;, &#x201c;a sub 3 equals 4 times factorial divided by 2 times factorial&#x201d;, &#x201c;a sub 4 equals 5 times factorial divided by 3 times factorial&#x201d;, &#x201c;a sub 5 equals 6 times factorial divided by 4 times factorial&#x201d;. The fourth row reads, &#x201c;a sub 1 equals 2 times g time 1 divided by 1&#x201d;, &#x201c;a sub 2 equals 3 times g times 2 times g times 1 divided by 1&#x201d;, &#x201c;a sub 3 equals 4 times g times 3 times g times 2 times g times 1 divided by 2 times g times 1&#x201d;, &#x201c;a sub 4 equals 5 times g times 4 times g times 3 times g times 2 times g times 1 divided by 3 g times 2 times g times 1&#x201d;, and &#x201c;a sub 5 equals 6 times g times 5 times g times 4 times g times 3 times g times 2 times g times 1 divided by 4 times g times 3 times g times 2 times g times 1&#x201d;. The fifth row reads &#x201c;a sub 1 equals 2&#x201d;, &#x201c;a sub 2 equals 6&#x201d;, &#x201c;a sub 3 equals 12&#x201d;, &#x201c;a sub 4 equals 20&#x201d;, &#x201c;a sub 5 equals 30&#x201d;.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_005_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows five columns and five rows. The first row shows the sequence &#x201c;nth term equals n plus 1 times factorial divided by n minus 1 times factorial&#x201d; written five times. The second row is &#x201c;a sub 1 equals 1 plus 1 times factorial divided by 1 minus 1 times factorial&#x201d;, &#x201c;a sub 2 equals 2 plus 1 times factorial divided by 2 minus 1 times factorial&#x201d;, &#x201c;a sub 3 equals 3 plus 1 times factorial divided by 3 minus 1 times factorial&#x201d;, &#x201c;a sub 4 equals 4 plus 1 times factorial divided by 4 minus 1 times factorial&#x201d;, &#x201c;a sub 5 equals 5 plus 1 times factorial divided by 5 minus 1 times factorial&#x201d;. The third row reads &#x201c;a sub 1 equals 2 times factorial divided by 0 times factorial&#x201d;, &#x201c;a sub 2 equals 3 times factorial divided by 1 times factorial&#x201d;, &#x201c;a sub 3 equals 4 times factorial divided by 2 times factorial&#x201d;, &#x201c;a sub 3 equals 4 times factorial divided by 2 times factorial&#x201d;, &#x201c;a sub 4 equals 5 times factorial divided by 3 times factorial&#x201d;, &#x201c;a sub 5 equals 6 times factorial divided by 4 times factorial&#x201d;. The fourth row reads, &#x201c;a sub 1 equals 2 times g time 1 divided by 1&#x201d;, &#x201c;a sub 2 equals 3 times g times 2 times g times 1 divided by 1&#x201d;, &#x201c;a sub 3 equals 4 times g times 3 times g times 2 times g times 1 divided by 2 times g times 1&#x201d;, &#x201c;a sub 4 equals 5 times g times 4 times g times 3 times g times 2 times g times 1 divided by 3 g times 2 times g times 1&#x201d;, and &#x201c;a sub 5 equals 6 times g times 5 times g times 4 times g times 3 times g times 2 times g times 1 divided by 4 times g times 3 times g times 2 times g times 1&#x201d;. The fifth row reads &#x201c;a sub 1 equals 2&#x201d;, &#x201c;a sub 2 equals 6&#x201d;, &#x201c;a sub 3 equals 12&#x201d;, &#x201c;a sub 4 equals 20&#x201d;, &#x201c;a sub 5 equals 30&#x201d;.\" \/><\/span><\/p>\n<p id=\"fs-id1167836621846\">The first five terms of the sequence are 2, 6, 12, 20, and 30.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829789958\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833023070\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833023072\">\n<p id=\"fs-id1167833023074\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26fe67bcbc3b56df48fca2d9d1415f72_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"91\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836398976\">\n<p id=\"fs-id1167836398978\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7ed095dfb2b284082b3d3bffb3473464_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#48;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"102\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836613306\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836613309\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836613312\">\n<p id=\"fs-id1167836622307\">Write the first five terms of the sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-28c87ec71ae87728de52e3ebc60792eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#33;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"91\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833056115\">\n<p id=\"fs-id1167833056118\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e82aaa9040550110e3cb39159523738_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"82\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167829719000\">\n<h3 data-type=\"title\">Find the Partial Sum<\/h3>\n<p id=\"fs-id1167829596997\">Sometimes in applications, rather than just list the terms, it is important for us to add the terms of a sequence. Rather than just connect the terms with plus signs, we can use <span data-type=\"term\">summation notation<\/span>.<\/p>\n<p id=\"fs-id1167836732726\">For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84e3845f2a697c1168c4fc2d0b3c1094_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#51;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#52;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"171\" style=\"vertical-align: -4px;\" \/> can be written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2f2eaa8459cb148fb18301f389cfcc7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#123;&#97;&#125;&#95;&#123;&#105;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -6px;\" \/> We read this as \u201cthe sum of <em data-effect=\"italics\">a<\/em> sub <em data-effect=\"italics\">i<\/em> from <em data-effect=\"italics\">i<\/em> equals one to five.\u201d The symbol <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58f087ff7f86b39fa0be9f1ec7f83572_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"18\" style=\"vertical-align: -5px;\" \/> means to add and the <em data-effect=\"italics\">i<\/em> is the index of summation. The 1 tells us where to start (initial value) and the 5 tells us where to end (terminal value).<\/p>\n<div data-type=\"note\" id=\"fs-id1167832982280\">\n<div data-type=\"title\">Summation Notation<\/div>\n<p id=\"fs-id1167836391189\">The sum of the first <em data-effect=\"italics\">n<\/em> terms of a sequence whose <em data-effect=\"italics\">n<\/em>th term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1d5f11b142b357c0576e6792d1cef0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> is written in summation notation as:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167836626906\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc2c02c6c4a389e04d630bd4e431330f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#123;&#97;&#125;&#95;&#123;&#105;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#51;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#52;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#53;&#125;&#43;&#92;&#100;&#111;&#116;&#115;&#32;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"336\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1167829719007\">The <em data-effect=\"italics\">i<\/em> is the index of summation and the 1 tells us where to start and the <em data-effect=\"italics\">n<\/em> tells us where to end.<\/p>\n<\/div>\n<p id=\"fs-id1167833019032\">When we add a finite number of terms, we call the sum a <span data-type=\"term\">partial sum<\/span>.<\/p>\n<div data-type=\"example\" id=\"fs-id1167833019038\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836561241\">\n<div data-type=\"problem\" id=\"fs-id1167836561243\">\n<p id=\"fs-id1167836561245\">Expand the partial sum and find its value: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-beb5cfa87a551a24cc94920ea57e8330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#50;&#105;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836684163\">\n<p id=\"fs-id1167836389436\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f79b88bea1ee4380a87062fbf9e1f94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#50;&#105;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#32;&#49;&#44;&#32;&#50;&#44;&#32;&#51;&#44;&#32;&#52;&#44;&#32;&#53;&#32;&#105;&#110;&#32;&#111;&#114;&#100;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&middot;&#49;&#43;&#50;&middot;&#50;&#43;&#50;&middot;&#51;&#43;&#50;&middot;&#52;&#43;&#50;&middot;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#43;&#52;&#43;&#54;&#43;&#56;&#43;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#100;&#100;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#50;&#105;&#61;&#51;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"244\" width=\"580\" style=\"vertical-align: -117px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829747135\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167833396844\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833396846\">\n<p id=\"fs-id1167833396848\">Expand the partial sum and find its value: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c630d436a8b924332b01820d17d99e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#51;&#105;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829719048\">\n<p id=\"fs-id1167829719050\">45<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836415595\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836287270\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836287272\">\n<p id=\"fs-id1167836287274\">Expand the partial sum and find its value: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5af475b4fbfc3ed3e7e67e24f2c1c104_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#52;&#105;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"64\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832984037\">\n<p id=\"fs-id1167832984039\">60<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836450406\">The index does not always have to be <em data-effect=\"italics\">i<\/em> we can use any letter, but <em data-effect=\"italics\">i<\/em> and <em data-effect=\"italics\">k<\/em> are commonly used. The index does not have to start with 1 either\u2014it can start and end with any positive integer.