Contents

Preface
1Foundations
- Introduction
- 1.1Use the Language of Algebra
- 1.2Integers
- 1.3Fractions
- 1.4Decimals
- 1.5Properties of Real Numbers
- Chapter Review
  Key Terms
  Key Concepts
- Exercises
  Review Exercises
  Practice Test
2Solving Linear Equations
3Graphs and Functions
4Systems of Linear Equations
5Polynomials and Polynomial Functions
6Factoring
7Rational Expressions and Functions
8Roots and Radicals
9Quadratic Equations and Functions
10Exponential and Logarithmic Functions
11Conics
12Sequences, Series and Binomial Theorem
- Introduction
- 12.1Sequences
- 12.2Arithmetic Sequences
- 12.3Geometric Sequences and Series
- 12.4Binomial Theorem
- Chapter Review
  Key Terms
Answer Key
Index

Geometric Sequences and Series

Learning Objectives

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

Determine if a sequence is geometric
Find the general term (nth term) of a geometric sequence
Find the sum of the first $n n$ terms of a geometric sequence
Find the sum of an infinite geometric series
Apply geometric sequences and series in the real world

Be Prepared 12.3

Before you get started, take this readiness quiz.

Simplify: $\frac{24}{32} . \frac{24}{32} .$
If you missed this problem, review Example 1.24.
Evaluate: ⓐ $3^{4} 3^{4}$ ⓑ ${(\frac{1}{2})}^{4} . {(\frac{1}{2})}^{4} .$
If you missed this problem, review Example 1.19.
If $f (x) = 4 \cdot 3^{x}, f (x) = 4 \cdot 3^{x},$ find ⓐ $f (1) f (1)$ ⓑ $f (2) f (2)$ ⓒ $f (3) . f (3) .$
If you missed this problem, review Example 3.49.

Determine if a Sequence is Geometric

We are now ready to look at the second special type of sequence, the geometric sequence.

A sequence is called a geometric sequence if the ratio between consecutive terms is always the same. The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.

Geometric Sequence

A geometric sequence is a sequence where the ratio between consecutive terms is always the same.

The ratio between consecutive terms, $\frac{a_{n}}{a_{n - 1}}, \frac{a_{n}}{a_{n - 1}},$ is r, the common ratio. n is greater than or equal to two.

Consider these sequences.

This figure shows two sets of sequences where r is the common ratio.

Example 12.21

Determine if each sequence is geometric. If so, indicate the common ratio.

ⓐ $4, 8, 16, 32, 64, 128, … 4, 8, 16, 32, 64, 128,$

ⓑ $−2, 6, −12, 36, −72, 216, … −2, 6, −12, 36, −72, 216,$

Show solution

Solution

To determine if the sequence is geometric, we find the ratio of the consecutive terms shown.

ⓐ
$\begin{matrix} \begin{matrix} Find the ratio of \\ the consecutive terms. \end{matrix} & \begin{matrix} 4, & 8, & 16, & 32, & 64, & 128, … \\ \frac{8}{4} & \frac{16}{8} & \frac{32}{16} & \frac{64}{32} & \frac{128}{64} \\ 2 & 2 & 2 & 2 & 2 \end{matrix} \\ The sequence is geometric. The common ratio is r = 2 . \end{matrix} \begin{matrix} \begin{matrix} Find the ratio of & the consecutive terms. \end{matrix} \end{matrix}$

ⓑ
$\begin{matrix} \begin{matrix} Find the ratio of \\ the consecutive terms. \end{matrix} & \begin{matrix} −2, & 6, & −12, & 36, & −72, & 216, … \\ \frac{6}{−2} & \frac{−12}{6} & \frac{36}{−12} & \frac{−72}{36} & \frac{216}{−72} \\ −3 & −2 & −3 & −2 & −3 \end{matrix} \\ The sequence is not geometric. There is no common ratio. \end{matrix} \begin{matrix} \begin{matrix} Find the ratio of & the consecutive terms. \end{matrix} \end{matrix}$

