{"id":15719,"date":"2019-09-05T12:08:23","date_gmt":"2019-09-05T16:08:23","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/arithmetic-sequences\/"},"modified":"2019-09-05T12:08:23","modified_gmt":"2019-09-05T16:08:23","slug":"arithmetic-sequences","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/chapter\/arithmetic-sequences\/","title":{"raw":"Arithmetic Sequences","rendered":"Arithmetic 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>Determine if a sequence is arithmetic<\/li><li>Find the general term (\\(n\\)th term) of an arithmetic sequence<\/li><li>Find the sum of the first \\(n\\) terms of an arithmetic sequence<\/li><\/ul><\/div><div data-type=\"note\" id=\"fs-id1167835327387\" class=\"be-prepared\"><p id=\"fs-id1167835375480\">Before you get started, take this readiness quiz.<\/p><ol id=\"fs-id1167835421842\" type=\"1\"><li>Evaluate \\(4n-1\\) 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>Solve the system of equations: \\(\\left\\{\\begin{array}{l}x+y=7\\\\ 3x+4y=23\\end{array}.\\)<span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e#fs-id1167834121139\" class=\"autogenerated-content\">(Figure)<\/a>.<\/li><li>If \\(f\\left(n\\right)=\\frac{n}{2}\\left(3n+5\\right),\\) find \\(f\\left(1\\right)+f\\left(20\\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-id1167835235945\"><h3 data-type=\"title\">Determine if a Sequence is Arithmetic<\/h3><p id=\"fs-id1167835421169\">The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.<\/p><p id=\"fs-id1167835308916\">An <span data-type=\"term\">arithmetic sequence<\/span> is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence, \\({a}_{n}-{a}_{n-1},\\) is <em data-effect=\"italics\">d<\/em>, the <span data-type=\"term\">common difference<\/span>, for <em data-effect=\"italics\">n<\/em> greater than or equal to two.<\/p><div data-type=\"note\" id=\"fs-id1167830705774\"><div data-type=\"title\">Arithmetic Sequence<\/div><p id=\"fs-id1167834449214\">An <strong data-effect=\"bold\">arithmetic sequence<\/strong> is a sequence where the difference between consecutive terms is always the same.<\/p><p id=\"fs-id1167835329986\">The difference between consecutive terms, \\({a}_{n}-{a}_{n-1},\\) is <em data-effect=\"italics\">d<\/em>, the <strong data-effect=\"bold\">common difference<\/strong>, for <em data-effect=\"italics\">n<\/em> greater than or equal to two.<\/p><\/div><span data-type=\"media\" id=\"fs-id1167835187194\" data-alt=\"This figure has two rows and three columns. The first row reads &#x201c;7&#x201d;, &#x201c;10&#x201d;,&#x201d;13&#x201d;, &#x201c;16&#x201d;, &#x201c;19&#x201d;, &#x201c;22&#x201d;, and an ellipsis, &#x201c;10 minus 7, divided by 3&#x201d;, &#x201c;13 minus 10, divided by 3&#x201d;, &#x201c;16 minus 13, divided by 3&#x201d;, nth term equals nth term minus 1 divided by d&#x201d;\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has two rows and three columns. The first row reads &#x201c;7&#x201d;, &#x201c;10&#x201d;,&#x201d;13&#x201d;, &#x201c;16&#x201d;, &#x201c;19&#x201d;, &#x201c;22&#x201d;, and an ellipsis, &#x201c;10 minus 7, divided by 3&#x201d;, &#x201c;13 minus 10, divided by 3&#x201d;, &#x201c;16 minus 13, divided by 3&#x201d;, nth term equals nth term minus 1 divided by d&#x201d;\" \/><\/span><p id=\"fs-id1167835304119\">In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic.<\/p><div data-type=\"example\" id=\"fs-id1167835420956\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167832041539\"><div data-type=\"problem\" id=\"fs-id1167835326267\"><p id=\"fs-id1167831959293\">Determine if each sequence is arithmetic. If so, indicate the common difference.<\/p><p id=\"fs-id1167832076355\"><span class=\"token\">\u24d0<\/span>\\(5,9,13,17,21,25\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><p id=\"fs-id1167834214180\"><span class=\"token\">\u24d1<\/span>\\(4,9,12,17,20,25\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><p id=\"fs-id1167835420225\"><span class=\"token\">\u24d2<\/span>\\(10,3,-4,-11,-18,-25\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835354025\"><p id=\"fs-id1167835339364\">To determine if the sequence is arithmetic, we find the difference of the consecutive terms shown.<\/p><p id=\"fs-id1167835368778\"><span class=\"token\">\u24d0<\/span><span data-type=\"newline\"><br \/><\/span>\\(\\begin{array}{cccccc}\\begin{array}{}\\\\ \\\\ \\\\ \\\\ \\text{Find the difference of}\\hfill \\\\ \\text{the consecutive terms.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\begin{array}{cccccccccc}\\hfill 5,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}9,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}13,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}17\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}21,\\hfill &amp; \\hfill \\phantom{\\rule{0.6em}{0ex}}25,\\dots \\hfill \\\\ \\\\ \\\\ \\hfill 9-5\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}13-9\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}17-13\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}21-17\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}25-21\\hfill &amp; &amp; \\\\ \\hfill 4\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}4\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}4\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}4\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}4\\hfill &amp; &amp; \\end{array}\\hfill \\\\ \\\\ \\\\ &amp; &amp; &amp; &amp; &amp; \\text{The sequence is arithmetic. The common difference is}\\phantom{\\rule{0.2em}{0ex}}d=4\\text{.}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167834094786\"><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span>\\(\\begin{array}{cccccc}\\begin{array}{}\\\\ \\\\ \\\\ \\\\ \\text{Find the difference of}\\hfill \\\\ \\text{the consecutive terms.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\begin{array}{cccccccccc}\\hfill 4,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}9,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}12,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}17\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}20,\\hfill &amp; \\hfill \\phantom{\\rule{0.6em}{0ex}}25,\\dots \\hfill \\\\ \\\\ \\\\ \\hfill 9-4\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}12-9\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}17-12\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}20-17\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}25-20\\hfill &amp; &amp; \\\\ \\hfill 2\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}3\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}5\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}3\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}5\\hfill &amp; &amp; \\end{array}\\hfill \\\\ \\\\ \\\\ &amp; &amp; &amp; &amp; &amp; \\begin{array}{c}\\text{The sequence is not arithmetic as all the differences between}\\hfill \\\\ \\text{the consecutive terms are not the same.}\\hfill \\\\ \\text{There is no common difference.}\\hfill \\end{array}\\hfill \\end{array}\\)<\/p><p id=\"fs-id1167832055415\"><span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/><\/span>\\(\\begin{array}{cccccc}\\begin{array}{}\\\\ \\\\ \\\\ \\\\ \\text{Find the difference of}\\hfill \\\\ \\text{the consecutive terms.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; &amp; &amp; \\hfill \\begin{array}{ccccccccccc}\\hfill 10,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}3,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-4,\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-11\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-18,\\hfill &amp; \\hfill \\phantom{\\rule{0.6em}{0ex}}-25,\\dots \\hfill &amp; \\\\ \\\\ \\\\ \\hfill 3-10\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-4-3\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-11-\\left(-4\\right)\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-18-\\left(-11\\right)\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-25-\\left(-18\\right)\\hfill &amp; &amp; \\\\ \\hfill -7\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-7\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-7\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-7\\hfill &amp; &amp; \\hfill \\phantom{\\rule{1em}{0ex}}-7\\hfill &amp; &amp; \\end{array}\\hfill \\\\ \\\\ \\\\ &amp; &amp; &amp; &amp; &amp; \\text{The sequence is arithmetic. The common difference is}\\phantom{\\rule{0.2em}{0ex}}d=-7\\text{.}\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834222032\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834556290\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834063445\"><p id=\"fs-id1167835332114\">Determine if each sequence is arithmetic. If so, indicate the common difference.<\/p><p id=\"fs-id1167835328888\"><span class=\"token\">\u24d0<\/span>\\(9,20,31,42,53,64\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<span class=\"token\">\u24d1<\/span>\\(12,6,0,-6,-12,-18\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<span class=\"token\">\u24d2<\/span>\\(7,1,10,4,13,7\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835301773\"><p id=\"fs-id1167832056814\"><span class=\"token\">\u24d0<\/span> The sequence is arithmetic with common difference \\(d=11\\). <span class=\"token\">\u24d1<\/span> The sequence is arithmetic with common difference \\(d=-6\\).<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span> The sequence is not arithmetic as all the differences between the consecutive terms are not the same.<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834138030\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835367810\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835305719\"><p id=\"fs-id1167834403016\">Determine if each sequence is arithmetic. If so, indicate the common difference.<\/p><p id=\"fs-id1167832067000\"><span class=\"token\">\u24d0<\/span>\\(-4,4,2,10,8,16\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<span class=\"token\">\u24d1<\/span>\\(-3,-1,1,3,5,7\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<span class=\"token\">\u24d2<\/span>\\(7,2,-3,-8,-13,-18\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835198777\"><p id=\"fs-id1167835337130\"><span class=\"token\">\u24d0<\/span> The sequence is not arithmetic as all the differences between the consecutive terms are not the same. <span class=\"token\">\u24d1<\/span> The sequence is arithmetic with common difference \\(d=2.\\)<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span> The sequence is arithmetic with common difference \\(d=-5.\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835341287\">If we know the first term, \\({a}_{1},\\) and the common difference, <em data-effect=\"italics\">d<\/em>, we can list a finite number of terms of the sequence.<\/p><div data-type=\"example\" id=\"fs-id1167831944141\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835416627\"><div data-type=\"problem\" id=\"fs-id1167835254902\"><p id=\"fs-id1167835336301\">Write the first five terms of the sequence where the first term is 5 and the common difference is \\(d=-6.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835422706\"><p id=\"fs-id1167834064148\">We start with the first term and add the common difference. Then we add the common difference to that result to get the next term, and so on.<\/p><div data-type=\"equation\" id=\"fs-id1167832056274\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{ccccccccccccccc}\\hfill {a}_{1}\\hfill &amp; &amp; &amp; \\hfill {a}_{2}\\hfill &amp; &amp; &amp; \\hfill {a}_{3}\\hfill &amp; &amp; &amp; \\hfill {a}_{4}\\hfill &amp; &amp; &amp; \\hfill {a}_{5}\\hfill &amp; &amp; \\\\ \\hfill 5\\hfill &amp; &amp; &amp; \\hfill 5+\\left(-6\\right)\\hfill &amp; &amp; &amp; \\hfill -1+\\left(-6\\right)\\hfill &amp; &amp; &amp; \\hfill -7+\\left(-6\\right)\\hfill &amp; &amp; &amp; \\hfill -13+\\left(-6\\right)\\hfill \\\\ &amp; &amp; &amp; \\hfill -1\\hfill &amp; &amp; &amp; \\hfill -7\\hfill &amp; &amp; &amp; \\hfill -13\\hfill &amp; &amp; &amp; \\hfill -19\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167831010120\">The sequence is \\(5,-1,-7,-13,-19\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834194151\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167832052324\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834433837\"><p id=\"fs-id1167835337752\">Write the first five terms of the sequence where the first term is 7 and the common difference is \\(d=-4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835261566\"><p id=\"fs-id1167831909186\">\\(7,3,-1,-5,-9\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835329130\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834189547\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826978682\"><p id=\"fs-id1167835181090\">Write the first five terms of the sequence where the first term is 11 and the common difference is \\(d=-8.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835162835\"><p id=\"fs-id1167835509099\">\\(11,3,-5,-13,-21\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834157026\"><h3 data-type=\"title\">Find the General Term (<em data-effect=\"italics\">n<\/em>th Term) of an Arithmetic Sequence<\/h3><p id=\"fs-id1167835191878\">Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.<\/p><p id=\"fs-id1167834063240\">Let\u2019s write the first few terms of a sequence where the first term is \\({a}_{1}\\) and the common difference is <em data-effect=\"italics\">d<\/em>. We will then look for a pattern.<\/p><p id=\"fs-id1167834423838\">As we look for a pattern we see that each term starts with \\({a}_{1}\\).<\/p><span data-type=\"media\" id=\"fs-id1167835340070\" data-alt=\"This figures shows an image of a sequence.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figures shows an image of a sequence.\" \/><\/span><p id=\"fs-id1167835346003\">The first term adds 0<em data-effect=\"italics\">d<\/em> to the \\({a}_{1}\\), the second term adds 1<em data-effect=\"italics\">d<\/em>, the third term adds 2<em data-effect=\"italics\">d<\/em>, the fourth term adds 3<em data-effect=\"italics\">d<\/em>, and the fifth term adds 4<em data-effect=\"italics\">d<\/em>. The number of <em data-effect=\"italics\">d<\/em>s that were added to \\({a}_{1}\\) is one less than the number of the term. This leads us to the following<\/p><div data-type=\"equation\" id=\"fs-id1167835280772\" class=\"unnumbered\" data-label=\"\">\\({a}_{n}={a}_{1}+\\left(n-1\\right)d\\)<\/div><div data-type=\"note\" id=\"fs-id1167835327327\"><div data-type=\"title\">General Term (<em data-effect=\"italics\">n<\/em>th term) of an Arithmetic Sequence<\/div><p id=\"fs-id1167835194069\">The general term of an arithmetic sequence with first term \\({a}_{1}\\) and the common difference <em data-effect=\"italics\">d<\/em> is<\/p><div data-type=\"equation\" id=\"fs-id1167835350396\" class=\"unnumbered\" data-label=\"\">\\({a}_{n}={a}_{1}+\\left(n-1\\right)d\\)<\/div><\/div><p id=\"fs-id1167835369032\">We will use this formula in the next example to find the 15<sup>th<\/sup> term of a sequence.<\/p><div data-type=\"example\" id=\"fs-id1167835309566\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835355850\"><div data-type=\"problem\" id=\"fs-id1167835173305\"><p id=\"fs-id1167834191748\">Find the fifteenth term of a sequence where the first term is 3 and the common difference is 6.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835237077\"><p id=\"fs-id1167835369908\">\\(\\begin{array}{cccc}\\text{To find the fifteenth term,}\\phantom{\\rule{0.2em}{0ex}}{a}_{15},\\phantom{\\rule{0.2em}{0ex}}\\text{use the}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{a}_{n}={a}_{1}+\\left(n-1\\right)d\\hfill \\\\ \\text{formula with}\\phantom{\\rule{0.2em}{0ex}}{a}_{1}=3\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d=6\\text{.}\\hfill &amp; &amp; &amp; \\\\ \\\\ \\\\ \\text{Substitute in the values.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{a}_{15}=3+\\left(15-1\\right)6\\hfill \\\\ \\\\ \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{a}_{15}=3+\\left(14\\right)6\\hfill \\\\ \\\\ \\\\ &amp; &amp; &amp; \\hfill \\phantom{\\rule{3em}{0ex}}{a}_{15}=87\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835375533\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835513406\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831239689\"><p id=\"fs-id1167835350380\">Find the twenty-seventh term of a sequence where the first term is 7 and the common difference is 9.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835418188\"><p id=\"fs-id1167835361810\">241<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835303825\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835347370\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834120366\"><p id=\"fs-id1167835307170\">Find the eighteenth term of a sequence where the first term is 13 and the common difference is \\(-7\\).<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835362393\"><p id=\"fs-id1167835332053\">\\(-106\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1167835243861\">Sometimes we do not know the first term and we must use other given information to find it before we find the requested term.<\/p><div data-type=\"example\" id=\"fs-id1167831883365\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167834185861\"><div data-type=\"problem\" id=\"fs-id1167835258413\"><p id=\"fs-id1167835214784\">Find the twelfth term of a sequence where the seventh term is 10 and the common difference is \\(-2\\). Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835418145\"><p id=\"fs-id1167832056903\">\\(\\begin{array}{cccc}\\text{To first find the first term,}\\phantom{\\rule{0.2em}{0ex}}{a}_{1},\\phantom{\\rule{0.2em}{0ex}}\\text{use the}\\hfill &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{n}={a}_{1}+\\left(n-1\\right)d\\hfill \\\\ \\text{formula with}\\phantom{\\rule{0.2em}{0ex}}{a}_{7}=10,n=7,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d=-2.\\hfill &amp; &amp; &amp; \\\\ \\text{Substitute in the values.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}10={a}_{1}+\\left(7-1\\right)\\left(-2\\right)\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}10={a}_{1}+\\left(6\\right)\\left(-2\\right)\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}10={a}_{1}-12\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{1}=22\\hfill \\\\ \\\\ \\\\ \\text{Find the twelfth term,}\\phantom{\\rule{0.2em}{0ex}}{a}_{12},\\phantom{\\rule{0.2em}{0ex}}\\text{using the}\\hfill &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{n}={a}_{1}+\\left(n-1\\right)d\\hfill \\\\ \\text{formula with}\\phantom{\\rule{0.2em}{0ex}}{a}_{1}=22,n=12,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}d=-2\\text{.}\\hfill &amp; &amp; &amp; \\\\ \\text{Substitute in the values.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{12}=22+\\left(12-1\\right)\\left(-2\\right)\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{12}=22+\\left(11\\right)\\left(-2\\right)\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{12}=0\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}\\text{The twelfth term of the sequence is 0,}\\phantom{\\rule{0.2em}{0ex}}{a}_{12}=0.