<\/p>\n<div data-type=\"example\" id=\"fs-id1167829747210\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167829747212\">\n<div data-type=\"problem\" id=\"fs-id1167829747214\">\n<p id=\"fs-id1167829747216\">Expand the partial sum and find its value: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b54178d76b3e08be8351dd78c6264227_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#48;&#125;&#94;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#107;&#33;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"67\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829747192\">\n<p id=\"fs-id1167829747195\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9505b4ad472592cb12771365a20b935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#48;&#125;&#94;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#107;&#33;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#32;&#48;&#44;&#32;&#49;&#44;&#32;&#50;&#44;&#32;&#51;&#44;&#32;&#105;&#110;&#32;&#111;&#114;&#100;&#101;&#114;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#48;&#33;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#33;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#33;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#33;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#69;&#118;&#97;&#108;&#117;&#97;&#116;&#101;&#32;&#116;&#104;&#101;&#32;&#102;&#97;&#99;&#116;&#111;&#114;&#105;&#97;&#108;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#43;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#54;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#105;&#109;&#112;&#108;&#105;&#102;&#121;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#48;&#125;&#94;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#107;&#33;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"156\" width=\"512\" style=\"vertical-align: -73px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167833158242\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836610913\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836610915\">\n<p id=\"fs-id1167836610917\">Expand the partial sum and find its value: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5590a126485cb480f15ac1e23764b01d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#48;&#125;&#94;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#107;&#33;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"67\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836534087\">\n<p id=\"fs-id1167836534089\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e7be055a86dc72d43ac2d84c9076941f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#54;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"14\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829717879\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829717882\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836540528\">\n<p id=\"fs-id1167836540530\">Expand the partial sum and find its value: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1df7e2eab3fae3ce49841d5a801c2c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#48;&#125;&#94;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#107;&#33;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"67\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829747803\">\n<p id=\"fs-id1167829747805\">8<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167836294722\">\n<h3 data-type=\"title\">Use Summation Notation to Write a Sum<\/h3>\n<p id=\"fs-id1167836294727\">In the last two examples, we went from summation notation to writing out the sum. Now we will start with a sum and change it to summation notation. This is very similar to finding the general term of a sequence. We will need to look at the terms and find a pattern. Often the patterns involve multiples or powers.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836688410\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836688412\">\n<div data-type=\"problem\" id=\"fs-id1167829720741\">\n<p id=\"fs-id1167829720743\">Write the sum using summation notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-243d294e4ac9c41e47429a298211483d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"142\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829809741\">\n<p id=\"fs-id1167829809743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9bb29b9060f32df8c68341710ae97e8a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#125;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#53;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#110;&#92;&#116;&#101;&#120;&#116;&#123;&#58;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#50;&#44;&#51;&#44;&#52;&#44;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#87;&#101;&#32;&#108;&#111;&#111;&#107;&#32;&#102;&#111;&#114;&#32;&#97;&#32;&#112;&#97;&#116;&#116;&#101;&#114;&#110;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#101;&#114;&#109;&#115;&#58;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#110;&#117;&#109;&#101;&#114;&#97;&#116;&#111;&#114;&#115;&#32;&#97;&#114;&#101;&#32;&#97;&#108;&#108;&#32;&#111;&#110;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#50;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#116;&#116;&#101;&#114;&#110;&#58;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#44;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#100;&#101;&#110;&#111;&#109;&#105;&#110;&#97;&#116;&#111;&#114;&#115;&#32;&#97;&#114;&#101;&#32;&#116;&#104;&#101;&#32;&#99;&#111;&#117;&#110;&#116;&#105;&#110;&#103;&#32;&#110;&#117;&#109;&#98;&#101;&#114;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#114;&#111;&#109;&#32;&#111;&#110;&#101;&#32;&#116;&#111;&#32;&#102;&#105;&#118;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#115;&#117;&#109;&#32;&#119;&#114;&#105;&#116;&#116;&#101;&#110;&#32;&#105;&#110;&#32;&#115;&#117;&#109;&#109;&#97;&#116;&#105;&#111;&#110;&#32;&#110;&#111;&#116;&#97;&#116;&#105;&#111;&#110;&#32;&#105;&#115;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#61;&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#110;&#125;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"156\" width=\"725\" style=\"vertical-align: -73px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829627520\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836605384\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836605386\">\n<p id=\"fs-id1167836605388\">Write the sum using summation notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f0867e234533a4844667df762dbca01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#50;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"156\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833061159\">\n<p id=\"fs-id1167833061162\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f4d8256cc6980105ac92ebec1f274522_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836531526\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836531529\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836531531\">\n<p id=\"fs-id1167833014902\">Write the sum using summation notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa328f1514c9cc9a2fc6ddec58fe6184_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"155\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829908624\">\n<p id=\"fs-id1167829908626\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-30e4788fa99b1a0787995a351b8bcb01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"66\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165923635499\">When the terms of a sum have negative coefficients, we must carefully analyze the pattern of the signs.