ⓒ
$\begin{matrix} 27, 9, 3, 1, \frac{1}{3}, \frac{1}{9}, … \\ \begin{matrix} Find the ratio of \\ the consecutive terms. \end{matrix} & \begin{matrix} \frac{9}{27} & \frac{3}{9} & \frac{1}{3} & \frac{\frac{1}{3}}{1} & \frac{\frac{1}{9}}{\frac{1}{3}} \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \end{matrix} \\ The sequence is geometric. The common ratio is r = \frac{1}{3} . \end{matrix} \begin{matrix} \begin{matrix} Find the ratio of \\ the consecutive terms. \end{matrix} \end{matrix}$

Try It 12.41

Determine if each sequence is geometric. If so indicate the common ratio.

ⓐ $7, 21, 63, 189, 567, 1,701, … 7, 21, 63, 189, 567, 1,701,$

ⓑ $64, 16, 4, 1, \frac{1}{4}, \frac{1}{16}, … 64, 16, 4, 1, \frac{1}{4}, \frac{1}{16},$

Try It 12.42

Determine if each sequence is geometric. If so indicate the common ratio.

ⓐ $−150, −30, −15, −5, - \frac{5}{2}, 0, … −150, −30, −15, −5, - \frac{5}{2}, 0,$

ⓑ $5, 10, 20, 40, 80, 160, … 5, 10, 20, 40, 80, 160,$

If we know the first term, $a_{1}, a_{1},$ and the common ratio, r, we can list a finite number of terms of the sequence.

Example 12.22

Write the first five terms of the sequence where the first term is 3 and the common ratio is $r = −2 . r = −2 .$

Show solution

Solution

We start with the first term and multiply it by the common ratio. Then we multiply that result by the common ratio to get the next term, and so on.

\begin{matrix} a_{1} & a_{2} & a_{3} & a_{4} & a_{5} \\ 3 & 3 \cdot (−2) & -6\cdot(-2) & 12\cdot(-2) & -24\cdot(-2) \\ −6 & 12 & −24 & 48 \end{matrix} \begin{matrix} a_{1} & a_{2} a_{3} a_{4} a_{5} \\ 3 & 3 \cdot (−2) -6\cdot(-2) 12\cdot(-2) -24\cdot(-2) \\ −6 12 −24 48 \end{matrix}

The sequence is $3, −6, 12, −24, 48, … 3, −6, 12, −24, 48,$

Try It 12.43

Write the first five terms of the sequence where the first term is 7 and the common ratio is $r = −3 . r = −3 .$

Try It 12.44

Write the first five terms of the sequence where the first term is 6 and the common ratio is $r = −4 . r = −4 .$

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

Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence.

Let’s write the first few terms of the sequence where the first term is $a_{1} a_{1}$ and the common ratio is r. We will then look for a pattern.

This figure shows an image of a geometric sequence.

As we look for a pattern in the five terms above, we see that each of the terms starts with $a_{1} . a_{1} .$

The first term, $a_{1}, a_{1},$ is not multiplied by any r. In the second term, the $a_{1} a_{1}$ is multiplied by r. In the third term, the $a_{1} a_{1}$ is multiplied by r two times ( $r \cdot r r \cdot r$ or $r^{2} r^{2}$ ). In the fourth term, the $a_{1} a_{1}$ is multiplied by r three times ( $r \cdot r \cdot r r \cdot r \cdot r$ or $r^{3} r^{3}$ ) and in the fifth term, the $a_{1} a_{1}$ is multiplied by r four times. In each term, the number of times $a_{1} a_{1}$ is multiplied by r is one less than the number of the term. This leads us to the following

a_{n} = a_{1} r^{n - 1} a_{n} = a_{1} r^{n - 1}

General Term (nth term) of a Geometric Sequence

The general term of a geometric sequence with first term $a_{1} a_{1}$ and the common ratio r is

a_{n} = a_{1} r^{n - 1} a_{n} = a_{1} r^{n - 1}

We will use this formula in the next example to find the fourteenth term of a sequence.