\\hfill \\\\ \\\\ \\\\ \\text{To find the general term, substitute}\\hfill &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{n}={a}_{1}+\\left(n-1\\right)d\\hfill \\\\ \\text{the values into the formula.}\\hfill &amp; &amp; &amp; \\\\ &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{n}=22+\\left(n-1\\right)\\left(-2\\right)\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}{a}_{n}=22-2n+2\\hfill \\\\ &amp; &amp; &amp; \\phantom{\\rule{3em}{0ex}}\\text{The general term is}\\phantom{\\rule{0.2em}{0ex}}{a}_{n}=-2n+24.\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835418132\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835342802\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835307172\"><p id=\"fs-id1167835358419\">Find the eleventh term of a sequence where the ninth term is 8 and the common difference is \\(-3.\\) Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835353069\"><p id=\"fs-id1167835366527\">\\({a}_{11}=2.\\) The general term is \\({a}_{n}=-3n+35.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834194524\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835346520\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834535686\"><p id=\"fs-id1167835173222\">Find the nineteenth term of a sequence where the fifth term is 1 and the common difference is \\(-4.\\) Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835352738\"><p id=\"fs-id1167834196928\">\\({a}_{19}=-55.\\) The general term is \\({a}_{n}=-4n+21.\\)<\/p><\/div><\/div><\/div><p id=\"fs-id1165926635835\">Sometimes the information given leads us to two equations in two unknowns. We then use our methods for solving systems of equations to find the values needed.<\/p><div data-type=\"example\" id=\"fs-id1167835304642\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835282964\"><div data-type=\"problem\" id=\"fs-id1167832060198\"><p id=\"fs-id1167826967445\">Find the first term and common difference of a sequence where the fifth term is 19 and the eleventh term is 37. Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835240133\"><p id=\"fs-id1167834141929\">Since we know two terms, we can make a system of equations using the formula for the general term.<\/p><table id=\"fs-id1167835348864\" summary=\"The formula nth term equals a 1 plus n minus 1 times d. We know the value of a 5 and a 11 because a 5 equals a 1 plus 5 minus 1 times d, and a 11 equals a 1 plus 11 minus 1 times d. We know that 19 equals a sub 1 plus 5 minus 1 times d, and 37 equals a sub 1 plus 11 minus 1 times, so we then substitute in the values a sub 5 equals 19 and a sub 11 equals 37. Simplify to get 19 equals a sub 1 plus 4 d, and 37 equals a sub 1 plus 10 d. Next we prepare to eliminate the a sub 1 term by multiplying the top equation by negative 1 to get 18 equals 6 d, and 3 equals d. Next, substitute d equals 3 back into the first equation to get 19 equals a sub 1 plus 4 times g 3, then solve for a sub 1 to get 19 equals a sub 1 plus 12, then 7 equals a. Use the formula with a sub 1 equals 7 and d equals 3. We then substitute in the values to get a sub n equals 7 plus the sum of n minus 1 times 3. Simplify to get a sub n equals 7 plus 3 n minus 3, then a sub n equals 3 n plus 4. The first term is a sub 1 equals 7 and the common difference is d equals 3. The general term of the sequence is a sub n equals 3 n plus 4.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345460\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">We know the value of \\({a}_{5}\\) and \\({a}_{11},\\) so we will use \\(n=5\\) and \\(n=11.\\)<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835236275\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Substitute in the values, \\({a}_{5}=19\\) and \\({a}_{11}=37.\\)<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509141\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835349126\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Prepare to eliminate the \\({a}_{1}\\) term by multiplying the top equation by \\(-1.\\)<span data-type=\"newline\"><br \/><\/span>Add the equations.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835321742\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><span data-type=\"newline\"><br \/><\/span>Substituting \\(d=3\\) back into the first equation.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"newline\"><br \/><\/span><span data-type=\"media\" id=\"fs-id1167835361932\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Solve for \\({a}_{1}.\\)<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834053512\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\"><span data-type=\"newline\"><br \/><\/span>Use the formula with \\({a}_{1}=7\\) and \\(d=3.\\)<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"newline\"><br \/><\/span><span data-type=\"media\" id=\"fs-id1167830704126\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Substitute in the values.<\/td><td><\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834186195\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Simplify.<\/td><td>\\(\\phantom{\\rule{2.5em}{0ex}}\\)<\/td><td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834279285\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td><\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\">The first term is \\({a}_{1}=7.\\)<span data-type=\"newline\"><br \/><\/span>The common difference is \\(d=3.\\)<\/td><\/tr><tr><td><\/td><td colspan=\"2\" data-valign=\"top\" data-align=\"center\"><p id=\"fs-id1167835343210\">The general term of the sequence is \\({a}_{n}=3n+4.\\)<\/p><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835257393\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835311257\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835351997\"><p id=\"fs-id1167831920536\">Find the first term and common difference of a sequence where the fourth term is 17 and the thirteenth term is 53. Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835352755\"><p id=\"fs-id1167835352757\">\\({a}_{1}=5,\\)\\(d=4.\\) The general term is \\({a}_{n}=4n+1.\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835376657\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835306222\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835367833\"><p id=\"fs-id1167835350706\">Find the first term and common difference of a sequence where the third term is 2 and the twelfth term is \\(-25.\\) Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835226713\"><p id=\"fs-id1167835226716\">\\({a}_{1}=8,\\)\\(d=-3.\\) The general term is \\({a}_{n}=-3n+11.\\)<\/p><\/div><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835319496\"><h3 data-type=\"title\">Find the Sum of the First <em data-effect=\"italics\">n<\/em> Terms of an Arithmetic Sequence<\/h3><p id=\"fs-id1167835353364\">As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum, \\({S}_{n},\\) of the first \\(n\\) terms of any arithmetic sequence is written as \\({S}_{n}={a}_{1}+{a}_{2}+{a}_{3}+...+{a}_{n}.\\) To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.<\/p><p id=\"fs-id1167834196804\">We can develop this new formula by first writing the sum by starting with the first term, \\({a}_{1},\\) and keep adding a <em data-effect=\"italics\">d<\/em> to get the next term as:<\/p><div data-type=\"equation\" id=\"fs-id1167835595644\" class=\"unnumbered\" data-label=\"\">\\({S}_{n}={a}_{1}+\\left({a}_{1}+d\\right)+\\left({a}_{1}+2d\\right)+\\dots +{a}_{n}.\\)<\/div><p id=\"fs-id1167834432133\">We can also reverse the order of the terms and write the sum by starting with \\({a}_{n}\\) and keep subtracting <em data-effect=\"italics\">d<\/em> to get the next term as<\/p><div data-type=\"equation\" id=\"fs-id1167832065716\" class=\"unnumbered\" data-label=\"\">\\({S}_{n}={a}_{n}+\\left({a}_{n}\u2013d\\right)+\\left({a}_{n}\u20132d\\right)+\\dots +{a}_{1}.\\)<\/div><p id=\"fs-id1167826783889\">If we add these two expressions for the sum of the first <em data-effect=\"italics\">n<\/em> terms of an arithmetic sequence, we can derive a formula for the sum of the first <em data-effect=\"italics\">n<\/em> terms of any arithmetic series.<\/p><div data-type=\"equation\" id=\"fs-id1167834190612\" class=\"unnumbered\" data-label=\"\">\\(\\begin{array}{c}\\underset{\\text{_________________________________________________________}}{\\begin{array}{ccccccccccc}\\hfill {S}_{n}&amp; =\\hfill &amp; {a}_{1}\\hfill &amp; \\phantom{\\rule{1.9em}{0ex}}+\\hfill &amp; \\left({a}_{1}+d\\right)\\hfill &amp; +\\hfill &amp; \\left({a}_{1}+2d\\right)\\hfill &amp; +\\hfill &amp; \\dots \\hfill &amp; +\\hfill &amp; {a}_{n}\\hfill \\\\ +{S}_{n}\\hfill &amp; =\\hfill &amp; {a}_{n}\\hfill &amp; \\phantom{\\rule{1.9em}{0ex}}+\\hfill &amp; \\left({a}_{n}-d\\right)\\hfill &amp; +\\hfill &amp; \\left({a}_{n}-2d\\right)\\hfill &amp; +\\hfill &amp; \\dots \\hfill &amp; +\\hfill &amp; {a}_{1}\\hfill \\end{array}}\\hfill \\\\ \\\\ \\phantom{\\rule{1.8em}{0ex}}2{S}_{n}\\phantom{\\rule{0.6em}{0ex}}=\\phantom{\\rule{0.2em}{0ex}}\\left({a}_{1}+{a}_{n}\\right)+\\phantom{\\rule{0.05em}{0ex}}\\left({a}_{1}+{a}_{n}\\right)\\phantom{\\rule{0.5em}{0ex}}+\\phantom{\\rule{0.5em}{0ex}}\\left({a}_{1}+{a}_{n}\\right)\\phantom{\\rule{0.6em}{0ex}}+\\phantom{\\rule{0.5em}{0ex}}\\dots \\phantom{\\rule{0.5em}{0ex}}+\\phantom{\\rule{0.5em}{0ex}}\\left({a}_{1}+{a}_{n}\\right)\\hfill \\end{array}\\)<\/div><p id=\"fs-id1167835236742\">Because there are <em data-effect=\"italics\">n<\/em> sums of \\(\\left({a}_{1}+{a}_{n}\\right)\\) on the right side of the equation, we rewrite the right side as \\(n\\left({a}_{1}+{a}_{n}\\right).\\)<\/p><div data-type=\"equation\" id=\"fs-id1167835340841\" class=\"unnumbered\" data-label=\"\">\\(2{S}_{n}=n\\left({a}_{1}+{a}_{n}\\right)\\)<\/div><p id=\"fs-id1167835297398\">We divide by two to solve for \\({S}_{n}.\\)<\/p><div data-type=\"equation\" id=\"fs-id1167835479285\" class=\"unnumbered\" data-label=\"\">\\({S}_{n}=\\frac{n}{2}\\left({a}_{1}+{a}_{n}\\right)\\)<\/div><p id=\"fs-id1167834184657\">This give us a general formula for the sum of the first <em data-effect=\"italics\">n<\/em> terms of an arithmetic sequence.<\/p><div data-type=\"note\" id=\"fs-id1167835366292\"><div data-type=\"title\">Sum of the First <em data-effect=\"italics\">n<\/em> Terms of an Arithmetic Sequence<\/div><p id=\"fs-id1167834064520\">The sum, \\({S}_{n},\\) of the first <em data-effect=\"italics\">n<\/em> terms of an arithmetic sequence is<\/p><div data-type=\"equation\" id=\"fs-id1167835332490\" class=\"unnumbered\" data-label=\"\">\\({S}_{n}=\\frac{n}{2}\\left({a}_{1}+{a}_{n}\\right)\\)<\/div><p id=\"fs-id1167835339098\">where \\({a}_{1}\\) is the first term and \\({a}_{n}\\) is the <em data-effect=\"italics\">n<\/em>th term.<\/p><\/div><p id=\"fs-id1167831847164\">We apply this formula in the next example where the first few terms of the sequence are given.<\/p><div data-type=\"example\" id=\"fs-id1167831847167\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835354048\"><div data-type=\"problem\" id=\"fs-id1167835354050\"><p id=\"fs-id1167835354052\">Find the sum of the first 30 terms of the arithmetic sequence: 8, 13, 18, 23, 28, \u2026<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835236244\"><p id=\"fs-id1167835355709\">To find the sum, we will use the formula \\({S}_{n}=\\frac{n}{2}\\left({a}_{1}+{a}_{n}\\right).\\) We know \\({a}_{1}=8\\), \\(d=5\\) and \\(n=30,\\) but we need to find \\({a}_{n}\\) in order to use the sum formula.<\/p><p id=\"fs-id1167835380271\">\\(\\begin{array}{cccccc}&amp; &amp; &amp; \\hfill {a}_{n}&amp; =\\hfill &amp; {a}_{1}+\\left(n-1\\right)d\\hfill \\\\ \\text{Find}\\phantom{\\rule{0.2em}{0ex}}{a}_{n}\\phantom{\\rule{0.2em}{0ex}}\\text{where}\\phantom{\\rule{0.2em}{0ex}}{a}_{1}=8,d=5\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}n=30.\\hfill &amp; &amp; &amp; \\hfill {a}_{30}&amp; =\\hfill &amp; 8+\\left(30-1\\right)5\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill {a}_{30}&amp; =\\hfill &amp; 8+\\left(29\\right)5\\hfill \\\\ &amp; &amp; &amp; \\hfill {a}_{30}&amp; =\\hfill &amp; 153\\hfill \\\\ \\\\ \\\\ \\begin{array}{c}\\text{Knowing}\\phantom{\\rule{0.2em}{0ex}}{a}_{1}=8,n=30,\\phantom{\\rule{0.2em}{0ex}}\\text{and}\\phantom{\\rule{0.2em}{0ex}}{a}_{30}=153,\\hfill \\\\ \\text{use the sum formula.}\\hfill \\end{array}\\hfill &amp; &amp; &amp; \\hfill {S}_{n}&amp; =\\hfill &amp; \\frac{n}{2}\\left({a}_{1}+{a}_{n}\\right)\\hfill \\\\ \\text{Substitute in the values.}\\hfill &amp; &amp; &amp; \\hfill {S}_{30}&amp; =\\hfill &amp; \\frac{30}{2}\\left(8+153\\right)\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill {S}_{30}&amp; =\\hfill &amp; 15\\left(161\\right)\\hfill \\\\ \\text{Simplify.}\\hfill &amp; &amp; &amp; \\hfill {S}_{30}&amp; =\\hfill &amp; 2,415\\hfill \\end{array}\\)<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167826880408\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835335560\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835335562\"><p id=\"fs-id1167835335565\">Find the sum of the first 30 terms of the arithmetic sequence: 5, 9, 13, 17, 21, \u2026<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835338305\"><p id=\"fs-id1167835234738\">1,890<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834062236\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167834062240\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834062242\"><p id=\"fs-id1167835596050\">Find the sum of the first 30 terms of the arithmetic sequence: 7, 10, 13, 16, 19, \u2026<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834537845\"><p id=\"fs-id1167834537847\">1,515<\/p><\/div><\/div><\/div><p id=\"fs-id1167835479176\">In the next example, we are given the general term for the sequence and are asked to find the sum of the first 50 terms.<\/p><div data-type=\"example\" id=\"fs-id1167835280727\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835280730\"><div data-type=\"problem\" id=\"fs-id1167835280732\"><p id=\"fs-id1167835230447\">Find the sum of the first 50 terms of the arithmetic sequence whose general term is \\({a}_{n}=3n-4.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835229121\"><p id=\"fs-id1167835229124\">To find the sum, we will use the formula \\({S}_{n}=\\frac{n}{2}\\left({a}_{1}+{a}_{n}\\right).\\) We know \\(n=50,\\) but we need to find \\({a}_{1}\\) and \\({a}_{n}\\) in order to use the sum formula.<\/p><table id=\"fs-id1167831908831\" summary=\"This figure shows how we find the sum of the formula a sub n equals 3 times n minus 4. First we find a sub 1 by substituting n equals 1 to get a sub 1 equals 3 times g times 1 minus 4. We then get a sub 1 equals negative 1. Next we find a sub n by substituting n equals 50 into the formula a sub n equals 3 n minus 4. We get a sub 50 equals 3 times g times 50 minus 4. Simplify to get a sub 50 equals 146. Knowing that n equals 50, a sub 1 equals negative 1, and a sub 50 equals 146, use the sum formula to get s sub n equals n divided by 2 times a sub 1 plus a sub n. Substitute the values to get s sub 50 equals 50 divided by 2 times negative 1 plus 146, and simplify to s sub 50 equals 25 times 145. Simplify again to get 3625.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td><\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228893\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Find \\({a}_{1},\\) by substituting \\(n=1.\\)<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835241370\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-valign=\"top\" data-align=\"left\">Find \\({a}_{n}\\) by substituting \\(n=50.\\)<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835356180\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834064558\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Knowing \\(n=50,\\)\\({a}_{1}=-1,\\) and \\({a}_{50}=146\\) use the sum formula.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834459182\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Substitute in the values.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832075937\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835514385\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835319309\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835343611\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835343614\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832086944\"><p id=\"fs-id1167832086946\">Find the sum of the first 50 terms of the arithmetic sequence whose general term is \\({a}_{n}=2n-5.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835319256\"><p id=\"fs-id1167835319258\">2,300<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834449065\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835239514\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835239516\"><p id=\"fs-id1167835239518\">Find the sum of the first 50 terms of the arithmetic sequence whose general term is \\({a}_{n}=4n+3.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831847148\"><p id=\"fs-id1167835339146\">5,250<\/p><\/div><\/div><\/div><p id=\"fs-id1167835238526\">In the next example we are given the sum in summation notation. To add all the terms would be tedious, so we extract the information needed to use the formula to find the sum of the first <em data-effect=\"italics\">n<\/em> terms.<\/p><div data-type=\"example\" id=\"fs-id1167835236622\" class=\"textbox textbox--examples\"><div data-type=\"exercise\" id=\"fs-id1167835236624\"><div data-type=\"problem\" id=\"fs-id1167835367026\"><p id=\"fs-id1167835367028\">Find the sum: \\(\\sum _{i=1}^{25}\\left(4i+7\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835346145\"><p id=\"fs-id1167834433019\">To find the sum, we will use the formula \\({S}_{n}=\\frac{n}{2}\\left({a}_{1}+{a}_{n}\\right).\\) We know \\(n=25,\\) but we need to find \\({a}_{1}\\) and \\({a}_{n}\\) in order to use the sum formula.<\/p><table id=\"fs-id1167834422890\" summary=\"This figure shows how to expand the summation notation the sum of 4 sub i plus 7 from i equals 1 to 25. We simplify to equal 11 plus 15 plus 19 plus ellipsis plus 107. Next identify a sub 1 which equals 11 and a sub 25 equals 107. Knowing that n equals 25, a sub 1 equals 11, and a sub 25 equals 107, use the sum formula to get s sub n equals n divided by 2 times a sub 1 plus a sub n. Substitute the values to get s sub 25 equals 25 divided by 2 times 11 plus 107, and simplify to 25 divided by 2 times 118. Simplify again to get 1475.\" class=\"unnumbered unstyled\" data-label=\"\"><tbody><tr><td data-align=\"left\">Expand the summation notation.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835325691\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835307318\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Identify \\({a}_{1}.\\)<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834539248\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Identify \\({a}_{25}.