Sequences, Series and Binomial Theorem

# Binomial Theorem

### Learning Objectives

By the end of this section, you will be able to:

• Use Pascal’s Triangle to expand a binomial
• Evaluate a binomial coefficient
• Use the Binomial Theorem to expand a binomial

Before you get started, take this readiness quiz.

1. Simplify:
If you missed this problem, review (Figure).
2. Expand:
If you missed this problem, review (Figure).
3. Expand:
If you missed this problem, review (Figure).

### Use Pascal’s Triangle to Expand a Binomial

In our previous work, we have squared binomials either by using FOIL or by using the Binomial Squares Pattern. We can also say that we expanded

To expand we recognize that this is and multiply.

To find a method that is less tedious that will work for higher expansions like we again look for patterns in some expansions.

Number of terms First term Last term
2
3
4
5
6
n

Notice the first and last terms show only one variable. Recall that so we could rewrite the first and last terms to include both variables. For example, we could expand to show each term with both variables.

Generally, we don’t show the zero exponents, just as we usually write x rather than 1x.

Patterns in the expansion of
• The number of terms is
• The first term is and the last term is
• The exponents on a decrease by one on each term going left to right.
• The exponents on b increase by one on each term going left to right.
• The sum of the exponents on any term is n.

Let’s look at an example to highlight the last three patterns.

From the patterns we identified, we see the variables in the expansion of would be

To find the coefficients of the terms, we write our expansions again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients.

The array to the right is called Pascal’s Triangle. Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.

This triangle gives the coefficients of the terms when we expand binomials.

Pascal’s Triangle

In the next example, we will use this triangle and the patterns we recognized to expand the binomial.

Use Pascal’s Triangle to expand

We know the variables for this expansion will follow the pattern we identified. The nonzero exponents of x will start at six and decrease to one. The nonzero exponents of y will start at one and increase to six. The sum of the exponents in each term will be six. In our pattern, and

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\begin{array}{c}{\left(a+b\right)}^{n}={a}^{n}+___{a}^{n-1}{b}^{1}+___{a}^{n-2}{b}^{2}+...+___{a}^{1}{b}^{n-1}+{b}^{n}\hfill \\ {\left(x+y\right)}^{6}={x}^{6}+___{x}^{5}{y}^{1}+___{x}^{4}{y}^{2}+___{x}^{3}{y}^{3}+___{x}^{2}{y}^{4}+___{x}^{1}{y}^{5}+{y}^{6}\hfill \end{array}

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Use Pascal’s Triangle to expand

Use Pascal’s Triangle to expand

In the next example we want to expand a binomial with one variable and one constant. We need to identify the a and b to carefully apply the pattern.

Use Pascal’s Triangle to expand

We identify the a and b of the pattern.

In our pattern, and

We know the variables for this expansion will follow the pattern we identified. The sum of the exponents in each term will be five.

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\begin{array}{c}{\left(a+b\right)}^{n}={a}^{n}+___{a}^{n-1}{b}^{1}+___{a}^{n-2}{b}^{2}+...+___{a}^{1}{b}^{n-1}+{b}^{n}\\ {\left(x+3\right)}^{5}={x}^{5}+___{x}^{4}·{3}^{1}+___{x}^{3}·{3}^{2}+___{x}^{2}·{3}^{3}+___{x}^{1}·{3}^{4}+{3}^{5}\end{array}

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Use Pascal’s Triangle to expand

Use Pascal’s Triangle to expand

In the next example, the binomial is a difference and the first term has a constant times the variable. Once we identify the a and b of the pattern, we must once again carefully apply the pattern.

Use Pascal’s Triangle to expand

We identify the a and b of the pattern.

In our pattern, and

Use Pascal’s Triangle to expand

Use Pascal’s Triangle to expand

### Evaluate a Binomial Coefficient

While Pascal’s Triangle is one method to expand a binomial, we will also look at another method. Before we get to that, we need to introduce some more factorial notation. This notation is not only used to expand binomials, but also in the study and use of probability.

To find the coefficients of the terms of expanded binomials, we will need to be able to evaluate the notation which is called a binomial coefficient. We read as “n choose r” or “n taken r at a time”.

Binomial Coefficient

A binomial coefficient where r and n are integers with is defined as

We read as “n choose r” or “n taken r at a time”.