Example 12.23

Find the fourteenth term of a sequence where the first term is 64 and the common ratio is $r = \frac{1}{2} . r = \frac{1}{2} .$

Show solution

Solution

$\begin{matrix} To find the fourteenth term, a_{14}, & a_{n} = a_{1} r^{n - 1} \\ use the formula with a_{1} = 64 and r = \frac{1}{2} . \\ Substitute in the values. & a_{14} = 64 {(\frac{1}{2})}^{14 - 1} \\ Simplify. & a_{14} = 64 {(\frac{1}{2})}^{13} \\ a_{14} = \frac{1}{128} \end{matrix} \begin{matrix} To find the fourteenth term, \\ use the formula with \\ Substitute in the values. \\ Simplify. \end{matrix}$

Try It 12.45

Find the thirteenth term of a sequence where the first term is 81 and the common ratio is $r = \frac{1}{3} . r = \frac{1}{3} .$

Try It 12.46

Find the twelfth term of a sequence where the first term is 256 and the common ratio is $r = \frac{1}{4} . r = \frac{1}{4} .$

Sometimes we do not know the common ratio and we must use the given information to find it before we find the requested term.

Example 12.24

Find the twelfth term of the sequence 3, 6, 12, 24, 48, 96, … Find the general term for the sequence.

Show solution

Solution

To find the twelfth term, we use the formula, $a_{n} = a_{1} r^{n - 1}, a_{n} = a_{1} r^{n - 1},$ and so we need to first determine $a_{1} a_{1}$ and the common ratio r.

$\begin{matrix} 3, 6, 12, 24, 48, 96, … \\ The first term is three. & a_{1} = 3 \\ Find the common ratio. & \begin{matrix} \frac{6}{3} & \frac{12}{6} & \frac{24}{12} & \frac{48}{24} & \frac{96}{48} \\ 2 & 2 & 2 & 2 & 2 \end{matrix} \\ The common ratio is r = 2 . \\ To find the twelfth term, a_{12}, use the & a_{n} = a_{1} r^{n - 1} \\ formula with a_{1} = 3 and r = 2 . \\ Substitute in the values. & a_{12} = 3 \cdot 2^{12 - 1} \\ Simplify. & a_{12} = 3 \cdot 2^{11} \\ a_{12} = 6,144 \\ Find the general term. & a_{n} = a_{1} r^{n - 1} \\ We use the formula with a_{1} = 3 and r = 2 . & a_{n} = 3 {(2)}^{n - 1} \end{matrix} \begin{matrix} The first term is three. \\ Find the common ratio. & To find the twelfth term, & formula with & Substitute in the values. & Simplify. & Find the general term. & We use the formula with \end{matrix}$

Try It 12.47

Find the ninth term of the sequence 6, 18, 54, 162, 486, 1,458, … Then find the general term for the sequence.

Try It 12.48

Find the eleventh term of the sequence 7, 14, 28, 56, 112, 224, … Then find the general term for the sequence.

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

We found the sum of both general sequences and arithmetic sequence. We will now do the same for geometric sequences. The sum, $S_{n}, S_{n},$ of the first n terms of a geometric sequence is written as $S_{n} = a_{1} + a_{2} + a_{3} + ... + a_{n} . S_{n} = a_{1} + a_{2} + a_{3} + ... + a_{n} .$ We can write this sum by starting with the first term, $a_{1}, a_{1},$ and keep multiplying by r to get the next term as:

S_{n} = a_{1} + a_{1} r + a_{1} r^{2} + ... + a_{1} r^{n - 1} S_{n} = a_{1} + a_{1} r + a_{1} r^{2} + ... + a_{1} r^{n - 1}

Let’s also multiply both sides of the equation by r.

r S_{n} = a_{1} r + a_{1} r^{2} + a_{1} r^{3} + ... + a_{1} r^{n} r S_{n} = a_{1} r + a_{1} r^{2} + a_{1} r^{3} + ... + a_{1} r^{n}

Next, we subtract these equations. We will see that when we subtract, all but the first term of the top equation and the last term of the bottom equation subtract to zero.