\\)<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831872401\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Knowing \\(n=25,\\)\\({a}_{1}=11,\\) and \\({a}_{25}=107\\)<span data-type=\"newline\"><br \/><\/span>use the sum formula.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831872560\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Substitute in the values.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834345657\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834557211\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><tr><td data-align=\"left\">Simplify.<\/td><td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835379571\" data-alt=\".\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td><\/tr><\/tbody><\/table><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167835381406\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167835381410\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835351308\"><p id=\"fs-id1167835351311\">Find the sum: \\(\\sum _{i=1}^{30}\\left(6i-4\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167832036168\"><p id=\"fs-id1167832036170\">2,670<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167831066311\" class=\"try\"><div data-type=\"exercise\" id=\"fs-id1167831066315\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831910744\"><p id=\"fs-id1167831910746\">Find the sum: \\(\\sum _{i=1}^{35}\\left(5i-3\\right).\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834183812\"><p id=\"fs-id1167834183814\">3,045<\/p><\/div><\/div><\/div><div data-type=\"note\" id=\"fs-id1167834552520\" class=\"media-2\"><p id=\"fs-id1167834552524\">Access these online resources for additional instruction and practice with arithmetic sequences<\/p><ul id=\"fs-id1167835479514\" data-display=\"block\"><li><a href=\"https:\/\/openstax.org\/l\/37ArithSequenc\">Arithmetic Sequences<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37AritSeqnthter\">Arithmetic Sequences: A Formula for the \u2018n-th\u2019 Term<\/a><\/li><li><a href=\"https:\/\/openstax.org\/l\/37ArithSeries\">Arithmetic Series<\/a><\/li><\/ul><\/div><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835351252\"><h3 data-type=\"title\">Key Concepts<\/h3><ul id=\"fs-id1167835351260\" data-bullet-style=\"bullet\"><li><strong data-effect=\"bold\">General Term (<em data-effect=\"italics\">n<\/em>th term) of an Arithmetic Sequence<\/strong><span data-type=\"newline\"><br \/><\/span> The general term of an arithmetic sequence with first term \\({a}_{1}\\) and the common difference <em data-effect=\"italics\">d<\/em> is<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1167835360226\" class=\"unnumbered\" data-label=\"\">\\({a}_{n}={a}_{1}+\\left(n-1\\right)d\\)<\/div><\/li><li><strong data-effect=\"bold\">Sum of the First <em data-effect=\"italics\">n<\/em> Terms of an Arithmetic Sequence<\/strong><span data-type=\"newline\"><br \/><\/span> The sum, \\({S}_{n},\\) of the first <em data-effect=\"italics\">n<\/em> terms of an arithmetic sequence, where \\({a}_{1}\\) is the first term and \\({a}_{n}\\) is the <em data-effect=\"italics\">n<\/em>th term is<span data-type=\"newline\"><br \/><\/span> <div data-type=\"equation\" id=\"fs-id1167834537731\" class=\"unnumbered\" data-label=\"\">\\({S}_{n}=\\frac{n}{2}\\left({a}_{1}+{a}_{n}\\right)\\)<\/div><\/li><\/ul><\/div><div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167832036180\"><div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167832036184\"><h4 data-type=\"title\">Practice Makes Perfect<\/h4><p id=\"fs-id1167834448987\"><strong data-effect=\"bold\">Determine if a Sequence is Arithmetic<\/strong><\/p><p id=\"fs-id1167834537735\">In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.<\/p><div data-type=\"exercise\" id=\"fs-id1167834537739\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834537741\"><p id=\"fs-id1167834537743\">\\(4,12,20,28,36,44\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834552550\"><p id=\"fs-id1167834552552\">The sequence is arithmetic with common difference \\(d=8.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835344522\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835344524\"><p id=\"fs-id1167835344526\">\\(-7,-2,3,8,13,18\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830703159\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830703162\"><p id=\"fs-id1167834431030\">\\(-15,-16,3,12,21,30\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826996781\"><p id=\"fs-id1167826996783\">The sequence is not arithmetic.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831922087\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831922089\"><p id=\"fs-id1167831922091\">\\(11,5,-1,-7-13,-19\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835284952\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835332220\"><p id=\"fs-id1167835332222\">\\(8,5,2,-1,-4,-7\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831921263\"><p id=\"fs-id1167831921265\">The sequence is arithmetic with common difference \\(d=-3.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832067132\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832067134\"><p id=\"fs-id1167832067136\">\\(15,5,-5,-15,-25,-35\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><p id=\"fs-id1167835280309\">In the following exercises, write the first five terms of each sequence with the given first term and common difference.<\/p><div data-type=\"exercise\" id=\"fs-id1167835280314\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835311941\"><p id=\"fs-id1167835311943\">\\({a}_{1}=11\\) and \\(d=7\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834537714\"><p id=\"fs-id1167834537716\">\\(11,18,25,32,39\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167832087084\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167832087086\"><p id=\"fs-id1167832087088\">\\({a}_{1}=18\\) and \\(d=9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835340208\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835340210\"><p id=\"fs-id1167835340212\">\\({a}_{1}=-7\\) and \\(d=4\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834063502\"><p id=\"fs-id1167834063504\">\\(-7,-3,1,5,9\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834301167\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834301169\"><p id=\"fs-id1167834301171\">\\({a}_{1}=-8\\) and \\(d=5\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834584370\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834584372\"><p id=\"fs-id1167831921597\">\\({a}_{1}=14\\) and \\(d=-9\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826994172\"><p id=\"fs-id1167826994174\">\\(14,5,-4,-13,-22\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831920468\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831920470\"><p id=\"fs-id1167831920472\">\\({a}_{1}=-3\\) and \\(d=-3\\)<\/p><\/div><\/div><p id=\"fs-id1167826994248\"><strong data-effect=\"bold\">Find the General Term (<em data-effect=\"italics\">n<\/em>th Term) of an Arithmetic Sequence<\/strong><\/p><p id=\"fs-id1167835515402\">In the following exercises, find the term described using the information provided.<\/p><div data-type=\"exercise\" id=\"fs-id1167835515405\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835515407\"><p id=\"fs-id1167835515409\">Find the twenty-first term of a sequence where the first term is three and the common difference is eight.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835515414\"><p id=\"fs-id1167835515417\">\\(163\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834063455\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834063457\"><p id=\"fs-id1167834063459\">Find the twenty-third term of a sequence where the first term is six and the common difference is four.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831872319\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831872321\"><p id=\"fs-id1167831872323\">Find the thirtieth term of a sequence where the first term is \\(-14\\) and the common difference is five.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831872333\"><p id=\"fs-id1167831872335\">\\(131\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835479014\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835479016\"><p id=\"fs-id1167835479018\">Find the fortieth term of a sequence where the first term is \\(-19\\) and the common difference is seven.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835479491\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835479493\"><p id=\"fs-id1167835479495\">Find the sixteenth term of a sequence where the first term is 11 and the common difference is \\(-6.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835479506\"><p id=\"fs-id1167835479508\">\\(-79\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831921343\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831921345\"><p id=\"fs-id1167831921347\">Find the fourteenth term of a sequence where the first term is eight and the common difference is \\(-3.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835301613\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835301615\"><p id=\"fs-id1167835301617\">Find the twentieth term of a sequence where the fifth term is \\(-4\\) and the common difference is \\(-2.\\) Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831071387\"><p id=\"fs-id1167831071389\">\\({a}_{20}=-34.\\) The general term is \\({a}_{n}=-2n+6.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835514340\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835514342\"><p id=\"fs-id1167835514344\">Find the thirteenth term of a sequence where the sixth term is \\(-1\\) and the common difference is \\(-4.\\) Give the formula for the general term.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835361599\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835361602\"><p id=\"fs-id1167835361604\">Find the eleventh term of a sequence where the third term is 19 and the common difference is five. Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835361609\"><p id=\"fs-id1167835361611\">\\({a}_{11}=59.\\) The general term is \\({a}_{n}=5n+4.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835332338\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835332340\"><p id=\"fs-id1167835332342\">Find the fifteenth term of a sequence where the tenth term is 17 and the common difference is seven. Give the formula for the general term.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835280338\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835280340\"><p id=\"fs-id1167835280342\">Find the eighth term of a sequence where the seventh term is \\(-8\\) and the common difference is \\(-5.\\) Give the formula for the general term.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834534420\"><p id=\"fs-id1167834534422\">\\({a}_{8}=-13.\\) The general term is \\({a}_{n}=-5n+27.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835280403\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835280405\"><p id=\"fs-id1167835280407\">Find the fifteenth term of a sequence where the tenth term is \\(-11\\) and the common difference is \\(-3.\\) Give the formula for the general term.<\/p><\/div><\/div><p id=\"fs-id1167830705162\">In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.<\/p><div data-type=\"exercise\" id=\"fs-id1167830705166\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830705169\"><p id=\"fs-id1167830705171\">The second term is 14 and the thirteenth term is 47.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830705175\"><p id=\"fs-id1167830705177\">\\({a}_{1}=11,\\)\\(d=3.\\) The general term is \\({a}_{n}=3n+8.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834194337\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834194339\"><p id=\"fs-id1167834194341\">The third term is 18 and the fourteenth term is 73.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167826996712\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826996714\"><p id=\"fs-id1167834063517\">The second term is 13 and the tenth term is \\(-51.\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834063527\"><p id=\"fs-id1167834063530\">\\({a}_{1}=21,\\)\\(d=-8.\\) The general term is \\({a}_{n}=-8n+29.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834535636\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834535638\"><p id=\"fs-id1167834535641\">The third term is four and the tenth term is \\(-38\\).<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834194386\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834194388\"><p id=\"fs-id1167834194390\">The fourth term is \\(-6\\) and the fifteenth term is 27.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167834194400\"><p id=\"fs-id1167834194402\">\\({a}_{1}=-15,\\)\\(d=3.\\) The general term is \\({a}_{n}=3n-18.\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835360260\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835360262\"><p id=\"fs-id1167835360264\">The third term is \\(-13\\) and the seventeenth term is 15.<\/p><\/div><\/div><p id=\"fs-id1167831872364\"><strong data-effect=\"bold\">Find the Sum of the First <em data-effect=\"italics\">n<\/em> Terms of an Arithmetic Sequence<\/strong><\/p><p id=\"fs-id1167831872373\">In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.<\/p><div data-type=\"exercise\" id=\"fs-id1167831872377\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831872379\"><p id=\"fs-id1167831872381\">\\(11,14,17,20,23\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167827967193\"><p id=\"fs-id1167827967195\">1,635<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167827967200\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167827967202\"><p id=\"fs-id1167827967204\">\\(12,18,24,30,36\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835515365\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835515367\"><p id=\"fs-id1167835515369\">\\(8,5,2,-1,-4\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835352290\"><p id=\"fs-id1167835352292\">\\(-1,065\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835360173\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835360175\"><p id=\"fs-id1167835360177\">\\(16,10,4,-2,-8\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834449015\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834449017\"><p id=\"fs-id1167834449019\">\\(-17,-15,-13,-11,-9\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835589628\"><p id=\"fs-id1167835589630\">360<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835589635\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835589637\"><p id=\"fs-id1167835589639\">\\(-15,-12,-9,-6,-3\\text{,}\\phantom{\\rule{0.2em}{0ex}}\\text{\u2026}\\)<\/p><\/div><\/div><p id=\"fs-id1167826996724\">In the following exercises, find the sum of the first 50 terms of the arithmetic sequence whose general term is given.<\/p><div data-type=\"exercise\" id=\"fs-id1167826996728\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167826996730\"><p id=\"fs-id1167826996732\">\\({a}_{n}=5n-1\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167826996754\"><p id=\"fs-id1167826996756\">6,325<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834448936\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834448938\"><p id=\"fs-id1167834448941\">\\({a}_{n}=2n+7\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834448969\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834448972\"><p id=\"fs-id1167834448974\">\\({a}_{n}=-3n+5\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831025348\"><p id=\"fs-id1167831025350\">\u20133,575<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831025355\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831025357\"><p id=\"fs-id1167831025360\">\\({a}_{n}=-4n+3\\)<\/p><\/div><\/div><p id=\"fs-id1167834583346\">In the following exercises, find each sum.<\/p><div data-type=\"exercise\" id=\"fs-id1167834583349\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834583351\"><p id=\"fs-id1167834583353\">\\(\\sum _{i=1}^{40}\\left(8i-7\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830704023\"><p id=\"fs-id1167830704025\">6,280<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830704030\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830704032\"><p id=\"fs-id1167830704034\">\\(\\sum _{i=1}^{45}\\left(7i-5\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835284910\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835284912\"><p id=\"fs-id1167835284914\">\\(\\sum _{i=1}^{50}\\left(3i+6\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835511978\"><p id=\"fs-id1167835511980\">4,125<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835511985\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835511988\"><p id=\"fs-id1167835511990\">\\(\\sum _{i=1}^{25}\\left(4i+3\\right)\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167834583425\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167834583427\"><p id=\"fs-id1167834583429\">\\(\\sum _{i=1}^{35}\\left(-6i-2\\right)\\)<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167835512721\"><p id=\"fs-id1167835512723\">\\(-3,850\\)<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167835512736\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167835512738\"><p id=\"fs-id1167835512740\">\\(\\sum _{i=1}^{30}\\left(-5i+1\\right)\\)<\/p><\/div><\/div><\/div><div class=\"writing\" data-depth=\"2\" id=\"fs-id1167830703940\"><h4 data-type=\"title\">Writing Exercises<\/h4><div data-type=\"exercise\" id=\"fs-id1167830703947\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167830703950\"><p id=\"fs-id1167830703952\">In your own words, explain how to determine whether a sequence is arithmetic.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167830703956\"><p id=\"fs-id1167830703958\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167830703963\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831066248\"><p id=\"fs-id1167831066250\">In your own words, explain how the first two terms are used to find the tenth term. Show an example to illustrate your explanation.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831066263\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831066265\"><p id=\"fs-id1167831066267\">In your own words, explain how to find the general term of an arithmetic sequence.<\/p><\/div><div data-type=\"solution\" id=\"fs-id1167831066271\"><p id=\"fs-id1167831066273\">Answers will vary.<\/p><\/div><\/div><div data-type=\"exercise\" id=\"fs-id1167831066279\" class=\"material-set-2\"><div data-type=\"problem\" id=\"fs-id1167831066281\"><p id=\"fs-id1167831066283\">In your own words, explain how to find the sum of the first \\(n\\) terms of an arithmetic sequence without adding all the terms.