<\/p>\n<div data-type=\"example\" id=\"fs-id1167836507518\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167836507520\">\n<div data-type=\"problem\" id=\"fs-id1167836507522\">\n<p id=\"fs-id1167836507524\">Write the sum using summation notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b08d6fb849d766d47a73ec8e588ba60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#43;&#56;&#45;&#50;&#55;&#43;&#54;&#52;&#45;&#49;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"184\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836530770\">\n<table id=\"fs-id1167836530772\" summary=\"This figure shows a table with seven columns and six rows. The first row reads the summation notation as &#x201c;negative 1 plus 8 minus 27 plus 64 minus 125&#x201d;. The second row reads &#x201c;n&#x201d;, then &#x201c;1&#x201d;, &#x201c;2&#x201d;, &#x201c;3&#x201d;, &#x201c;4&#x201d; and &#x201c;5&#x201d;. The third row reads, &#x201c;We look for a pattern in the terms&#x201d;, &#x201c;Terms&#x201d;, &#x201c;-1&#x201d;, &#x201c;negative 8&#x201d;, &#x201c;negative 27&#x201d;, &#x201c;64&#x201d;, and &#x201c;negative 125&#x201d;. The fourth row reads, &#x201c;The signs of the terms alternate, and the odd terms are negative&#x201d;, &#x201c;Pattern&#x201d;, &#x201c;negative 1 times the power of 1 times g times 1, negative 1 squared times g times 8, negative 1 cubed times g times 27, negative 1 to the power of 4 times g times 64, negative 1 to the power of 5 times g times 125. The fifth row reads, &#x201c;The numbers are the cubes of the counting numbers from 1 to 5&#x201d;, &#x201c;Pattern&#x201d;, &#x201c;negative 1 to the power of 1 times g times l cubed, negative 1 to the power of 2, times g times 2 cubed, negative 1 to the power of 3 times g times 3 cubed, negative 1 to the power of 4 times g times 4 cubed, negative 1 to the power of 5 times g times 5 cubed&#x201d;. The sixth row reads &#x201c;Pattern is negative 1 to the power of n times g times n to the power of 3. The seventh row is &#x201c;The sum written as summation notation is negative 1 plus 8 minus 27 plus 64 minus 124 equals the sum of a sub i from i equals 1 to 5 times negative 1 to the power of n times g times n to the power of 3.&#x201d;\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836433359\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836685110\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">We look for a pattern in the terms.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836728372\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The signs of the terms alternate,<span data-type=\"newline\"><br \/><\/span>and the odd terms are negative.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167829741797\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">The numbers are the cubes of the<span data-type=\"newline\"><br \/><\/span>counting numbers from one to five.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836444781\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167836646292\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_009f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\">The sum written in summation notation is<\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e13481c3d66144b270b6e4a8a2c61c1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#43;&#56;&#45;&#50;&#55;&#43;&#54;&#52;&#45;&#49;&#50;&#53;&#61;&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#123;&#110;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"328\" style=\"vertical-align: -6px;\" \/><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167836542582\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167836542586\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836542588\">\n<p id=\"fs-id1167836492236\">Write each sum using summation notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5b6951fed35ab125683d3c9904e9ea68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#52;&#43;&#57;&#45;&#49;&#54;&#43;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"152\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836363682\">\n<p id=\"fs-id1167836363219\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-244047cc87bcb5385697be9db9a7aa9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"130\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829627922\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167829627926\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829627928\">\n<p id=\"fs-id1167829627930\">Write each sum using summation notation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ab32ecc3a0f27bf6d3371c8a9abac2d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#52;&#45;&#54;&#43;&#56;&#45;&#49;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"157\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833048339\">\n<p id=\"fs-id1167833048341\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-56911d313225c310a829a2838f789641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#50;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167829597663\" class=\"media-2\">\n<p id=\"fs-id1167829597666\">Access this online resource for additional instruction and practice with sequences.<\/p>\n<ul id=\"fs-id1167836321416\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37serseqfindpat\">Series and Sequences-Finding Patterns<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836363337\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167836363344\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">Factorial Notation<\/strong><span data-type=\"newline\"><br \/><\/span> If <em data-effect=\"italics\">n<\/em> is a positive integer, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da0ef996f36e1b32a0f26f6e896e1771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/> is<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1167833017967\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1c194e7a5cb15b1c08a61cecc7fbf8d0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#33;&#61;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"273\" style=\"vertical-align: -4px;\" \/><\/div>\n<p><span data-type=\"newline\"><br \/><\/span> We define <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b00e8fb86b6f4891f9d2fbed6a7670b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> as 1, so <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e4cedf8c2d5ce209b20e6b15a01111e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#33;&#61;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"45\" style=\"vertical-align: -1px;\" \/><\/li>\n<li><strong data-effect=\"bold\">Summation Notation<\/strong><span data-type=\"newline\"><br \/><\/span> The sum of the first <em data-effect=\"italics\">n<\/em> terms of a sequence whose <em data-effect=\"italics\">n<\/em>th term <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e1d5f11b142b357c0576e6792d1cef0b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"17\" style=\"vertical-align: -3px;\" \/> is written in summation notation as:<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1167836692030\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc2c02c6c4a389e04d630bd4e431330f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#110;&#125;&#123;&#97;&#125;&#95;&#123;&#105;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#51;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#52;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#53;&#125;&#43;&#92;&#100;&#111;&#116;&#115;&#32;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"336\" style=\"vertical-align: -6px;\" \/><\/div>\n<p><span data-type=\"newline\"><br \/><\/span> The <em data-effect=\"italics\">i<\/em> is the index of summation and the 1 tells us where to start and the <em data-effect=\"italics\">n<\/em> tells us where to end.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167836692470\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167836692473\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167836692479\"><strong data-effect=\"bold\">Write the First Few Terms of a Sequence<\/strong><\/p>\n<p id=\"fs-id1167836542434\">In the following exercises, write the first five terms of the sequence whose general term is given.