$\begin{matrix} \begin{matrix} \underset{____________________________________________________}{\begin{matrix} S_{n} & = & a_{1} + a_{1} r + a_{1} r^{2} + a_{1} r^{3} + \dots + a_{1} r^{n - 1} \\ r S_{n} & = & a_{1} r + a_{1} r^{2} + a_{1} r^{3} + \dots + a_{1} r^{n - 1} + a_{1} r^{n} \end{matrix}} \\ S_{n} - r S_{n} = a_{1} {−a}_{1} r^{n} \end{matrix} \\ We factor both sides. & S_{n} (1 - r) = a_{1} (1 - r^{n}) \\ \begin{matrix} To obtain the formula for S_{n}, \\ divide both sides by (1 - r) . \end{matrix} & S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} \end{matrix} \begin{matrix} \begin{matrix} \underset{____________________________________________________}{\begin{matrix} = & a_{1} + a_{1} r + a_{1} r^{2} + a_{1} r^{3} + \dots + a_{1} r^{n - 1} \\ = \end{matrix}} \\ S_{n} - r S_{n} \end{matrix} \\ We factor both sides. & S_{n} (1 - r) = a_{1} (1 - r^{n}) & \begin{matrix} To obtain the formula for \\ divide both sides by \end{matrix} \end{matrix}$

Sum of the First n Terms of a Geometric Series

The sum, $S_{n}, S_{n},$ of the first n terms of a geometric sequence is

S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}

where $a_{1} a_{1}$ is the first term and r is the common ratio, and r is not equal to one.

We apply this formula in the next example where the first few terms of the sequence are given. Notice the sum of a geometric sequence typically gets very large when the common ratio is greater than one.

Example 12.25

Find the sum of the first 20 terms of the geometric sequence 7, 14, 28, 56, 112, 224, …

Show solution

Solution

To find the sum, we will use the formula $S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} . S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} .$ We know $a_{1} = 7, a_{1} = 7,$ $r = 2, r = 2,$ and $n = 20 . n = 20 .$

$\begin{matrix} \begin{matrix} Knowing a_{1} = 7, r = 2, and n = 20, \\ use the sum formula. \end{matrix} & S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} \\ Substitute in the values. & S_{20} = \frac{7 (1 - 2^{20})}{1 - 2} \\ Simplify. & S_{20} = 7,340,025 \end{matrix} \begin{matrix} \begin{matrix} Knowing \\ use the sum formula. \end{matrix} \\ Substitute in the values. \\ Simplify. \end{matrix}$

Try It 12.49

Find the sum of the first 20 terms of the geometric sequence 3, 6, 12, 24, 48, 96, …

Try It 12.50

Find the sum of the first 20 terms of the geometric sequence 6, 18, 54, 162, 486, 1,458, …

In the next example, we are given the sum in summation notation. While adding all the terms might be possible, most often it is easiest to use the formula to find the sum of the first n terms.

To use the formula, we need r. We can find it by writing out the first few terms of the sequence and find their ratio. Another option is to realize that in summation notation, a sequence is written in the form $\sum_{i = 1}^{k} a {(r)}^{i}, \sum_{i = 1}^{k} a {(r)}^{i},$ where r is the common ratio.

Example 12.26

Find the sum: $\sum_{i = 1}^{15} 2 {(3)}^{i} . \sum_{i = 1}^{15} 2 {(3)}^{i} .$

Show solution

Solution

To find the sum, we will use the formula $S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}, S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r},$ which requires $a_{1} a_{1}$ and r. We will write out a few of the terms, so we can get the needed information.