<\/p><\/div><\/div><\/div><div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167831066300\"><h4 data-type=\"title\">Self Check<\/h4><p id=\"fs-id1167835370651\"><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-id1167835370667\" data-alt=\"This figure shows a chart with four rows and four columns. The first row is the header row and reads, &#x201c;I can&#x201d;, &#x201c;Confidently&#x201d;, &#x201c;With some help&#x201d;, and &#x201c;No, I don&#x2019;t get it!&#x201d; The column, beginning with second row reads 1. Determine if a Sequence is Arithmetic, 2. Find the General Term (nth term) of Arithmetic Sequence, and 3. Find the sum of the first Terms of an Arithmetic Sequence&#x201d;. The remaining columns are blank.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a chart with four rows and four columns. The first row is the header row and reads, &#x201c;I can&#x201d;, &#x201c;Confidently&#x201d;, &#x201c;With some help&#x201d;, and &#x201c;No, I don&#x2019;t get it!&#x201d; The column, beginning with second row reads 1. Determine if a Sequence is Arithmetic, 2. Find the General Term (nth term) of Arithmetic Sequence, and 3. Find the sum of the first Terms of an Arithmetic Sequence&#x201d;. The remaining columns are blank.\" \/><\/span><p id=\"fs-id1167835370675\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/p><\/div><\/div><div data-type=\"glossary\" class=\"textbox shaded\"><h3 data-type=\"glossary-title\">Glossary<\/h3><dl id=\"fs-id1167835370688\"><dt>arithmetic sequence<\/dt><dd id=\"fs-id1167835370694\">An arithmetic sequence is a sequence where the difference between consecutive terms is constant.<\/dd><\/dl><dl id=\"fs-id1167835370699\"><dt>common difference<\/dt><dd id=\"fs-id1167835370704\">The difference between consecutive terms in an arithmetic sequence, \\({a}_{n}-{a}_{n-1},\\) is <em data-effect=\"italics\">d<\/em>, the common difference, for <em data-effect=\"italics\">n<\/em> greater than or equal to two.<\/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>Determine if a sequence is arithmetic<\/li>\n<li>Find the general term (<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;\" \/>th term) of an arithmetic sequence<\/li>\n<li>Find the sum of the first <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;\" \/> terms of an arithmetic sequence<\/li>\n<\/ul>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835327387\" class=\"be-prepared\">\n<p id=\"fs-id1167835375480\">Before you get started, take this readiness quiz.<\/p>\n<ol id=\"fs-id1167835421842\" type=\"1\">\n<li>Evaluate <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-638ca352f5fd1fb480141ef4320cfd7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#110;&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"49\" style=\"vertical-align: -1px;\" \/> 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>Solve the system of equations: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4fa015aa5467bc74a1629a3a85f644b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#125;&#120;&#43;&#121;&#61;&#55;&#92;&#92;&#32;&#51;&#120;&#43;&#52;&#121;&#61;&#50;&#51;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"43\" width=\"132\" style=\"vertical-align: -17px;\" \/><span data-type=\"newline\"><br \/><\/span> If you missed this problem, review <a href=\"\/contents\/a236420d-b79a-43e3-8f7c-1b8e18c5045e#fs-id1167834121139\" 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-4833517d9bb865903c0b177437e38756_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;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#110;&#43;&#53;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"148\" style=\"vertical-align: -6px;\" \/> find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2c4a584d13f700feefc922a0731aa33f_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;&#48;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"110\" 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-id1167835235945\">\n<h3 data-type=\"title\">Determine if a Sequence is Arithmetic<\/h3>\n<p id=\"fs-id1167835421169\">The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.<\/p>\n<p id=\"fs-id1167835308916\">An <span data-type=\"term\">arithmetic sequence<\/span> is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89e3f4cf74e1b833f8b956476ff8239c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#45;&#123;&#97;&#125;&#95;&#123;&#110;&#45;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"80\" style=\"vertical-align: -4px;\" \/> is <em data-effect=\"italics\">d<\/em>, the <span data-type=\"term\">common difference<\/span>, for <em data-effect=\"italics\">n<\/em> greater than or equal to two.<\/p>\n<div data-type=\"note\" id=\"fs-id1167830705774\">\n<div data-type=\"title\">Arithmetic Sequence<\/div>\n<p id=\"fs-id1167834449214\">An <strong data-effect=\"bold\">arithmetic sequence<\/strong> is a sequence where the difference between consecutive terms is always the same.<\/p>\n<p id=\"fs-id1167835329986\">The difference between consecutive terms, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89e3f4cf74e1b833f8b956476ff8239c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#45;&#123;&#97;&#125;&#95;&#123;&#110;&#45;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"80\" style=\"vertical-align: -4px;\" \/> is <em data-effect=\"italics\">d<\/em>, the <strong data-effect=\"bold\">common difference<\/strong>, for <em data-effect=\"italics\">n<\/em> greater than or equal to two.<\/p>\n<\/div>\n<p><span data-type=\"media\" id=\"fs-id1167835187194\" data-alt=\"This figure has two rows and three columns. The first row reads &#x201c;7&#x201d;, &#x201c;10&#x201d;,&#x201d;13&#x201d;, &#x201c;16&#x201d;, &#x201c;19&#x201d;, &#x201c;22&#x201d;, and an ellipsis, &#x201c;10 minus 7, divided by 3&#x201d;, &#x201c;13 minus 10, divided by 3&#x201d;, &#x201c;16 minus 13, divided by 3&#x201d;, nth term equals nth term minus 1 divided by d&#x201d;\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_001_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure has two rows and three columns. The first row reads &#x201c;7&#x201d;, &#x201c;10&#x201d;,&#x201d;13&#x201d;, &#x201c;16&#x201d;, &#x201c;19&#x201d;, &#x201c;22&#x201d;, and an ellipsis, &#x201c;10 minus 7, divided by 3&#x201d;, &#x201c;13 minus 10, divided by 3&#x201d;, &#x201c;16 minus 13, divided by 3&#x201d;, nth term equals nth term minus 1 divided by d&#x201d;\" \/><\/span><\/p>\n<p id=\"fs-id1167835304119\">In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835420956\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167832041539\">\n<div data-type=\"problem\" id=\"fs-id1167835326267\">\n<p id=\"fs-id1167831959293\">Determine if each sequence is arithmetic. If so, indicate the common difference.<\/p>\n<p id=\"fs-id1167832076355\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-84352a057612aadb776368d479332f9a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#57;&#44;&#49;&#51;&#44;&#49;&#55;&#44;&#50;&#49;&#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=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167834214180\"><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-99b60e4b5966affb0d49b0dd12860808_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#57;&#44;&#49;&#50;&#44;&#49;&#55;&#44;&#50;&#48;&#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=\"132\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"fs-id1167835420225\"><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ced6776a8d118caabdba37fee434d5c5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#48;&#44;&#51;&#44;&#45;&#52;&#44;&#45;&#49;&#49;&#44;&#45;&#49;&#56;&#44;&#45;&#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=\"187\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835354025\">\n<p id=\"fs-id1167835339364\">To determine if the sequence is arithmetic, we find the difference of the consecutive terms shown.<\/p>\n<p id=\"fs-id1167835368778\"><span class=\"token\">\u24d0<\/span><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7335bc464cae23f3f836b6ed8ce4355e_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;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#51;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#49;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#53;&#44;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#45;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#51;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#55;&#45;&#49;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#49;&#45;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#53;&#45;&#50;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#115;&#101;&#113;&#117;&#101;&#110;&#99;&#101;&#32;&#105;&#115;&#32;&#97;&#114;&#105;&#116;&#104;&#109;&#101;&#116;&#105;&#99;&#46;&#32;&#84;&#104;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#105;&#115;&#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;&#100;&#61;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#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=\"192\" width=\"793\" style=\"vertical-align: -91px;\" \/><\/p>\n<p id=\"fs-id1167834094786\"><span class=\"token\">\u24d1<\/span><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-739c4661274db54c27b418aa1ba87008_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;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#52;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#57;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#48;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#53;&#44;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#57;&#45;&#52;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#50;&#45;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#55;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#48;&#45;&#49;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#53;&#45;&#50;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#115;&#101;&#113;&#117;&#101;&#110;&#99;&#101;&#32;&#105;&#115;&#32;&#110;&#111;&#116;&#32;&#97;&#114;&#105;&#116;&#104;&#109;&#101;&#116;&#105;&#99;&#32;&#97;&#115;&#32;&#97;&#108;&#108;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#115;&#32;&#98;&#101;&#116;&#119;&#101;&#101;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#32;&#97;&#114;&#101;&#32;&#110;&#111;&#116;&#32;&#116;&#104;&#101;&#32;&#115;&#97;&#109;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#114;&#101;&#32;&#105;&#115;&#32;&#110;&#111;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#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=\"234\" width=\"793\" style=\"vertical-align: -111px;\" \/><\/p>\n<p id=\"fs-id1167832055415\"><span class=\"token\">\u24d2<\/span><span data-type=\"newline\"><br \/><\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-53cc63c1d220b07670b4d23183dc3dfb_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;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#111;&#102;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#99;&#111;&#110;&#115;&#101;&#99;&#117;&#116;&#105;&#118;&#101;&#32;&#116;&#101;&#114;&#109;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#49;&#48;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#51;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#52;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#56;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#48;&#46;&#54;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#53;&#44;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#51;&#45;&#49;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#52;&#45;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#49;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#49;&#56;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#50;&#53;&#45;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#49;&#56;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#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;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#115;&#101;&#113;&#117;&#101;&#110;&#99;&#101;&#32;&#105;&#115;&#32;&#97;&#114;&#105;&#116;&#104;&#109;&#101;&#116;&#105;&#99;&#46;&#32;&#84;&#104;&#101;&#32;&#99;&#111;&#109;&#109;&#111;&#110;&#32;&#100;&#105;&#102;&#102;&#101;&#114;&#101;&#110;&#99;&#101;&#32;&#105;&#115;&#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;&#100;&#61;&#45;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#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=\"181\" width=\"976\" style=\"vertical-align: -91px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834222032\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834556290\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834063445\">\n<p id=\"fs-id1167835332114\">Determine if each sequence is arithmetic. If so, indicate the common difference.<\/p>\n<p id=\"fs-id1167835328888\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-26e303e21519e0a1f9a4f7c7d3878424_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#44;&#50;&#48;&#44;&#51;&#49;&#44;&#52;&#50;&#44;&#53;&#51;&#44;&#54;&#52;&#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=\"141\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f516586209028e2494c7c9c0b0cbdad1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#44;&#54;&#44;&#48;&#44;&#45;&#54;&#44;&#45;&#49;&#50;&#44;&#45;&#49;&#56;&#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=\"16\" width=\"164\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5f7b68bc881638a7641901d8c9ad7989_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#44;&#49;&#44;&#49;&#48;&#44;&#52;&#44;&#49;&#51;&#44;&#55;&#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=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835301773\">\n<p id=\"fs-id1167832056814\"><span class=\"token\">\u24d0<\/span> The sequence is arithmetic with common difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-27fd00f082031aef8c5c6fb57bd1c338_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"50\" style=\"vertical-align: -1px;\" \/>. <span class=\"token\">\u24d1<\/span> The sequence is arithmetic with common difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a899d11b33501bef67985bacd8cb64df_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/>.<span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span> The sequence is not arithmetic as all the differences between the consecutive terms are not the same.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834138030\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835367810\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835305719\">\n<p id=\"fs-id1167834403016\">Determine if each sequence is arithmetic. If so, indicate the common difference.<\/p>\n<p id=\"fs-id1167832067000\"><span class=\"token\">\u24d0<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83b3e2a5b9c6a985effb9525e562b623_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#44;&#52;&#44;&#50;&#44;&#49;&#48;&#44;&#56;&#44;&#49;&#54;&#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=\"16\" width=\"127\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d1<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b0c50bd61c36ccc3934801a7faf5bb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#45;&#49;&#44;&#49;&#44;&#51;&#44;&#53;&#44;&#55;&#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=\"123\" style=\"vertical-align: -4px;\" \/><span class=\"token\">\u24d2<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7ed6f29a4cf9e57fd947726b0114f4d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#44;&#50;&#44;&#45;&#51;&#44;&#45;&#56;&#44;&#45;&#49;&#51;&#44;&#45;&#49;&#56;&#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=\"170\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835198777\">\n<p id=\"fs-id1167835337130\"><span class=\"token\">\u24d0<\/span> The sequence is not arithmetic as all the differences between the consecutive terms are not the same. <span class=\"token\">\u24d1<\/span> The sequence is arithmetic with common difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe6fbbbfed400276f2f0becc57869deb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/><span data-type=\"newline\"><br \/><\/span><span class=\"token\">\u24d2<\/span> The sequence is arithmetic with common difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5cd47a2de95d5a26ae68f08ada0d00da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835341287\">If we know the first term, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd6d5e05e9022f1ee8855acce1c293c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: -4px;\" \/> and the common difference, <em data-effect=\"italics\">d<\/em>, we can list a finite number of terms of the sequence.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831944141\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835416627\">\n<div data-type=\"problem\" id=\"fs-id1167835254902\">\n<p id=\"fs-id1167835336301\">Write the first five terms of the sequence where the first term is 5 and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c5ea655b8235e08df8e4933578587935_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835422706\">\n<p id=\"fs-id1167834064148\">We start with the first term and add the common difference. Then we add the common difference to that result to get the next term, and so on.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167832056274\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c2f8721bac12b7bd3614dfed315a355_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;&#99;&#99;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#50;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#51;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#52;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#53;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#53;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#55;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#45;&#49;&#57;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"53\" width=\"529\" style=\"vertical-align: -23px;\" \/><\/div>\n<p id=\"fs-id1167831010120\">The sequence is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cdf392ef6921f8f3e2d4e6d4ab148ce8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#44;&#45;&#49;&#44;&#45;&#55;&#44;&#45;&#49;&#51;&#44;&#45;&#49;&#57;&#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=\"153\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834194151\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167832052324\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834433837\">\n<p id=\"fs-id1167835337752\">Write the first five terms of the sequence where the first term is 7 and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b976d3a363db98a552726a27fc7d123_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"60\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835261566\">\n<p id=\"fs-id1167831909186\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5dea2eac818c5da0a323b570d86dc42a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#44;&#51;&#44;&#45;&#49;&#44;&#45;&#53;&#44;&#45;&#57;&#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=\"122\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835329130\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834189547\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826978682\">\n<p id=\"fs-id1167835181090\">Write the first five terms of the sequence where the first term is 11 and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc47993a78dc950d063d1343c5bb9ef8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835162835\">\n<p id=\"fs-id1167835509099\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-18ca20c4eb7aae740f771610049cdcec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#44;&#51;&#44;&#45;&#53;&#44;&#45;&#49;&#51;&#44;&#45;&#50;&#49;&#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=\"147\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167834157026\">\n<h3 data-type=\"title\">Find the General Term (<em data-effect=\"italics\">n<\/em>th Term) of an Arithmetic Sequence<\/h3>\n<p id=\"fs-id1167835191878\">Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.