Evaluate:

We will use the definition of a binomial coefficient,

Evaluate each binomial coefficient:

6 1 1 35

Evaluate each binomial coefficient:

2 1 1 6

In the previous example, parts (a), (b), (c) demonstrate some special properties of binomial coefficients.

Properties of Binomial Coefficients

### Use the Binomial Theorem to Expand a Binomial

We are now ready to use the alternate method of expanding binomials. The Binomial Theorem uses the same pattern for the variables, but uses the binomial coefficient for the coefficient of each term.

Binomial Theorem

For any real numbers a and b, and positive integer n,

Use the Binomial Theorem to expand

We identify the a and b of the pattern.

In our pattern, and

We use the Binomial Theorem.

Substitute in the values and

Simplify the exponents.

Evaluate the coefficients. Remember,

Use the Binomial Theorem to expand

Use the Binomial Theorem to expand

Notice that when we expanded in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal’s Triangle.

The next example, the binomial is a difference. When the binomial is a difference, we must be careful in identifying the values we will use in the pattern.

Use the Binomial Theorem to expand

We identify the a and b of the pattern.

In our pattern, and

We use the Binomial Theorem.

Substitute in the values and

Simplify the exponents and evaluate the coefficients. Remember,

Use the Binomial Theorem to expand

Use the Binomial Theorem to expand

Things can get messy when both terms have a coefficient and a variable.

Use the Binomial Theorem to expand

We identify the a and b of the pattern.

In our pattern, and

We use the Binomial Theorem.

Substitute in the values and

Simplify the exponents.

Evaluate the coefficients. Remember,

Use the Binomial Theorem to expand

Use the Binomial Theorem to expand

The real beauty of the Binomial Theorem is that it gives a formula for any particular term of the expansion without having to compute the whole sum. Let’s look for a pattern in the Binomial Theorem.

Notice, that in each case the exponent on the b is one less than the number of the term. The term is the term where the exponent of b is r. So we can use the format of the term to find the value of a specific term.

Find a Specific Term in a Binomial Expansion

The term in the expansion of is

Find the fourth term of

 In our pattern, and We are looking for the fourth term. Write the formula. Substitute in the values, and Simplify. Simplify.

Find the third term of

Find the fifth term of

Find the coefficient of the term of

 In our pattern, then and We are looking for the coefficient of the term.    Since and we know Write the formula. Substitute in the values, and Simplify. Simplify. Simplify. The coefficient of the term is 2268.

Find the coefficient of the term of

7,168

Find the coefficient of the term of

280

Access these online resources for additional instruction and practice with sequences.

### Key Concepts

• Patterns in the expansion of
• The number of terms is
• The first term is and the last term is
• The exponents on a decrease by one on each term going left to right.
• The exponents on b increase by one on each term going left to right.
• The sum of the exponents on any term is n.
• Pascal’s Triangle
• Binomial Coefficient: A binomial coefficient where r and n are integers with is defined as

We read as “n choose r” or “n taken r at a time”.

• Properties of Binomial Coefficients
• Binomial Theorem: For any real numbers a, b, and positive integer n,

### Section Exercises

#### Practice Makes Perfect

Use Pascal’s Triangle to Expand a Binomial

In the following exercises, expand each binomial using Pascal’s Triangle.

Evaluate a Binomial Coefficient

In the following exercises, evaluate.

7 1 1 45

4 1 1 55

Use the Binomial Theorem to Expand a Binomial

In the following exercises, expand each binomial.

In the following exercises, find the indicated term in the expansion of the binomial.

Sixth term of

Fifth term of

Fourth term of

Seventh term of

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.

term of

term of

112

term of

term of

324

term of

term of

30,618

#### Writing Exercises

In your own words explain how to find the rows of the Pascal’s Triangle. Write the first five rows of Pascal’s Triangle.

In your own words, explain the pattern of exponents for each variable in the expansion of.

In your own words, explain the difference between and

In your own words, explain how to find a specific term in the expansion of a binomial without expanding the whole thing. Use an example to help explain.

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

### Chapter Review Exercises

#### Sequences

Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

Find a Formula for the General Term (nth Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

Use Summation Notation to write a Sum

In the following exercises, write each sum using summation notation.

#### Arithmetic Sequences

Determine if a Sequence is Arithmetic

In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.