Write out the first few terms.
Identify $a_{1} a_{1}$ .
Find the common ratio.
Knowing $a_{1} = 6, a_{1} = 6,$ $r = 3, r = 3,$ and $n = 15, n = 15,$ use the sum formula.
Substitute in the values.
Simplify.

Try It 12.51

Find the sum: $\sum_{i = 1}^{15} 6 {(2)}^{i} . \sum_{i = 1}^{15} 6 {(2)}^{i} .$

Try It 12.52

Find the sum: $\sum_{i = 1}^{10} 5 {(2)}^{i} . \sum_{i = 1}^{10} 5 {(2)}^{i} .$

Find the Sum of an Infinite Geometric Series

If we take a geometric sequence and add the terms, we have a sum that is called a geometric series. An infinite geometric series is an infinite sum whose first term is $a_{1} a_{1}$ and common ratio is r and is written

a_{1} + a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n - 1} + \dots a_{1} + a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n - 1} + \dots

Infinite Geometric Series

An infinite geometric series is an infinite sum whose first term is $a_{1} a_{1}$ and common ratio is r and is written

a_{1} + a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n - 1} + \dots a_{1} + a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n - 1} + \dots

We know how to find the sum of the first n terms of a geometric series using the formula, $S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} . S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} .$ But how do we find the sum of an infinite sum?

Let’s look at the infinite geometric series $3 + 6 + 12 + 24 + 48 + 96 + \dots . 3 + 6 + 12 + 24 + 48 + 96 + \dots .$ Each term gets larger and larger so it makes sense that the sum of the infinite number of terms gets larger. Let’s look at a few partial sums for this series. We see $a_{1} = 3 a_{1} = 3$ and $r = 2 r = 2$

\begin{matrix} S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} & S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} & S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{10} & = & \frac{3 (1 - 2^{10})}{1 - 2} & S_{30} & = & \frac{3 (1 - 2^{30})}{1 - 2} & S_{50} & = & \frac{3 (1 - 2^{50})}{1 - 2} \\ S_{10} & = & 3,069 & S_{30} & = & 3,221,225,469 & S_{50} & \approx & 3.38 \times 10^{15} \end{matrix} \begin{matrix} S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} & = \frac{a_{1} (1 - r^{n})}{1 - r} = \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{10} & = & \frac{3 (1 - 2^{10})}{1 - 2} & = \frac{3 (1 - 2^{30})}{1 - 2} = \frac{3 (1 - 2^{50})}{1 - 2} \\ S_{10} & = & 3,069 & = 3,221,225,469 \approx 3.38 \end{matrix}

As n gets larger and larger, the sum gets larger and larger. This is true when $| r | \geq 1 | r | \geq 1$ and we call the series divergent. We cannot find a sum of an infinite geometric series when $| r | \geq 1 . | r | \geq 1 .$

Let’s look at an infinite geometric series whose common ratio is a fraction less than one,
$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \dots \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \dots$ . Here the terms get smaller and smaller as n gets larger. Let’s look at a few finite sums for this series. We see $a_{1} = \frac{1}{2} a_{1} = \frac{1}{2}$ and $r = \frac{1}{2} . r = \frac{1}{2} .$

\begin{matrix} S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} & S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} & S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{10} & = & \frac{\frac{1}{2} (1 - {(\frac{1}{2})}^{10})}{1 - \frac{1}{2}} & S_{20} & = & \frac{\frac{1}{2} (1 - {(\frac{1}{2})}^{20})}{1 - \frac{1}{2}} & S_{30} & = & \frac{\frac{1}{2} (1 - {(\frac{1}{2})}^{30})}{1 - \frac{1}{2}} \\ S_{10} & \approx & .9990234375 & S_{20} & \approx & 0.9999990463 & S_{30} & \approx & 0.9999999991 \end{matrix} \begin{matrix} S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} & = \frac{a_{1} (1 - r^{n})}{1 - r} = \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{10} & = & \frac{\frac{1}{2} (1 - {(\frac{1}{2})}^{10})}{1 - \frac{1}{2}} & = \frac{\frac{1}{2} (1 - {(\frac{1}{2})}^{20})}{1 - \frac{1}{2}} = \frac{\frac{1}{2} (1 - {(\frac{1}{2})}^{30})}{1 - \frac{1}{2}} & S_{10} & \approx & .9990234375 & \approx 0.9999990463 \approx 0.9999999991 \end{matrix}