<\/p>\n<p id=\"fs-id1167834063240\">Let\u2019s write the first few terms of a sequence where the first term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> and the common difference is <em data-effect=\"italics\">d<\/em>. We will then look for a pattern.<\/p>\n<p id=\"fs-id1167834423838\">As we look for a pattern we see that each term starts with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p><span data-type=\"media\" id=\"fs-id1167835340070\" data-alt=\"This figures shows an image of a sequence.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_002_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figures shows an image of a sequence.\" \/><\/span><\/p>\n<p id=\"fs-id1167835346003\">The first term adds 0<em data-effect=\"italics\">d<\/em> to the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/>, the second term adds 1<em data-effect=\"italics\">d<\/em>, the third term adds 2<em data-effect=\"italics\">d<\/em>, the fourth term adds 3<em data-effect=\"italics\">d<\/em>, and the fifth term adds 4<em data-effect=\"italics\">d<\/em>. The number of <em data-effect=\"italics\">d<\/em>s that were added to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> is one less than the number of the term. This leads us to the following<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835280772\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf70a8479d7942787021a2893d7d8126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"note\" id=\"fs-id1167835327327\">\n<div data-type=\"title\">General Term (<em data-effect=\"italics\">n<\/em>th term) of an Arithmetic Sequence<\/div>\n<p id=\"fs-id1167835194069\">The general term of an arithmetic sequence with first term <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> and the common difference <em data-effect=\"italics\">d<\/em> is<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835350396\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf70a8479d7942787021a2893d7d8126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<p id=\"fs-id1167835369032\">We will use this formula in the next example to find the 15<sup>th<\/sup> term of a sequence.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835309566\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835355850\">\n<div data-type=\"problem\" id=\"fs-id1167835173305\">\n<p id=\"fs-id1167834191748\">Find the fifteenth term of a sequence where the first term is 3 and the common difference is 6.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835237077\">\n<p id=\"fs-id1167835369908\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1073edc99c922bea6e244a45a5c69dd4_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;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#102;&#116;&#101;&#101;&#110;&#116;&#104;&#32;&#116;&#101;&#114;&#109;&#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;&#123;&#97;&#125;&#95;&#123;&#49;&#53;&#125;&#44;&#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;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#109;&#117;&#108;&#97;&#32;&#119;&#105;&#116;&#104;&#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;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#51;&#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;&#97;&#110;&#100;&#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;&#100;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#53;&#125;&#61;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#53;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#54;&#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;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#53;&#125;&#61;&#51;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#54;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#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;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#53;&#125;&#61;&#56;&#55;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"238\" width=\"552\" style=\"vertical-align: -114px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835375533\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835513406\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831239689\">\n<p id=\"fs-id1167835350380\">Find the twenty-seventh term of a sequence where the first term is 7 and the common difference is 9.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835418188\">\n<p id=\"fs-id1167835361810\">241<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835303825\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835347370\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834120366\">\n<p id=\"fs-id1167835307170\">Find the eighteenth term of a sequence where the first term is 13 and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00c6d30c5f7439a21caf981437f64be1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835362393\">\n<p id=\"fs-id1167835332053\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-62a4c088b5274bc69993dcf083e927a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#48;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"40\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835243861\">Sometimes we do not know the first term and we must use other given information to find it before we find the requested term.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831883365\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167834185861\">\n<div data-type=\"problem\" id=\"fs-id1167835258413\">\n<p id=\"fs-id1167835214784\">Find the twelfth term of a sequence where the seventh term is 10 and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17c33e2329e29a62a80ad2b547b4753d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/>. Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835418145\">\n<p id=\"fs-id1167832056903\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-453d87ce24fd3faced4cedbc6c29d25d_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;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#102;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#102;&#105;&#114;&#115;&#116;&#32;&#116;&#101;&#114;&#109;&#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;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#44;&#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;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#109;&#117;&#108;&#97;&#32;&#119;&#105;&#116;&#104;&#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;&#123;&#97;&#125;&#95;&#123;&#55;&#125;&#61;&#49;&#48;&#44;&#110;&#61;&#55;&#44;&#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;&#97;&#110;&#100;&#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;&#100;&#61;&#45;&#50;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#48;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#48;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#49;&#48;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#45;&#49;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#50;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#116;&#119;&#101;&#108;&#102;&#116;&#104;&#32;&#116;&#101;&#114;&#109;&#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;&#123;&#97;&#125;&#95;&#123;&#49;&#50;&#125;&#44;&#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;&#117;&#115;&#105;&#110;&#103;&#32;&#116;&#104;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#109;&#117;&#108;&#97;&#32;&#119;&#105;&#116;&#104;&#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;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#50;&#50;&#44;&#110;&#61;&#49;&#50;&#44;&#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;&#97;&#110;&#100;&#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;&#100;&#61;&#45;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#50;&#125;&#61;&#50;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#50;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#50;&#125;&#61;&#50;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#49;&#50;&#125;&#61;&#48;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#116;&#119;&#101;&#108;&#102;&#116;&#104;&#32;&#116;&#101;&#114;&#109;&#32;&#111;&#102;&#32;&#116;&#104;&#101;&#32;&#115;&#101;&#113;&#117;&#101;&#110;&#99;&#101;&#32;&#105;&#115;&#32;&#48;&#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;&#123;&#97;&#125;&#95;&#123;&#49;&#50;&#125;&#61;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#111;&#32;&#102;&#105;&#110;&#100;&#32;&#116;&#104;&#101;&#32;&#103;&#101;&#110;&#101;&#114;&#97;&#108;&#32;&#116;&#101;&#114;&#109;&#44;&#32;&#115;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#32;&#105;&#110;&#116;&#111;&#32;&#116;&#104;&#101;&#32;&#102;&#111;&#114;&#109;&#117;&#108;&#97;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#50;&#50;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#50;&#50;&#45;&#50;&#110;&#43;&#50;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#51;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#84;&#104;&#101;&#32;&#103;&#101;&#110;&#101;&#114;&#97;&#108;&#32;&#116;&#101;&#114;&#109;&#32;&#105;&#115;&#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;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#50;&#110;&#43;&#50;&#52;&#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=\"457\" width=\"786\" style=\"vertical-align: -223px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835418132\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835342802\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835307172\">\n<p id=\"fs-id1167835358419\">Find the eleventh term of a sequence where the ninth term is 8 and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa36400e576c82fd4847d2da37b1d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835353069\">\n<p id=\"fs-id1167835366527\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e253cae17038419f0ec9daa294563235_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#49;&#125;&#61;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"61\" style=\"vertical-align: -4px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9f5fcf56c5b96c4f578eacda20628db4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#51;&#110;&#43;&#51;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834194524\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835346520\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834535686\">\n<p id=\"fs-id1167835173222\">Find the nineteenth term of a sequence where the fifth term is 1 and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e344351eb8a6a075b3d1df3943b3d637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835352738\">\n<p id=\"fs-id1167834196928\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d9e93743caccd9a8b2feb7768efbec8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#57;&#125;&#61;&#45;&#53;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"84\" style=\"vertical-align: -4px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-519f1ae7da50371a1253c00de9ac0405_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#52;&#110;&#43;&#50;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165926635835\">Sometimes the information given leads us to two equations in two unknowns. We then use our methods for solving systems of equations to find the values needed.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835304642\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835282964\">\n<div data-type=\"problem\" id=\"fs-id1167832060198\">\n<p id=\"fs-id1167826967445\">Find the first term and common difference of a sequence where the fifth term is 19 and the eleventh term is 37. Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835240133\">\n<p id=\"fs-id1167834141929\">Since we know two terms, we can make a system of equations using the formula for the general term.<\/p>\n<table id=\"fs-id1167835348864\" summary=\"The formula nth term equals a 1 plus n minus 1 times d. We know the value of a 5 and a 11 because a 5 equals a 1 plus 5 minus 1 times d, and a 11 equals a 1 plus 11 minus 1 times d. We know that 19 equals a sub 1 plus 5 minus 1 times d, and 37 equals a sub 1 plus 11 minus 1 times, so we then substitute in the values a sub 5 equals 19 and a sub 11 equals 37. Simplify to get 19 equals a sub 1 plus 4 d, and 37 equals a sub 1 plus 10 d. Next we prepare to eliminate the a sub 1 term by multiplying the top equation by negative 1 to get 18 equals 6 d, and 3 equals d. Next, substitute d equals 3 back into the first equation to get 19 equals a sub 1 plus 4 times g 3, then solve for a sub 1 to get 19 equals a sub 1 plus 12, then 7 equals a. Use the formula with a sub 1 equals 7 and d equals 3. We then substitute in the values to get a sub n equals 7 plus the sum of n minus 1 times 3. Simplify to get a sub n equals 7 plus 3 n minus 3, then a sub n equals 3 n plus 4. The first term is a sub 1 equals 7 and the common difference is d equals 3. The general term of the sequence is a sub n equals 3 n plus 4.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835345460\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">We know the value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-22f073d383bac0d8703f9c331bd87c65_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#53;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"16\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e0a42a9f88e98199794f36789e248c37_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"28\" style=\"vertical-align: -4px;\" \/> so we will use <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6f852454e2c6c16d005bca006d38cbc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fe9c97fa7f7ec644e9a6ad529b5410da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835236275\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Substitute in the values, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bf2526074224dbde0141a44f700eaa0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#53;&#125;&#61;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-17bd408cd3c76a6e27181a03c9a04e18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#49;&#125;&#61;&#51;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"70\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835509141\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835349126\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Prepare to eliminate the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> term by multiplying the top equation by <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;\" \/><span data-type=\"newline\"><br \/><\/span>Add the equations.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835321742\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"newline\"><br \/><\/span>Substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c670fab3ff7b014c602cc0ebf5e374b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/> back into the first equation.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"newline\"><br \/><\/span><span data-type=\"media\" id=\"fs-id1167835361932\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0526020c87962a8a5a61ef3f77d6349c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: -4px;\" \/><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834053512\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"newline\"><br \/><\/span>Use the formula with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a9f839808d7050689cb1b4ae733a1867_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"50\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3de7278a447bec41bf67e06f751f5c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"newline\"><br \/><\/span><span data-type=\"media\" id=\"fs-id1167830704126\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003i_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Substitute in the values.<\/td>\n<td><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834186195\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003j_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Simplify.<\/td>\n<td><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41964c4995689dcf95c6c54d9bdef249_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"1\" width=\"1\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-valign=\"top\" data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834279285\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_003k_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\">The first term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-870e6fba9e217e176bd9a71ac620d563_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"54\" style=\"vertical-align: -4px;\" \/><span data-type=\"newline\"><br \/><\/span>The common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3de7278a447bec41bf67e06f751f5c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/td>\n<\/tr>\n<tr>\n<td><\/td>\n<td colspan=\"2\" data-valign=\"top\" data-align=\"center\">\n<p id=\"fs-id1167835343210\">The general term of the sequence is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08bec8dc4610818586b62bceb29ac0bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#51;&#110;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835257393\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835311257\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835351997\">\n<p id=\"fs-id1167831920536\">Find the first term and common difference of a sequence where the fourth term is 17 and the thirteenth term is 53. Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835352755\">\n<p id=\"fs-id1167835352757\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-07cf67dccc67f0251aaf82ef3da88741_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"54\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7395254a8a7e2922b12eaac8093e0888_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"46\" style=\"vertical-align: -1px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7e48ba7e1a4d29ae1a7b66b146f56b5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#52;&#110;&#43;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835376657\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835306222\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835367833\">\n<p id=\"fs-id1167835350706\">Find the first term and common difference of a sequence where the third term is 2 and the twelfth term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4e362d973b8b62a09dead6c6e5f8812d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"35\" style=\"vertical-align: 0px;\" \/> Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835226713\">\n<p id=\"fs-id1167835226716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-bc359b48baadbf26523a8e2221d82402_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#56;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8fb4bf4e0e359f436d2859e3c3a644f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d3f37cdeb841b7ecb83817313238b86_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#51;&#110;&#43;&#49;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1167835319496\">\n<h3 data-type=\"title\">Find the Sum of the First <em data-effect=\"italics\">n<\/em> Terms of an Arithmetic Sequence<\/h3>\n<p id=\"fs-id1167835353364\">As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a7a1b2fb13044ac205ba72c633be5e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -4px;\" \/> of the first <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;\" \/> terms of any arithmetic sequence is written as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4d401a137802bfb4bd607618ba29a88f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#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;&#46;&#46;&#46;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"219\" style=\"vertical-align: -4px;\" \/> To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.<\/p>\n<p id=\"fs-id1167834196804\">We can develop this new formula by first writing the sum by starting with the first term, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd6d5e05e9022f1ee8855acce1c293c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: -4px;\" \/> and keep adding a <em data-effect=\"italics\">d<\/em> to get the next term as:<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835595644\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6e25ff94aa569dd0fcce230d79fb304_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#50;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#100;&#111;&#116;&#115;&#32;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"323\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167834432133\">We can also reverse the order of the terms and write the sum by starting with <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;\" \/> and keep subtracting <em data-effect=\"italics\">d<\/em> to get the next term as<\/p>\n<div data-type=\"equation\" id=\"fs-id1167832065716\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-657c7685c002cdc88094482bae643970_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#45;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#45;&#50;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#100;&#111;&#116;&#115;&#32;&#43;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"326\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167826783889\">If we add these two expressions for the sum of the first <em data-effect=\"italics\">n<\/em> terms of an arithmetic sequence, we can derive a formula for the sum of the first <em data-effect=\"italics\">n<\/em> terms of any arithmetic series.<\/p>\n<div data-type=\"equation\" id=\"fs-id1167834190612\" class=\"unnumbered\" data-label=\"\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#117;&#110;&#100;&#101;&#114;&#115;&#101;&#116;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#95;&#125;&#125;&#123;&#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;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#50;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#43;&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#57;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#45;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#45;&#50;&#100;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#49;&#46;&#56;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#50;&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#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;&#61;&#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;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#48;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#100;&#111;&#116;&#115;&#32;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#43;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#123;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#43;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#125;\r\n\n<\/pre>\n<\/div>\n<p id=\"fs-id1167835236742\">Because there are <em data-effect=\"italics\">n<\/em> sums of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-da9a4580dc8c6f16ee0fe65be95d5a28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"69\" style=\"vertical-align: -4px;\" \/> on the right side of the equation, we rewrite the right side as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-83056f4be0506f62ce5cf505d58b2a92_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"92\" style=\"vertical-align: -4px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167835340841\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa88491c4c8c787cd41bfa25d7cc36de_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#110;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"136\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"fs-id1167835297398\">We divide by two to solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d7abec84c976c4fff9be3b7fd7f4ccde_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"24\" style=\"vertical-align: -3px;\" \/><\/p>\n<div data-type=\"equation\" id=\"fs-id1167835479285\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6f387f240a5512d3c85241eeb1e4f4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"129\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1167834184657\">This give us a general formula for the sum of the first <em data-effect=\"italics\">n<\/em> terms of an arithmetic sequence.<\/p>\n<div data-type=\"note\" id=\"fs-id1167835366292\">\n<div data-type=\"title\">Sum of the First <em data-effect=\"italics\">n<\/em> Terms of an Arithmetic Sequence<\/div>\n<p id=\"fs-id1167834064520\">The sum, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a7a1b2fb13044ac205ba72c633be5e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -4px;\" \/> of the first <em data-effect=\"italics\">n<\/em> terms of an arithmetic sequence is<\/p>\n<div data-type=\"equation\" id=\"fs-id1167835332490\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6f387f240a5512d3c85241eeb1e4f4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"129\" style=\"vertical-align: -6px;\" \/><\/div>\n<p id=\"fs-id1167835339098\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> is the first term and <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.<\/p>\n<\/div>\n<p id=\"fs-id1167831847164\">We apply this formula in the next example where the first few terms of the sequence are given.<\/p>\n<div data-type=\"example\" id=\"fs-id1167831847167\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835354048\">\n<div data-type=\"problem\" id=\"fs-id1167835354050\">\n<p id=\"fs-id1167835354052\">Find the sum of the first 30 terms of the arithmetic sequence: 8, 13, 18, 23, 28, \u2026<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835236244\">\n<p id=\"fs-id1167835355709\">To find the sum, we will use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-783fcbb99072ebcbf54b5c55fc20e6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"137\" style=\"vertical-align: -6px;\" \/> We know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2ec59e1f924513e496ad183953304f8e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"50\" style=\"vertical-align: -4px;\" \/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313a91835988d3729267cb47e609f373_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d61cf63405b4438ebac567523329209f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#51;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"56\" style=\"vertical-align: -4px;\" \/> but we need to find <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;\" \/> in order to use the sum formula.<\/p>\n<p id=\"fs-id1167835380271\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-274d336ddfe1ed68bb111ff6324add9e_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;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#100;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#70;&#105;&#110;&#100;&#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;&#123;&#97;&#125;&#95;&#123;&#110;&#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;&#119;&#104;&#101;&#114;&#101;&#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;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#56;&#44;&#100;&#61;&#53;&#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;&#97;&#110;&#100;&#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;&#110;&#61;&#51;&#48;&#46;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#51;&#48;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#56;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#48;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#53;&#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;&#123;&#97;&#125;&#95;&#123;&#51;&#48;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#56;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#57;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#97;&#125;&#95;&#123;&#51;&#48;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&#51;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#92;&#32;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#99;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#110;&#111;&#119;&#105;&#110;&#103;&#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;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#56;&#44;&#110;&#61;&#51;&#48;&#44;&#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;&#97;&#110;&#100;&#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;&#123;&#97;&#125;&#95;&#123;&#51;&#48;&#125;&#61;&#49;&#53;&#51;&#44;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#117;&#115;&#101;&#32;&#116;&#104;&#101;&#32;&#115;&#117;&#109;&#32;&#102;&#111;&#114;&#109;&#117;&#108;&#97;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#83;&#117;&#98;&#115;&#116;&#105;&#116;&#117;&#116;&#101;&#32;&#105;&#110;&#32;&#116;&#104;&#101;&#32;&#118;&#97;&#108;&#117;&#101;&#115;&#46;&#125;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#38;&#32;&#38;&#32;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#123;&#83;&#125;&#95;&#123;&#51;&#48;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#51;&#48;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#43;&#49;&#53;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#123;&#83;&#125;&#95;&#123;&#51;&#48;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#49;&#53;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#54;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#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;&#123;&#83;&#125;&#95;&#123;&#51;&#48;&#125;&#38;&#32;&#61;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#38;&#32;&#50;&#44;&#52;&#49;&#53;&#92;&#104;&#102;&#105;&#108;&#108;&#32;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"238\" width=\"547\" style=\"vertical-align: -114px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167826880408\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835335560\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835335562\">\n<p id=\"fs-id1167835335565\">Find the sum of the first 30 terms of the arithmetic sequence: 5, 9, 13, 17, 21, \u2026<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835338305\">\n<p id=\"fs-id1167835234738\">1,890<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834062236\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167834062240\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834062242\">\n<p id=\"fs-id1167835596050\">Find the sum of the first 30 terms of the arithmetic sequence: 7, 10, 13, 16, 19, \u2026<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537845\">\n<p id=\"fs-id1167834537847\">1,515<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835479176\">In the next example, we are given the general term for the sequence and are asked to find the sum of the first 50 terms.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835280727\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835280730\">\n<div data-type=\"problem\" id=\"fs-id1167835280732\">\n<p id=\"fs-id1167835230447\">Find the sum of the first 50 terms of the arithmetic 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-id1167835229121\">\n<p id=\"fs-id1167835229124\">To find the sum, we will use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-783fcbb99072ebcbf54b5c55fc20e6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"137\" style=\"vertical-align: -6px;\" \/> We know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02816f626e1bfc1c39ce6ad2603cab4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#53;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" style=\"vertical-align: -4px;\" \/> but we need to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> and <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;\" \/> in order to use the sum formula.<\/p>\n<table id=\"fs-id1167831908831\" summary=\"This figure shows how we find the sum of the formula a sub n equals 3 times n minus 4. First we find a sub 1 by substituting n equals 1 to get a sub 1 equals 3 times g times 1 minus 4. We then get a sub 1 equals negative 1. Next we find a sub n by substituting n equals 50 into the formula a sub n equals 3 n minus 4. We get a sub 50 equals 3 times g times 50 minus 4. Simplify to get a sub 50 equals 146. Knowing that n equals 50, a sub 1 equals negative 1, and a sub 50 equals 146, use the sum formula to get s sub n equals n divided by 2 times a sub 1 plus a sub n. Substitute the values to get s sub 50 equals 50 divided by 2 times negative 1 plus 146, and simplify to s sub 50 equals 25 times 145. Simplify again to get 3625.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834228893\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-fd6d5e05e9022f1ee8855acce1c293c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: -4px;\" \/> by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd7844b4064f00795a67aea525e1e439_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"47\" style=\"vertical-align: -1px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835241370\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\" data-align=\"left\">Find <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;\" \/> by substituting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-8e53dbc7bdc0c7e5d6b4ca17a64da6d1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#53;&#48;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835356180\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834064558\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Knowing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-02816f626e1bfc1c39ce6ad2603cab4e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#53;&#48;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-350cae271e17df8773db920224140980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#45;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"68\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2521c8005349f5741cdcabc33d1d833f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#53;&#48;&#125;&#61;&#49;&#52;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"75\" style=\"vertical-align: -3px;\" \/> use the sum formula.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834459182\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Substitute in the values.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167832075937\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835514385\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835319309\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_004h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835343611\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835343614\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832086944\">\n<p id=\"fs-id1167832086946\">Find the sum of the first 50 terms of the arithmetic 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-id1167835319256\">\n<p id=\"fs-id1167835319258\">2,300<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834449065\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835239514\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835239516\">\n<p id=\"fs-id1167835239518\">Find the sum of the first 50 terms of the arithmetic sequence whose general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a98033adb99867a384b8acb4491d626e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#52;&#110;&#43;&#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-id1167831847148\">\n<p id=\"fs-id1167835339146\">5,250<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835238526\">In the next example we are given the sum in summation notation. To add all the terms would be tedious, so we extract the information needed to use the formula to find the sum of the first <em data-effect=\"italics\">n<\/em> terms.<\/p>\n<div data-type=\"example\" id=\"fs-id1167835236622\" class=\"textbox textbox--examples\">\n<div data-type=\"exercise\" id=\"fs-id1167835236624\">\n<div data-type=\"problem\" id=\"fs-id1167835367026\">\n<p id=\"fs-id1167835367028\">Find the sum: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee32da051781b2761b0628d4ea1ee2bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#50;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#105;&#43;&#55;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"111\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835346145\">\n<p id=\"fs-id1167834433019\">To find the sum, we will use the formula <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-783fcbb99072ebcbf54b5c55fc20e6a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"137\" style=\"vertical-align: -6px;\" \/> We know <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-793ace59a77f29c480da18232c4270ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" style=\"vertical-align: -4px;\" \/> but we need to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> and <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;\" \/> in order to use the sum formula.<\/p>\n<table id=\"fs-id1167834422890\" summary=\"This figure shows how to expand the summation notation the sum of 4 sub i plus 7 from i equals 1 to 25. We simplify to equal 11 plus 15 plus 19 plus ellipsis plus 107. Next identify a sub 1 which equals 11 and a sub 25 equals 107. Knowing that n equals 25, a sub 1 equals 11, and a sub 25 equals 107, use the sum formula to get s sub n equals n divided by 2 times a sub 1 plus a sub n. Substitute the values to get s sub 25 equals 25 divided by 2 times 11 plus 107, and simplify to 25 divided by 2 times 118. Simplify again to get 1475.