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836542437\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836542439\">\n<p id=\"fs-id1167836732091\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bbd829a521a6188ae1964cb35f6dcf81_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#50;&#110;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833047542\">\n<p id=\"fs-id1167833047545\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7acda366c2fb62d4fdd703ea42de808f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#44;&#45;&#51;&#44;&#45;&#49;&#44;&#49;&#44;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836697712\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836697714\">\n<p id=\"fs-id1167836662577\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59508614d60c5a301b405f316badd957_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#53;&#110;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833030960\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833030962\">\n<p id=\"fs-id1167829593872\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8b3067926c1e43798bee6cff5267f7cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#51;&#110;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829586574\">\n<p id=\"fs-id1167829586576\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-58fff65afcb1f6b23b3a37f7c075470a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#55;&#44;&#49;&#48;&#44;&#49;&#51;&#44;&#49;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"103\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836433028\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836433030\">\n<p id=\"fs-id1167836622408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a886a2dc9933935cf7ae1f5b0c58e2bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#52;&#110;&#43;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836554862\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836554864\">\n<p id=\"fs-id1167836621806\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f3197f656c5b001f4152d4abc9ba5ad8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"91\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836626353\">\n<p id=\"fs-id1167836626355\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aaa3cd0434a018948f7b93c940c90fc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#55;&#44;&#49;&#49;&#44;&#49;&#57;&#44;&#51;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"102\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829747310\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836433289\">\n<p id=\"fs-id1167836433291\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3370b8b6b9d9fade8ec17fe2b86ccbe3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#51;&#125;&#94;&#123;&#110;&#125;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"90\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829745911\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836663089\">\n<p id=\"fs-id1167836663091\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1e7caa96cda6cddc77f46e05aab49f3f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#51;&#125;&#94;&#123;&#110;&#125;&#45;&#50;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"102\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833025530\">\n<p id=\"fs-id1167833025532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22e9f6a9768e6684bcd73f70474f2aae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#53;&#44;&#50;&#49;&#44;&#55;&#51;&#44;&#50;&#51;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833008151\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833008153\">\n<p id=\"fs-id1167833008155\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cb236d6db9f5458db7976f9c219390c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#45;&#51;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"102\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836518570\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836518572\">\n<p id=\"fs-id1167836518574\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0f7b1f39598de131f457b5c49b1e762f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"60\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829745278\">\n<p id=\"fs-id1167829745280\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17103d1ca69f9f4f43cdff3ebb61bec1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#57;&#125;&#44;&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#50;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836697736\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836697738\">\n<p id=\"fs-id1167836697740\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a1cb753e9525e271f1fea513736ba01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#51;&#125;&#94;&#123;&#110;&#125;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"60\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836728891\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836728893\">\n<p id=\"fs-id1167836728895\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71afb02613debd3378a4f98952e5e545_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#110;&#45;&#50;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"77\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829747157\">\n<p id=\"fs-id1167829747160\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3f887073953cd7c8f2fbde0aa44c2f16_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#125;&#123;&#56;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#57;&#125;&#123;&#49;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"88\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836610291\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836610293\">\n<p id=\"fs-id1167829788284\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26922cb6d07d901575079c5e3524c7b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#110;&#43;&#51;&#125;&#123;&#123;&#51;&#125;&#94;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"77\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836731549\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836731552\">\n<p id=\"fs-id1167836731554\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4714cfccd918f04e72cbab6234c7a8aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&middot;&#50;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836663177\">\n<p id=\"fs-id1167833018382\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d53078891beb9d9083102548f07a88d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#44;&#52;&#44;&#45;&#54;&#44;&#56;&#44;&#45;&#49;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"126\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836409447\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836409449\">\n<p id=\"fs-id1167836409451\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd129d854e9a2071c12d1cdf1d635af7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&middot;&#51;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836444704\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836444706\">\n<p id=\"fs-id1167836444708\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3c6e2475fdd795adf440aabd1a44d6f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#123;&#110;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836542558\">\n<p id=\"fs-id1167836542560\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-20ad02464b36a885bed08a2fc6d45520_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#45;&#52;&#44;&#57;&#44;&#45;&#49;&#54;&#44;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"119\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833056685\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833056687\">\n<p id=\"fs-id1167833056689\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-259f545ba2177cc378198e7da64b1eb2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#123;&#110;&#125;&#94;&#123;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833071792\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833071795\">\n<p id=\"fs-id1167836613274\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-987b666c31a3ed806019dba807aa065a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"96\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836630557\">\n<p id=\"fs-id1167836630559\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5e2d97a8bca624a320e82c3bfd29e266_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"122\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836688954\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836688957\">\n<p id=\"fs-id1167829719948\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-adc7a773691ee179f1d63734a6184c3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#125;&#123;&#50;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"96\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167833018343\"><strong data-effect=\"bold\">Find a Formula for the General Term (<em data-effect=\"italics\">n<\/em>th Term) of a Sequence<\/strong><\/p>\n<p id=\"fs-id1167836560597\">In the following exercises, find a general term for the sequence whose first five terms are shown.