Notice the sum gets larger and larger but also gets closer and closer to one. When $| r | < 1, | r | < 1,$ the expression $r^{n} r^{n}$ gets smaller and smaller. In this case, we call the series convergent. As n approaches infinity, (gets infinitely large), $r^{n} r^{n}$ gets closer and closer to zero. In our sum formula, we can replace the $r^{n} r^{n}$ with zero and then we get a formula for the sum, S, for an infinite geometric series when $| r | < 1 . | r | < 1 .$

\begin{matrix} S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{} & = & \frac{a_{1} (1 - 0)}{1 - r} \\ S_{} & = & \frac{a_{1}}{1 - r} \end{matrix} \begin{matrix} S_{n} & = & \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{} & = & \frac{a_{1} (1 - 0)}{1 - r} \\ S_{} & = & \frac{a_{1}}{1 - r} \end{matrix}

This formula gives us the sum of the infinite geometric sequence. Notice the S does not have the subscript n as in $S_{n} S_{n}$ as we are not adding a finite number of terms.

Sum of an Infinite Geometric Series

For an infinite geometric series whose first term is $a_{1} a_{1}$ and common ratio r,

$If | r | < 1, the sum is$

S_{} = \frac{a_{1}}{1 - r} S_{} = \frac{a_{1}}{1 - r}

$If | r | \geq 1, the infinite geometric series does not have a sum. We say the series diverges.$

Example 12.27

Find the sum of the infinite geometric series $54 + 18 + 6 + 2 + \frac{2}{3} + \frac{2}{9} + \dots 54 + 18 + 6 + 2 + \frac{2}{3} + \frac{2}{9} + \dots$

Show solution

Solution

To find the sum, we first have to verify that the common ratio $| r | < 1 | r | < 1$ and then we can use the sum formula $S_{} = \frac{a_{1}}{1 - r} . S_{} = \frac{a_{1}}{1 - r} .$

$\begin{matrix} Find the common ratio. & r & = & \frac{18}{54} & r & = & \frac{6}{18} \dots \\ r & = & \frac{1}{3} & r & = & \frac{1}{3} | r | < 1 \\ Identify a_{1} . & a_{1} & = & 54 \\ \begin{matrix} Knowing a_{1} = 54, r = \frac{1}{3}, \\ use the sum formula. \end{matrix} & S_{} & = & \frac{a_{1}}{1 - r} \\ Substitute in the values. & S_{} & = & \frac{54}{1 - \frac{1}{3}} \\ Simplify. & S_{} & = & 81 \end{matrix} \begin{matrix} Find the common ratio. & = \frac{18}{54} r = \frac{6}{18} \dots \\ = \frac{1}{3} r = \frac{1}{3} \\ Identify & = 54 & \begin{matrix} Knowing \\ use the sum formula. \end{matrix} & = \frac{a_{1}}{1 - r} & Substitute in the values. & = \frac{54}{1 - \frac{1}{3}} & Simplify. & = 81 \end{matrix}$

Try It 12.53

Find the sum of the infinite geometric series $48 + 24 + 12 + 6 + 3 + \frac{3}{2} + \dots 48 + 24 + 12 + 6 + 3 + \frac{3}{2} + \dots$

Try It 12.54

Find the sum of the infinite geometric series $64 + 16 + 4 + 1 + \frac{1}{4} + \frac{1}{16} + \dots 64 + 16 + 4 + 1 + \frac{1}{4} + \frac{1}{16} + \dots$

An interesting use of infinite geometric series is to write a repeating decimal as a fraction.