\" class=\"unnumbered unstyled\" data-label=\"\">\n<tbody>\n<tr>\n<td data-align=\"left\">Expand the summation notation.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835325691\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005a_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835307318\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005b_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Identify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0526020c87962a8a5a61ef3f77d6349c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: -4px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834539248\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005c_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Identify <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0a875a823996e90828a44639d10aafa1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#50;&#53;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"11\" width=\"28\" style=\"vertical-align: -3px;\" \/><\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831872401\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005d_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Knowing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-793ace59a77f29c480da18232c4270ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#50;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57565333a890dfd3573f3fe610f1240b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#49;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-97005d6de60200f8f6b7cc5129db27ea_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#50;&#53;&#125;&#61;&#49;&#48;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"75\" style=\"vertical-align: -3px;\" \/><span data-type=\"newline\"><br \/><\/span>use the sum formula.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167831872560\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005e_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Substitute in the values.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834345657\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005f_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167834557211\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005g_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<tr>\n<td data-align=\"left\">Simplify.<\/td>\n<td data-align=\"left\"><span data-type=\"media\" id=\"fs-id1167835379571\" data-alt=\".\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_005h_img.jpg\" data-media-type=\"image\/jpeg\" alt=\".\" \/><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167835381406\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167835381410\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835351308\">\n<p id=\"fs-id1167835351311\">Find the sum: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c6b34bbbba16d64232c391e753c654eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#51;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#105;&#45;&#52;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"111\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167832036168\">\n<p id=\"fs-id1167832036170\">2,670<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167831066311\" class=\"try\">\n<div data-type=\"exercise\" id=\"fs-id1167831066315\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831910744\">\n<p id=\"fs-id1167831910746\">Find the sum: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ecaeb14be9fa1d2c8f8f2c65f51ed674_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#51;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#53;&#105;&#45;&#51;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"111\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834183812\">\n<p id=\"fs-id1167834183814\">3,045<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"note\" id=\"fs-id1167834552520\" class=\"media-2\">\n<p id=\"fs-id1167834552524\">Access these online resources for additional instruction and practice with arithmetic sequences<\/p>\n<ul id=\"fs-id1167835479514\" data-display=\"block\">\n<li><a href=\"https:\/\/openstax.org\/l\/37ArithSequenc\">Arithmetic Sequences<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37AritSeqnthter\">Arithmetic Sequences: A Formula for the \u2018n-th\u2019 Term<\/a><\/li>\n<li><a href=\"https:\/\/openstax.org\/l\/37ArithSeries\">Arithmetic Series<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167835351252\">\n<h3 data-type=\"title\">Key Concepts<\/h3>\n<ul id=\"fs-id1167835351260\" data-bullet-style=\"bullet\">\n<li><strong data-effect=\"bold\">General Term (<em data-effect=\"italics\">n<\/em>th term) of an Arithmetic Sequence<\/strong><span data-type=\"newline\"><br \/><\/span> The general term of an arithmetic sequence with first term <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> and the common difference <em data-effect=\"italics\">d<\/em> is<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1167835360226\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cf70a8479d7942787021a2893d7d8126_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#110;&#45;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#100;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"149\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li><strong data-effect=\"bold\">Sum of the First <em data-effect=\"italics\">n<\/em> Terms of an Arithmetic Sequence<\/strong><span data-type=\"newline\"><br \/><\/span> The sum, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4a7a1b2fb13044ac205ba72c633be5e1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -4px;\" \/> of the first <em data-effect=\"italics\">n<\/em> terms of an arithmetic sequence, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-77648c75659a694d2bb0027b94be4c77_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: -4px;\" \/> is the first term and <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 is<span data-type=\"newline\"><br \/><\/span>\n<div data-type=\"equation\" id=\"fs-id1167834537731\" class=\"unnumbered\" data-label=\"\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b6f387f240a5512d3c85241eeb1e4f4f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#83;&#125;&#95;&#123;&#110;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#110;&#125;&#123;&#50;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#43;&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"129\" style=\"vertical-align: -6px;\" \/><\/div>\n<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox\" data-depth=\"1\" id=\"fs-id1167832036180\">\n<div class=\"practice-perfect\" data-depth=\"2\" id=\"fs-id1167832036184\">\n<h4 data-type=\"title\">Practice Makes Perfect<\/h4>\n<p id=\"fs-id1167834448987\"><strong data-effect=\"bold\">Determine if a Sequence is Arithmetic<\/strong><\/p>\n<p id=\"fs-id1167834537735\">In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834537739\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834537741\">\n<p id=\"fs-id1167834537743\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dd8177ac25dd862994c43670bdca6564_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#52;&#44;&#49;&#50;&#44;&#50;&#48;&#44;&#50;&#56;&#44;&#51;&#54;&#44;&#52;&#52;&#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=\"16\" width=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834552550\">\n<p id=\"fs-id1167834552552\">The sequence is arithmetic with common difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-35d38f1ec259104b14f50e7009f45345_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835344522\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835344524\">\n<p id=\"fs-id1167835344526\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-aa52101cf7fa6e58c914b0009e60cc97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#45;&#50;&#44;&#51;&#44;&#56;&#44;&#49;&#51;&#44;&#49;&#56;&#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=\"141\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830703159\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830703162\">\n<p id=\"fs-id1167834431030\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1309d145b9600406d03ccb2cea6b8d48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#44;&#45;&#49;&#54;&#44;&#51;&#44;&#49;&#50;&#44;&#50;&#49;&#44;&#51;&#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;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826996781\">\n<p id=\"fs-id1167826996783\">The sequence is not arithmetic.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831922087\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831922089\">\n<p id=\"fs-id1167831922091\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-353aa0f61378f3de7e562815509a16e0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#44;&#53;&#44;&#45;&#49;&#44;&#45;&#55;&#45;&#49;&#51;&#44;&#45;&#49;&#57;&#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=\"178\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835284952\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835332220\">\n<p id=\"fs-id1167835332222\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-71ca236a5d3731bb08d672a469108f49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#44;&#53;&#44;&#50;&#44;&#45;&#49;&#44;&#45;&#52;&#44;&#45;&#55;&#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=\"138\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831921263\">\n<p id=\"fs-id1167831921265\">The sequence is arithmetic with common difference <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b8fb4bf4e0e359f436d2859e3c3a644f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167832067132\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832067134\">\n<p id=\"fs-id1167832067136\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b272d698ebd116776aebdac1bf7e870b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#53;&#44;&#53;&#44;&#45;&#53;&#44;&#45;&#49;&#53;&#44;&#45;&#50;&#53;&#44;&#45;&#51;&#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=\"187\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167835280309\">In the following exercises, write the first five terms of each sequence with the given first term and common difference.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835280314\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835311941\">\n<p id=\"fs-id1167835311943\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d31b806337905424c62946186973e6a1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"58\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-6b51410672b8f8b8924fb023cafcee9f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834537714\">\n<p id=\"fs-id1167834537716\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-59a3b2f21827b19bc0a3fc85c3f106b9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#44;&#49;&#56;&#44;&#50;&#53;&#44;&#51;&#50;&#44;&#51;&#57;\" 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-id1167832087084\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167832087086\">\n<p id=\"fs-id1167832087088\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3dafa851db85c1da37e9a64e03be435b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#49;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-65212d32bffb37b0be80bcfd3c6a1e38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"42\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835340208\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835340210\">\n<p id=\"fs-id1167835340212\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dee9970536333a4b62d086ce021dad39_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#45;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"64\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7046b749c3c12381f487c8978b4fc364_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"42\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834063502\">\n<p id=\"fs-id1167834063504\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0da3d66ef81aac5411ba2a948fd97e38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#44;&#45;&#51;&#44;&#49;&#44;&#53;&#44;&#57;\" 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-id1167834301167\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834301169\">\n<p id=\"fs-id1167834301171\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-5644a9c972f3a6546dc1edde0dd3d92d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-313a91835988d3729267cb47e609f373_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834584370\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834584372\">\n<p id=\"fs-id1167831921597\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b121e0fd0d51eec89188c2fa4151a1a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"59\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4c476a43c765500f65c63a0f487fcb6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167826994172\">\n<p id=\"fs-id1167826994174\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e6d4391db3434701f08dcc8a4d9635c4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#52;&#44;&#53;&#44;&#45;&#52;&#44;&#45;&#49;&#51;&#44;&#45;&#50;&#50;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"142\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831920468\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831920470\">\n<p id=\"fs-id1167831920472\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f0b2dcc1fd4030f424280a7424c73a69_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"64\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-204dd4bdcdec3879abb8d0135a9491c8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826994248\"><strong data-effect=\"bold\">Find the General Term (<em data-effect=\"italics\">n<\/em>th Term) of an Arithmetic Sequence<\/strong><\/p>\n<p id=\"fs-id1167835515402\">In the following exercises, find the term described using the information provided.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167835515405\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835515407\">\n<p id=\"fs-id1167835515409\">Find the twenty-first term of a sequence where the first term is three and the common difference is eight.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835515414\">\n<p id=\"fs-id1167835515417\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9e27d329c6831cec0c0499927486f41f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834063455\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834063457\">\n<p id=\"fs-id1167834063459\">Find the twenty-third term of a sequence where the first term is six and the common difference is four.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831872319\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831872321\">\n<p id=\"fs-id1167831872323\">Find the thirtieth term of a sequence where the first term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4066673cb5834f8ce47cc6896c29963a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> and the common difference is five.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831872333\">\n<p id=\"fs-id1167831872335\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d4013094e0147c4c6192bcb6d85eaa7c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#51;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479014\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835479016\">\n<p id=\"fs-id1167835479018\">Find the fortieth term of a sequence where the first term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-0e22bf27371ae8e82f2cf9857cc59c29_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> and the common difference is seven.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835479491\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835479493\">\n<p id=\"fs-id1167835479495\">Find the sixteenth term of a sequence where the first term is 11 and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e45e9e13a8901226d7601cbfa15df84a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835479506\">\n<p id=\"fs-id1167835479508\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ccda9e46db63d8b18775183d7c8b666b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#55;&#57;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831921343\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831921345\">\n<p id=\"fs-id1167831921347\">Find the fourteenth term of a sequence where the first term is eight and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa36400e576c82fd4847d2da37b1d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835301613\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835301615\">\n<p id=\"fs-id1167835301617\">Find the twentieth term of a sequence where the fifth term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-00b9cce9021441b203ec0271d72e6ba2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"22\" style=\"vertical-align: -1px;\" \/> and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d36c434a1919e0aaa1a4125fdaa40853_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#50;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831071387\">\n<p id=\"fs-id1167831071389\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd4ea7b26c83c874d1bc867ec714e2b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#50;&#48;&#125;&#61;&#45;&#51;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"84\" style=\"vertical-align: -3px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c918b5e8328626b55c0ca9f0edcc26cb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#50;&#110;&#43;&#54;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"110\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835514340\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835514342\">\n<p id=\"fs-id1167835514344\">Find the thirteenth term of a sequence where the sixth term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7b34c01098c83fa602de54e9d74d63a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"21\" style=\"vertical-align: -1px;\" \/> and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e344351eb8a6a075b3d1df3943b3d637_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: -1px;\" \/> Give the formula for the general term.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835361599\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835361602\">\n<p id=\"fs-id1167835361604\">Find the eleventh term of a sequence where the third term is 19 and the common difference is five. Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835361609\">\n<p id=\"fs-id1167835361611\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-32ce0c63470557a82c574d80b08a4b1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#49;&#125;&#61;&#53;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"70\" style=\"vertical-align: -4px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-34793198804b9800b98fe8a464ffaab9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#53;&#110;&#43;&#52;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835332338\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835332340\">\n<p id=\"fs-id1167835332342\">Find the fifteenth term of a sequence where the tenth term is 17 and the common difference is seven. Give the formula for the general term.