The sequence is arithmetic with common difference

In the following exercises, write the first five terms of each arithmetic sequence with the given first term and common difference.

and

and

and

Find the General Term (nth Term) of an Arithmetic Sequence

In the following exercises, find the term described using the information provided.

Find the twenty-fifth term of a sequence where the first term is five and the common difference is three.

Find the thirtieth term of a sequence where the first term is 16 and the common difference is

Find the seventeenth term of a sequence where the first term is and the common difference is two.

In the following exercises, find the indicated term and give the formula for the general term.

Find the eighteenth term of a sequence where the fifth term is and the common difference is seven.

The general term is

Find the twenty-first term of a sequence where the seventh term is and the common difference is

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.

The fifth term is 17 and the fourteenth term is 53.

The general term is

The third term is and the sixteenth term is .

Find the Sum of the First n Terms of an Arithmetic Sequence

In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.

In the following exercises, find the sum of the first fifteen terms of the arithmetic sequence whose general term is given.

In the following exercises, find each sum.

#### Geometric Sequences and Series

Determine if a Sequence is Geometric

In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.

The sequence is not geometric.

The sequence is geometric with common ratio

In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.

and

and

and

Find the General Term (nth Term) of a Geometric Sequence

In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.

Find given and

Find given and

In the following exercises, find the indicated term of the given sequence. Find the general term of the sequence.

Find of the sequence,

The general term is

Find of the sequence,

Find the Sum of the First n terms of a Geometric Sequence

In the following exercises, find the sum of the first fifteen terms of each geometric sequence.

5,460

In the following exercises, find the sum

Find the Sum of an Infinite Geometric Series

In the following exercises, find the sum of each infinite geometric series.

In the following exercises, write each repeating decimal as a fraction.

Apply Geometric Sequences and Series in the Real World

In the following exercises, solve the problem.

What is the total effect on the economy of a government tax rebate of to each household in order to stimulate the economy if each household will spend of the rebate in goods and services?

Adam just got his first full-time job after graduating from high school at age 17. He decided to invest per month in an IRA (an annuity). The interest on the annuity is which is compounded monthly. How much will be in Adam’s account when he retires at his sixty-seventh birthday?

#### Binomial Theorem

Use Pascal’s Triangle to Expand a Binomial

In the following exercises, expand each binomial using Pascal’s Triangle.

Evaluate a Binomial Coefficient

In the following exercises, evaluate.

11 1 1 56

1 1 1 55

Use the Binomial Theorem to Expand a Binomial

In the following exercises, expand each binomial, using the Binomial Theorem.

In the following exercises, find the indicated term in the expansion of the binomial.

Seventh term of

Third term of

In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.

term of

135

term of

term of

280

### Practice Test

In the following exercises, write the first five terms of the sequence whose general term is given.

Find a general term for the sequence,

Expand the partial sum and find its value.

Write the following using summation notation.

Write the first five terms of the arithmetic sequence with the given first term and common difference. and

Find the twentieth term of an arithmetic sequence where the first term is two and the common difference is

Find the twenty-third term of an arithmetic sequence whose seventh term is and common difference is three. Then find a formula for the general term.

The general term is

Find the first term and common difference of an arithmetic sequence whose ninth term is and the sixteenth term is Then find a formula for the general term.

Find the sum of the first 25 terms of the arithmetic sequence,

Find the sum of the first 50 terms of the arithmetic sequence whose general term is

Find the sum.

In the following exercises, determine if the sequence is arithmetic, geometric, or neither. If arithmetic, then find the common difference. If geometric, then find the common ratio.

The sequence is geometric with common ratio

Write the first five terms of the geometric sequence with the given first term and common ratio. and

In the geometric sequence whose first term and common ratio are and find

Find of the geometric sequence, Then find a formula for the general term.

Find the sum of the first thirteen terms of the geometric sequence,

In the following exercises, find the sum.

Write the repeating decimal as a fraction.

Dave just got his first full-time job after graduating from high school at age 18. He decided to invest ?450 per month in an IRA (an annuity). The interest on the annuity is 6% which is compounded monthly. How much will be in Adam’s account when he retires at his sixty-fifth birthday?

Expand the binomial using Pascal’s Triangle.

Evaluate each binomial coefficient.

8 1 1 210

Expand the binomial using the Binomial Theorem.