Example 12.28

Write the repeating decimal $0. \bar{5} 0. \bar{5}$ as a fraction.

Show solution

Solution

$\begin{matrix} Rewrite the 0. \bar{5} showing the repeating five. & 0.5555555555555 … \\ Use place value to rewrite this as a sum. & 0.5 + 0.05 + 0.005 + 0.0005 + ... \\ This is an infinite geometric series. \\ Find the common ratio. & r = \frac{0.05}{0.5} r = \frac{0.005}{0.05} \dots \\ r = 0.1 r = 0.1 | r | < 1 \\ Identify a_{1} . & a_{1} = 0.5 \\ \begin{matrix} Knowing a_{1} = 0.5, r = 0.1, \\ use the sum formula. \end{matrix} & S_{} = \frac{a_{1}}{1 - r} \\ Substitute in the values. & S_{} = \frac{0.5}{1 - 0.1} \\ Simplify. & S_{} = \frac{0.5}{0.9} \\ Multiply numerator and denominator by 10. & S_{} = \frac{5}{9} \\ We are asked to find the fraction form. & 0. \bar{5} = \frac{5}{9} \end{matrix} \begin{matrix} Rewrite the & Use place value to rewrite this as a sum. & This is an infinite geometric series. & Find the common ratio. & Identify & \begin{matrix} Knowing \\ use the sum formula. \end{matrix} & Substitute in the values. & Simplify. & Multiply numerator and denominator by 10. & We are asked to find the fraction form. & 0. \bar{5} = \frac{5}{9} \end{matrix}$

Try It 12.55

Write the repeating decimal $0. \bar{4} 0. \bar{4}$ as a fraction.

Try It 12.56

Write the repeating decimal $0. \bar{8} 0. \bar{8}$ as a fraction.

Apply Geometric Sequences and Series in the Real World

One application of geometric sequences has to do with consumer spending. If a tax rebate is given to each household, the effect on the economy is many times the amount of the individual rebate.

Example 12.29

The government has decided to give a $1,000 tax rebate to each household in order to stimulate the economy. The government statistics say that each household will spend 80% of the rebate in goods and services. The businesses and individuals who benefitted from that 80% will then spend 80% of what they received and so on. The result is called the multiplier effect. What is the total effect of the rebate on the economy?

Show solution

Solution

Every time money goes into the economy, 80% of it is spent and is then in the economy to be spent. Again, 80% of this money is spent in the economy again. This situation continues and so leads us to an infinite geometric series.

1000 + 1000 (0.8) + 1000 {(0.8)}^{2} + \dots 1000 + 1000 (0.8) + 1000 {(0.8)}^{2} + \dots

Here the first term is 1,000, $a_{1} = 1000 . a_{1} = 1000 .$ The common ratio is $0.8, 0.8,$ $r = 0.8 . r = 0.8 .$ We can evaluate this sum since $0.8 < 1 . 0.8 < 1 .$ We use the formula for the sum on an infinite geometric series.

$\begin{matrix} S_{} & = & \frac{a_{1}}{1 - r} \\ Substitute in the values, a_{1} = 1,000 and r = 0.8 . & S_{} & = & \frac{1,000}{1 - 0.8} \\ Evaluate. & S_{} & = & 5,000 \end{matrix} \begin{matrix} = \frac{a_{1}}{1 - r} \\ Substitute in the values, & = \frac{1,000}{1 - 0.8} \\ Evaluate. & = 5,000 \end{matrix}$

The total effect of the $1,000 received by each household will be a $5,000 growth in the economy.

Try It 12.57

What is the total effect on the economy of a government tax rebate of $1,000 to each household in order to stimulate the economy if each household will spend 90% of the rebate in goods and services?