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836560600\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829599598\">\n<p id=\"fs-id1167829599600\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-955e78bd306cbcb036fe1322ec4a9275_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#44;&#49;&#54;&#44;&#50;&#52;&#44;&#51;&#50;&#44;&#52;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836530056\">\n<p id=\"fs-id1167836530058\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c1905bfb7dc479d0a11d2718a8e4ed86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#56;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"62\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836663456\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836663458\">\n<p id=\"fs-id1167836663460\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d007a17ad933f4007886ae061f9efa0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#44;&#49;&#52;&#44;&#50;&#49;&#44;&#50;&#56;&#44;&#51;&#53;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"116\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833369184\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833369187\">\n<p id=\"fs-id1167833369189\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59ae14e12b3ce3e8ae1772460f64841b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#44;&#55;&#44;&#56;&#44;&#57;&#44;&#49;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"89\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829746838\">\n<p id=\"fs-id1167829746840\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-70d620c059322f7f4dfeed1faf34763e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#110;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836433330\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836433332\">\n<p id=\"fs-id1167836433335\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-40f2cedfb70ca49550de7ecf03a5af6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#45;&#50;&#44;&#45;&#49;&#44;&#48;&#44;&#49;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"120\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836612792\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836612794\">\n<p id=\"fs-id1167836612796\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-351d544037dd6ca1af3532923b6b2544_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#101;&#125;&#94;&#123;&#51;&#125;&#44;&#123;&#101;&#125;&#94;&#123;&#52;&#125;&#44;&#123;&#101;&#125;&#94;&#123;&#53;&#125;&#44;&#123;&#101;&#125;&#94;&#123;&#54;&#125;&#44;&#123;&#101;&#125;&#94;&#123;&#55;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"115\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836728288\">\n<p id=\"fs-id1167836728290\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc294671e24db23ef6329c178ca61137_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#101;&#125;&#94;&#123;&#110;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"76\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829811395\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829811398\">\n<p id=\"fs-id1167829811400\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ef8a6dead44e7cdcdf7e9db52a26fa34_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#101;&#125;&#94;&#123;&#50;&#125;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#101;&#125;&#44;&#49;&#44;&#101;&#44;&#123;&#101;&#125;&#94;&#123;&#50;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"94\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742040\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836363311\">\n<p id=\"fs-id1167836363313\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-29f4cf1ae0fa639b946dd55b58bcf1c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#44;&#49;&#48;&#44;&#45;&#49;&#53;&#44;&#50;&#48;&#44;&#45;&#50;&#53;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836731605\">\n<p id=\"fs-id1167836731607\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e5d2c426af2f71624aa0759f6a2f35e7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#53;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836560420\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836560422\">\n<p id=\"fs-id1167836560424\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2b57f7146379e2484df481edd309f196_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#44;&#49;&#49;&#44;&#45;&#49;&#54;&#44;&#50;&#49;&#44;&#45;&#50;&#54;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836532531\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829752946\">\n<p id=\"fs-id1167829752948\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c96bdc836581e08c96969ced82b67b9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#56;&#44;&#45;&#50;&#55;&#44;&#54;&#52;&#44;&#45;&#49;&#50;&#53;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"156\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833019697\">\n<p id=\"fs-id1167833019699\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6b1fed7fb23bfb593efa55f3db638de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#123;&#110;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"106\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832945880\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832945882\">\n<p id=\"fs-id1167832945885\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a8d217f08fa53191edc5b955ca8394b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#45;&#53;&#44;&#49;&#48;&#44;&#45;&#49;&#55;&#44;&#50;&#54;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"134\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836477546\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836477548\">\n<p id=\"fs-id1167836477550\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a7974e75819c3abe56f02a5c4c1b5fb9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#44;&#52;&#44;&#45;&#54;&#44;&#56;&#44;&#45;&#49;&#48;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"130\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836559964\">\n<p id=\"fs-id1167836559966\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3cb3053e827985ce5ebb651a179d675d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#50;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829906696\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836293168\">\n<p id=\"fs-id1167836293170\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b93a55b4e2e1bca3cd67de2d6b10aaed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#45;&#51;&#44;&#53;&#44;&#45;&#55;&#44;&#57;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"107\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833380689\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833380691\">\n<p id=\"fs-id1167833380693\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-adea058cb701ccea093500a2abce0260_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#44;&#48;&#50;&#52;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"139\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836415616\">\n<p id=\"fs-id1167836635475\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f9d684a6f3f3e4ac296fab3f82038c3a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#52;&#125;&#94;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836386988\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836386990\">\n<p id=\"fs-id1167836386992\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dfd032fde1f040365720ea3a61bfd096_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#53;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"115\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829747828\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829747830\">\n<p id=\"fs-id1167829747832\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ffc28202b22beabba3e4cd68e9a18d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"157\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836728478\">\n<p id=\"fs-id1167836728480\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-baa248e3745f28e4fcb2ec2ebe17838d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"84\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829753358\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829753360\">\n<p id=\"fs-id1167829753362\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9b030cf70e8f7a8bf55609556312f637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#51;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#54;&#125;&#123;&#53;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"155\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836684137\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836684139\">\n<p id=\"fs-id1167833021620\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-be98cf6e77f8ba6378c15ebb89f3a665_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#50;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#52;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#56;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#49;&#54;&#125;&#44;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#51;&#50;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"171\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829711771\">\n<p id=\"fs-id1167829711773\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-efd25c98d033c320f9d42f0027451f70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#123;&#50;&#125;&#94;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"31\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829751436\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829751438\">\n<p id=\"fs-id1167829751440\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-24c7a844c7f644c994f1228a5d153fb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#50;&#55;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#54;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#50;&#53;&#125;&#44;&#92;&#116;&#101;&#120;&#116;&#123;&#8230;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"115\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167829720967\"><strong data-effect=\"bold\">Use Factorial Notation<\/strong><\/p>\n<p