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835280338\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835280340\">\n<p id=\"fs-id1167835280342\">Find the eighth term of a sequence where the seventh term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ca32393b1b5af7c55a95d89cf9d610f2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-a304c08841a2d59337cd8ea9a3811e93_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"26\" style=\"vertical-align: 0px;\" \/> Give the formula for the general term.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834534420\">\n<p id=\"fs-id1167834534422\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cd3b3eaad41700b3f4d0ccd7e6047636_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#56;&#125;&#61;&#45;&#49;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"77\" style=\"vertical-align: -3px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-590e8e489e02d43c7578d94b98cf1b28_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#53;&#110;&#43;&#50;&#55;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835280403\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835280405\">\n<p id=\"fs-id1167835280407\">Find the fifteenth term of a sequence where the tenth term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2981688f5b26fd15fa08c897afa741ad_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"30\" style=\"vertical-align: -1px;\" \/> and the common difference is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-cfa36400e576c82fd4847d2da37b1d56_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"26\" style=\"vertical-align: 0px;\" \/> Give the formula for the general term.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167830705162\">In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167830705166\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830705169\">\n<p id=\"fs-id1167830705171\">The second term is 14 and the thirteenth term is 47.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830705175\">\n<p id=\"fs-id1167830705177\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-57565333a890dfd3573f3fe610f1240b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#49;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3de7278a447bec41bf67e06f751f5c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3bcdd3cb3acb460cc1209f7e27936879_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#51;&#110;&#43;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"96\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834194337\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834194339\">\n<p id=\"fs-id1167834194341\">The third term is 18 and the fourteenth term is 73.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167826996712\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826996714\">\n<p id=\"fs-id1167834063517\">The second term is 13 and the tenth term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3b9e967f33251314a2a8f19cc71cbd15_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#53;&#49;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"35\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834063527\">\n<p id=\"fs-id1167834063530\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-910104a0a86778e17372066f1f3d7202_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#50;&#49;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-dc47993a78dc950d063d1343c5bb9ef8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#45;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"60\" style=\"vertical-align: 0px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f40cb9ce80bcc20182d409a65b20e158_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#56;&#110;&#43;&#50;&#57;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"119\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834535636\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834535638\">\n<p id=\"fs-id1167834535641\">The third term is four and the tenth term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-227517817f1fe7ddaf0c1f6403e5c6d7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#56;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"31\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834194386\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834194388\">\n<p id=\"fs-id1167834194390\">The fourth term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-4797c874a138ca175d7c2cd8b3ed9a98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#54;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"22\" style=\"vertical-align: 0px;\" \/> and the fifteenth term is 27.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167834194400\">\n<p id=\"fs-id1167834194402\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1f0afdd28838616a4b13c86d6407ccb8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#49;&#125;&#61;&#45;&#49;&#53;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\" \/><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-3de7278a447bec41bf67e06f751f5c8d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#100;&#61;&#51;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"46\" style=\"vertical-align: 0px;\" \/> The general term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-c40df129369d2f5bd390ba8dc7613c91_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#51;&#110;&#45;&#49;&#56;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"105\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835360260\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835360262\">\n<p id=\"fs-id1167835360264\">The third term is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ee7bea24371f97fb5673002404153494_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> and the seventeenth term is 15.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167831872364\"><strong data-effect=\"bold\">Find the Sum of the First <em data-effect=\"italics\">n<\/em> Terms of an Arithmetic Sequence<\/strong><\/p>\n<p id=\"fs-id1167831872373\">In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167831872377\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831872379\">\n<p id=\"fs-id1167831872381\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-e9c10b713ebc19b9aee67897320a2a8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#49;&#44;&#49;&#52;&#44;&#49;&#55;&#44;&#50;&#48;&#44;&#50;&#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=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167827967193\">\n<p id=\"fs-id1167827967195\">1,635<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167827967200\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167827967202\">\n<p id=\"fs-id1167827967204\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9c4bd072eccd1d053412b96352b006d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#50;&#44;&#49;&#56;&#44;&#50;&#52;&#44;&#51;&#48;&#44;&#51;&#54;&#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=\"16\" width=\"123\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835515365\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835515367\">\n<p id=\"fs-id1167835515369\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-41b3266ebfc5789ccb04fca53ebd81ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#44;&#53;&#44;&#50;&#44;&#45;&#49;&#44;&#45;&#52;&#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=\"108\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835352290\">\n<p id=\"fs-id1167835352292\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-d740eebadd3c37cd7743ddfc587784ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#44;&#48;&#54;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835360173\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835360175\">\n<p id=\"fs-id1167835360177\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-b5cece0fe445d6fa0f4f06476de87b58_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#54;&#44;&#49;&#48;&#44;&#52;&#44;&#45;&#50;&#44;&#45;&#56;&#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=\"16\" width=\"124\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834449015\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834449017\">\n<p id=\"fs-id1167834449019\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1391e94c8cdde6375100d70700bef541_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#55;&#44;&#45;&#49;&#53;&#44;&#45;&#49;&#51;&#44;&#45;&#49;&#49;&#44;&#45;&#57;&#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=\"184\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835589628\">\n<p id=\"fs-id1167835589630\">360<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835589635\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835589637\">\n<p id=\"fs-id1167835589639\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-08e9c206afbb3717720b346bc6215e7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#49;&#53;&#44;&#45;&#49;&#50;&#44;&#45;&#57;&#44;&#45;&#54;&#44;&#45;&#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=\"166\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167826996724\">In the following exercises, find the sum of the first 50 terms of the arithmetic sequence whose general term is given.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167826996728\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167826996730\">\n<p id=\"fs-id1167826996732\"><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 data-type=\"solution\" id=\"fs-id1167826996754\">\n<p id=\"fs-id1167826996756\">6,325<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448936\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834448938\">\n<p id=\"fs-id1167834448941\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-1a6666489af8a6528f3083df5f0b3811_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#50;&#110;&#43;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"92\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167834448969\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834448972\">\n<p id=\"fs-id1167834448974\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-039b892710d54abd14c52aad4ca7bc0c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#51;&#110;&#43;&#53;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"105\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831025348\">\n<p id=\"fs-id1167831025350\">\u20133,575<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831025355\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831025357\">\n<p id=\"fs-id1167831025360\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-090922418d72cee5e2078098b87a8ab4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#61;&#45;&#52;&#110;&#43;&#51;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"106\" style=\"vertical-align: -3px;\" \/><\/p>\n<\/div>\n<\/div>\n<p id=\"fs-id1167834583346\">In the following exercises, find each sum.<\/p>\n<div data-type=\"exercise\" id=\"fs-id1167834583349\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834583351\">\n<p id=\"fs-id1167834583353\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-ea30a9b6241e950116e7abd41436c64e_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;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#105;&#45;&#55;&#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-id1167830704023\">\n<p id=\"fs-id1167830704025\">6,280<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830704030\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830704032\">\n<p id=\"fs-id1167830704034\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-f7beae5bd9b92216ecd9100958889f2b_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;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#105;&#45;&#53;&#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-id1167835284910\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835284912\">\n<p id=\"fs-id1167835284914\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9ce7c2392c55afc5ff1403fa22b6b77c_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;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#51;&#105;&#43;&#54;&#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-id1167835511978\">\n<p id=\"fs-id1167835511980\">4,125<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835511985\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835511988\">\n<p id=\"fs-id1167835511990\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-7d0743a2d6303e035496e294f1f0a948_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#50;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#52;&#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>\n<div data-type=\"exercise\" id=\"fs-id1167834583425\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167834583427\">\n<p id=\"fs-id1167834583429\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-2901cd2568bb9a9ebdc28d412301ec49_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#51;&#53;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#54;&#105;&#45;&#50;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"117\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167835512721\">\n<p id=\"fs-id1167835512723\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-9bcb39830d5e6dac48c9f20cff88a6db_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#51;&#44;&#56;&#53;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"57\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167835512736\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167835512738\">\n<p id=\"fs-id1167835512740\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-16269436a7ec52dae277ab3b47328c7e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#117;&#109;&#32;&#95;&#123;&#105;&#61;&#49;&#125;&#94;&#123;&#51;&#48;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#53;&#105;&#43;&#49;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"117\" style=\"vertical-align: -6px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"writing\" data-depth=\"2\" id=\"fs-id1167830703940\">\n<h4 data-type=\"title\">Writing Exercises<\/h4>\n<div data-type=\"exercise\" id=\"fs-id1167830703947\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167830703950\">\n<p id=\"fs-id1167830703952\">In your own words, explain how to determine whether a sequence is arithmetic.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167830703956\">\n<p id=\"fs-id1167830703958\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167830703963\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831066248\">\n<p id=\"fs-id1167831066250\">In your own words, explain how the first two terms are used to find the tenth term. Show an example to illustrate your explanation.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831066263\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831066265\">\n<p id=\"fs-id1167831066267\">In your own words, explain how to find the general term of an arithmetic sequence.<\/p>\n<\/div>\n<div data-type=\"solution\" id=\"fs-id1167831066271\">\n<p id=\"fs-id1167831066273\">Answers will vary.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" id=\"fs-id1167831066279\" class=\"material-set-2\">\n<div data-type=\"problem\" id=\"fs-id1167831066281\">\n<p id=\"fs-id1167831066283\">In your own words, explain how to find the sum of the first <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;\" \/> terms of an arithmetic sequence without adding all the terms.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"2\" id=\"fs-id1167831066300\">\n<h4 data-type=\"title\">Self Check<\/h4>\n<p id=\"fs-id1167835370651\"><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-id1167835370667\" data-alt=\"This figure shows a chart with four rows and four columns. The first row is the header row and reads, &#x201c;I can&#x201d;, &#x201c;Confidently&#x201d;, &#x201c;With some help&#x201d;, and &#x201c;No, I don&#x2019;t get it!&#x201d; The column, beginning with second row reads 1. Determine if a Sequence is Arithmetic, 2. Find the General Term (nth term) of Arithmetic Sequence, and 3. Find the sum of the first Terms of an Arithmetic Sequence&#x201d;. The remaining columns are blank.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/uploads\/sites\/599\/2019\/09\/CNX_IntAlg_Figure_12_02_201_img.jpg\" data-media-type=\"image\/jpeg\" alt=\"This figure shows a chart with four rows and four columns. The first row is the header row and reads, &#x201c;I can&#x201d;, &#x201c;Confidently&#x201d;, &#x201c;With some help&#x201d;, and &#x201c;No, I don&#x2019;t get it!&#x201d; The column, beginning with second row reads 1. Determine if a Sequence is Arithmetic, 2. Find the General Term (nth term) of Arithmetic Sequence, and 3. Find the sum of the first Terms of an Arithmetic Sequence&#x201d;. The remaining columns are blank.\" \/><\/span><\/p>\n<p id=\"fs-id1167835370675\"><span class=\"token\">\u24d1<\/span> After reviewing this checklist, what will you do to become confident for all objectives?<\/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-id1167835370688\">\n<dt>arithmetic sequence<\/dt>\n<dd id=\"fs-id1167835370694\">An arithmetic sequence is a sequence where the difference between consecutive terms is constant.<\/dd>\n<\/dl>\n<dl id=\"fs-id1167835370699\">\n<dt>common difference<\/dt>\n<dd id=\"fs-id1167835370704\">The difference between consecutive terms in an arithmetic sequence, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-content\/ql-cache\/quicklatex.com-89e3f4cf74e1b833f8b956476ff8239c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#97;&#125;&#95;&#123;&#110;&#125;&#45;&#123;&#97;&#125;&#95;&#123;&#110;&#45;&#49;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"80\" style=\"vertical-align: -4px;\" \/> is <em data-effect=\"italics\">d<\/em>, the common difference, for <em data-effect=\"italics\">n<\/em> greater than or equal to two.<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":103,"menu_order":3,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-15719","chapter","type-chapter","status-publish","hentry"],"part":15662,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapters\/15719","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\/15719\/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\/15719\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/media?parent=15719"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/pressbooks\/v2\/chapter-type?post=15719"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/contributor?post=15719"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/algebraintermediate\/wp-json\/wp\/v2\/license?post=15719"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}