id=\"fs-id1167829720972\">In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167829720976\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829720978\">\n<p id=\"fs-id1167836398761\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0afc08931b9ab0efb9db542f0c805645_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#110;&#33;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829906646\">\n<p id=\"fs-id1167829906648\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-154f80dc95a7edd8d04f9c9696fd7f26_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#50;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"87\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836362732\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836362734\">\n<p id=\"fs-id1167836362736\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-487a2619b646c1f8c1e06506a700d72c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#110;&#33;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"57\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829745763\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829745765\">\n<p id=\"fs-id1167829745767\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d3f851b04492a7fae63632ff3af3742_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#51;&#110;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829753610\">\n<p id=\"fs-id1167829753612\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-68cfb9c066493606853c9dd16d081857_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#44;&#54;&#44;&#49;&#56;&#44;&#55;&#50;&#44;&#51;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"112\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833018403\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833018406\">\n<p id=\"fs-id1167833018408\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dea8efb85925f8d5a26293383a32cab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#50;&#110;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"66\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833025452\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833025454\">\n<p id=\"fs-id1167833025456\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d3c84a83f9b1833ba59f1e3bab908191_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829906197\">\n<p id=\"fs-id1167829906199\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95f3c2d89193a61fed8c1d5ee4638e12_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#50;&#52;&#44;&#55;&#50;&#48;&#44;&#52;&#48;&#51;&#50;&#48;&#44;&#51;&#54;&#50;&#56;&#56;&#48;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"192\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829742666\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829742668\">\n<p id=\"fs-id1167829742670\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8becd7264e4aa1bf566e6c8c4e1d0ddd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"79\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836289638\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836289640\">\n<p id=\"fs-id1167836289642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f79e8c9211fdd09b00deb73aaa705fb4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"29\" width=\"85\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833379764\">\n<p id=\"fs-id1167829904857\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f82adbbca102eef945f766ada6580fdc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"81\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836511165\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829850885\">\n<p id=\"fs-id1167829850887\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-95b16d11729aa98d3fee499c59fe1e75_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#33;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"85\" style=\"vertical-align: -9px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829627454\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829627456\">\n<p id=\"fs-id1167836699088\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-60900762630c976a0840358831070a6f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#33;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"60\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836621263\">\n<p id=\"fs-id1167836621266\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a868421d116ff783b2ecc684b3bd87af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"88\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836398785\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836398787\">\n<p id=\"fs-id1167836398789\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f29dacd62a504549e8d957a75414250a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#110;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#125;&#125;&#125;&#123;&#110;&#33;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"60\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829681252\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829681254\">\n<p id=\"fs-id1167829681257\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-626a830c2499c15b3f7e0908f2dd6eb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#50;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"27\" width=\"85\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836607396\">\n<p id=\"fs-id1167829936533\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-230b33a3eaae0d319fb5f908c9e2ea51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#56;&#125;&#123;&#51;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#53;&#125;&#123;&#50;&#125;&#44;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#52;&#52;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"103\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836662794\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836662797\">\n<p id=\"fs-id1167829754708\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-05e1f838f7b99a25d1d6f966812e14fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#33;&#125;&#123;&#50;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"85\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836433402\"><strong data-effect=\"bold\">Find the Partial Sum<\/strong><\/p>\n<p id=\"fs-id1167836433408\">In the following exercises, expand the partial sum and find its value.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836433411\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836433413\">\n<p id=\"fs-id1167836433415\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b97f7355cca80243380edef4f63aa8be_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#123;&#105;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"58\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829752801\">\n<p id=\"fs-id1167829752803\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a783bbf08b28c6520837db39f453f56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#52;&#43;&#57;&#43;&#49;&#54;&#43;&#50;&#53;&#61;&#53;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"188\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829808424\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829808427\">\n<p id=\"fs-id1167829808429\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-67081cb0d8b22a18ef40074460bfb5ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#123;&#105;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"58\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829741717\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829741719\">\n<p id=\"fs-id1167829741722\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9cacb6e04e4e312b71010278d6c9e28c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#54;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#105;&#43;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"103\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836635456\">\n<p id=\"fs-id1167836635459\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0426c431804a4a1ef64218183c72e6aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#43;&#55;&#43;&#57;&#43;&#49;&#49;&#43;&#49;&#51;&#43;&#49;&#53;&#61;&#54;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"230\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836528185\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836528187\">\n<p id=\"fs-id1167836528189\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea3d7e656617087282bc4c3322cf217a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#54;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#105;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"103\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836548302\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836548304\">\n<p id=\"fs-id1167836732280\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-36b730a951e30e087dc40b23599b5854_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#52;&#125;&#123;&#50;&#125;&#94;&#123;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833021801\">\n<p id=\"fs-id1167833021803\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fc867396a50a630529eb9a02f0ac8daf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#52;&#43;&#56;&#43;&#49;&#54;&#61;&#51;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"151\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836510199\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836510201\">\n<p id=\"fs-id1167836510204\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a1cbc8166b7c16e81a94ab77fd7b877_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#52;&#125;&#123;&#51;&#125;&#94;&#123;&#105;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836386938\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833381315\">\n<p id=\"fs-id1167833381317\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6638d13f07a771887e5353261975c9ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#48;&#125;&#94;&#123;&#51;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#107;&#33;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"62\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833202871\">\n<p id=\"fs-id1167833202873\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f62a8b5cc4141896e3e5db892ed93e4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#50;&#125;&#123;&#51;&#125;&#61;&#49;&#48;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"196\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836444835\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836433208\">\n<p id=\"fs-id1167836433210\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ab28d3bc84b81a918048c3e08d02a8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#48;&#125;&#94;&#123;&#52;&#125;&#45;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#107;&#33;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"75\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829905677\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829905679\">\n<p id=\"fs-id1167829905681\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5083ba8e691fefe0fd096bbd19da8d90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#107;&#92;&#108;&#101;&#102;&#116;&#40;&#107;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832994357\">\n<p id=\"fs-id1167832994359\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ae79295712315b50901fb4269d8411e8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#54;&#43;&#49;&#50;&#43;&#50;&#48;&#43;&#51;&#48;&#61;&#55;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"199\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833381075\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833381078\">\n<p id=\"fs-id1167833381080\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-09f353a7e33026af8dad753bc1ef3401_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#107;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#107;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"123\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836728850\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836728852\">\n<p id=\"fs-id1167836728854\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53617b5f0e8fb912705a22d9df2a4aca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#110;&#43;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"76\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836363431\">\n<p id=\"fs-id1167836363433\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-057172f1bbec78d1e591a3d3b8712861_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#52;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#53;&#125;&#123;&#54;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#55;&#49;&#125;&#123;&#50;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"178\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829755864\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829755867\">\n<p id=\"fs-id1167829755869\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0d098aa8bce126ff242cfc932da4126a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#52;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#110;&#43;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"76\" style=\"vertical-align: -8px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167836732252\"><strong data-effect=\"bold\">Use Summation Notation to write a Sum<\/strong><\/p>\n<p id=\"fs-id1167836732257\">In the following exercises, write each sum using summation notation.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167836732260\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836732262\">\n<p id=\"fs-id1167836732265\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6770692472410e8652578ff1f682111_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#57;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#49;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#52;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"164\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829627761\">\n<p id=\"fs-id1167829627763\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5102f60b1960e2ca8883742102f964f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#51;&#125;&#94;&#123;&#110;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"66\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833379856\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833379858\">\n<p id=\"fs-id1167833379860\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-590a717dea7ff953744e326c1707d2dd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#54;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#54;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"132\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829720089\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829720091\">\n<p id=\"fs-id1167829720093\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6ffd5f3f6eec18095940299420dbf2e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#56;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#55;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"164\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836450380\">\n<p id=\"fs-id1167836450383\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d15676da1fdcdef7772f714e9a3a373c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#123;&#110;&#125;&#94;&#123;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"66\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836331126\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836331128\">\n<p id=\"fs-id1167829693463\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f56e915406eddb0bd1366fe2ee67205_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#49;&#50;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#54;&#50;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"139\" style=\"vertical-align: -7px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833380085\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833380087\">\n<p id=\"fs-id1167833380089\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-edb185a693d517059fffbd10d969c494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#43;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#51;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"135\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829627310\">\n<p id=\"fs-id1167829627312\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4b7eeebbf9d102212cf844da22c4bcc9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#125;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"59\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829811078\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829811080\">\n<p id=\"fs-id1167829811082\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8eb749dbb4db01400ca10d3989a17470_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#50;&#125;&#43;&#49;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#52;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#125;&#123;&#53;&#125;&#43;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"167\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836728731\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836728733\">\n<p id=\"fs-id1167836728735\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdb81f7d2750fd49aac7a2aaf33dbb1c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#45;&#54;&#43;&#57;&#45;&#49;&#50;&#43;&#49;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"148\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836729430\">\n<p id=\"fs-id1167836729432\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b7f1a92b57a44a2045a6183749cd2421_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#53;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#43;&#49;&#125;&#51;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"132\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836626890\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836626892\">\n<p id=\"fs-id1167836626894\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f274f5b0acc0274637da464b8ada5857_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#43;&#49;&#48;&#45;&#49;&#53;&#43;&#50;&#48;&#45;&#50;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"179\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829905564\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829905566\">\n<p id=\"fs-id1167829905569\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f384c9c292283e5434903143649bfff2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#43;&#52;&#45;&#54;&#43;&#56;&#45;&#49;&#48;&#43;&#92;&#100;&#111;&#116;&#115;&#32;&#43;&#50;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"235\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167829905835\">\n<p id=\"fs-id1167829905837\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f78d816b182446b25ce4ea7c84659ab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#49;&#48;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#50;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"114\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836728528\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836728530\">\n<p id=\"fs-id1167836728532\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-db3724486fd2d260547bb75ac0f7b7b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#45;&#51;&#43;&#53;&#45;&#55;&#43;&#57;&#43;&#92;&#100;&#111;&#116;&#115;&#32;&#43;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"211\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167829790474\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167829790476\">\n<p id=\"fs-id1167829790478\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-af1e4605f9996856582cc9fee57fa60a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#43;&#49;&#54;&#43;&#49;&#56;&#43;&#50;&#48;&#43;&#50;&#50;&#43;&#50;&#52;&#43;&#50;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"253\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836732228\">\n<p id=\"fs-id1167836732231\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3a04256a769d103dd8424343b01001d2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#110;&#61;&#49;&#125;&#94;&#123;&#55;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#110;&#43;&#49;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"120\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833379616\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833379618\">\n<p id=\"fs-id1167833379620\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-93da585303a388b0053046a38d5e1dae_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#43;&#49;&#49;&#43;&#49;&#51;&#43;&#49;&#53;&#43;&#49;&#55;&#43;&#49;&#57;&#43;&#50;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"245\" style=\"vertical-align: -2px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167836730456\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167836730097\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836730099\">\n<p id=\"fs-id1167836730101\">In your own words, explain how to write the terms of a sequence when you know the formula. Show an example to illustrate your explanation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167836730107\">\n<p id=\"fs-id1167836730109\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167836730114\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167836730116\">\n<p id=\"fs-id1167836730118\">Which terms of the sequence are negative when the <em data-effect=\"italics\">n<\/em><sup>th<\/sup> term of the sequence is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a79b4e5c56d923972899d5822b5db10b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#94;&#123;&#110;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#43;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"154\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833381108\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833381110\">\n<p id=\"fs-id1167833381112\">In your own words, explain what is meant by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da0ef996f36e1b32a0f26f6e896e1771_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#33;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\" \/> Show some examples to illustrate your explanation.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167833381124\">\n<p id=\"fs-id1167833381126\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167833381132\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167833381134\">\n<p id=\"fs-id1167829904879\">Explain what each part of the notation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c85708437423c617742b81845494e631_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#123;&#49;&#50;&#125;&#50;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"66\" style=\"vertical-align: -6px;\" \/> means.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167829904916\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167829809598\"><span class=\"token\">\u24d0<\/span> After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167829809613\" data-alt=\"This figure shows a table with four columns and six rows. The first row is the header row and labels each column, &#x201c;I can&#x201d;, &#x201c;Confidently&#x201d;, &#x201c;With some help&#x201d;, and &#x201c;No I don&#x2019;t get it!&#x201d;. The first row in the second column reads, &#x201c;Write the first few terms of a sequence&#x201d;, the third row, first column reads, &#x201c;Find a Formula for the nth Term of a Sequence&#x201d;, the fourth row first column reads &#x201c;Use Factorial Notation, the fifth row, first column reads, Find the partial sum&#x201d;, and the last row, first column reads, &#x201c;Use Summation Notation to write a Sum&#x201d;. The remaining three columns and rows are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_01_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a table with four columns and six rows. The first row is the header row and labels each column, &#x201c;I can&#x201d;, &#x201c;Confidently&#x201d;, &#x201c;With some help&#x201d;, and &#x201c;No I don&#x2019;t get it!&#x201d;. The first row in the second column reads, &#x201c;Write the first few terms of a sequence&#x201d;, the third row, first column reads, &#x201c;Find a Formula for the nth Term of a Sequence&#x201d;, the fourth row first column reads &#x201c;Use Factorial Notation, the fifth row, first column reads, Find the partial sum&#x201d;, and the last row, first column reads, &#x201c;Use Summation Notation to write a Sum&#x201d;. The remaining three columns and rows are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167829809621\"><span class=\"token\">\u24d1<\/span> If most of your checks were:<\/p>\n<p id=\"fs-id1167829809628\">\u2026confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.<\/p>\n<p id=\"fs-id1167836513507\">\u2026with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?<\/p>\n<p id=\"fs-id1167836513518\">\u2026no &#8211; I don\u2019t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h3 data-type=\"glossary-title\">Glossary<\/h3>\n<dl id=\"fs-id1167836513531\">\n<dt>finite sequence<\/dt>\n<dd id=\"fs-id1167836513536\">A sequence with a domain that is limited to a finite number of counting numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836513540\">\n<dt>general term of a sequence<\/dt>\n<dd id=\"fs-id1167829809632\">The general term of the sequence is the formula for writing the <em data-effect=\"italics\">n<\/em>th term of the sequence. The <em data-effect=\"italics\">n<\/em>th term of the sequence, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-de303bf9b0d7d27838a9ca3ac356e6b3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"23\" style=\"vertical-align: -4px;\" \/> is the term in the <em data-effect=\"italics\">n<\/em>th position where <em data-effect=\"italics\">n<\/em> is a value in the domain.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167829742733\">\n<dt>infinite sequence<\/dt>\n<dd id=\"fs-id1167829742738\">A sequence whose domain is all counting numbers and there is an infinite number of counting numbers.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836731128\">\n<dt>partial sum<\/dt>\n<dd id=\"fs-id1167836731133\">When we add a finite number of terms of a sequence, we call the sum a partial sum.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167836731137\">\n<dt>sequence<\/dt>\n<dd id=\"fs-id1167836731141\">A sequence is a function whose domain is the counting numbers.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15689","chapter","type-chapter","status-publish","hentry"],"part":15662,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15689","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/users\/103"}],"version-history":[{"count":0,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15689\/revisions"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/parts\/15662"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15689\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=15689"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=15689"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=15689"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=15689"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}