Try It 12.58

What is the total effect on the economy of a government tax rebate of $500 to each household in order to stimulate the economy if each household will spend 85% of the rebate in goods and services?

We have looked at a compound interest formula where a principal, P, is invested at an interest rate, r, for t years. The new balance, A, is $A = P {(1 + \frac{r}{n})}^{n t} A = P {(1 + \frac{r}{n})}^{n t}$ when interest is compounded n times a year. This formula applies when a lump sum was invested upfront and tells us the value after a certain time period.

An annuity is an investment that is a sequence of equal periodic deposits. We will be looking at annuities that pay the interest at the time of the deposits. As we develop the formula for the value of an annuity, we are going to let $n = 1 . n = 1 .$ That means there is one deposit per year.

\begin{matrix} A & = & P {(1 + \frac{r}{n})}^{n t} \\ Let n = 1 . & A & = & P {(1 + \frac{r}{1})}^{1 t} \\ Simplify. & A & = & P {(1 + r)}^{t} \end{matrix} \begin{matrix} = P {(1 + \frac{r}{n})}^{n t} \\ Let & = P {(1 + \frac{r}{1})}^{1 t} \\ Simplify. & = P {(1 + r)}^{t} \end{matrix}

Suppose P dollars is invested at the end of each year. One year later that deposit is worth $P {(1 + r)}^{1} P {(1 + r)}^{1}$ dollars, and another year later it is worth $P {(1 + r)}^{2} P {(1 + r)}^{2}$ dollars. After t years, it will be worth $A = P {(1 + r)}^{t} A = P {(1 + r)}^{t}$ dollars.

	End of year 1	End of year 2	End of year 3
First Deposit P @ end of year 1	P	Amount 1 year later $P {(1 + r)}^{1} P {(1 + r)}^{1}$	Amount 2 years later $P {(1 + r)}^{2} P {(1 + r)}^{2}$
2nd Deposit P @ end of year 2		P	Amount 1 year later $P {(1 + r)}^{1} P {(1 + r)}^{1}$
3^rd Deposit P @ end of year 3			P

After three years, the value of the annuity is

P plus P times the quantity 1 plus r in parentheses, to the first power, plus P times the quantity 1 plus r, in parentheses, squared. This equals the money deposited at the end of year three, plus the money deposited at the end of year two, plus the money deposited at the end of year 1.

This a sum of the terms of a geometric sequence where the first term is P and the common ratio is $1 + r . 1 + r .$ We substitute these values into the sum formula. Be careful, we have two different uses of r. The r in the sum formula is the common ratio of the sequence. In this case, that is $1 + r 1 + r$ where r is the interest rate.

\begin{matrix} S_{t} & = & \frac{a_{1} (1 - r^{t})}{1 - r} \\ Substitute in the values. & S_{t} & = & \frac{P (1 - {(1 + r)}^{t})}{1 - (1 + r)} \\ Simplify. & S_{t} & = & \frac{P (1 - {(1 + r)}^{t})}{− r} \\ S_{t} & = & \frac{P ({(1 + r)}^{t} - 1)}{r} \end{matrix} \begin{matrix} = \frac{a_{1} (1 - r^{t})}{1 - r} \\ Substitute in the values. & = \frac{P (1 - {(1 + r)}^{t})}{1 - (1 + r)} \\ Simplify. & = \frac{P (1 - {(1 + r)}^{t})}{− r} \\ = \frac{P ({(1 + r)}^{t} - 1)}{r} \end{matrix}

Remember our premise was that one deposit was made at the end of each year.

We can adapt this formula for n deposits made per year and the interest is compounded n times a year.

Value of an Annuity with Interest Compounded $n n$ Times a Year

For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is

A_{t} = \frac{P ({(1 + \frac{r}{n})}^{n t} - 1)}{\frac{r}{n}} A_{t} = \frac{P ({(1 + \frac{r}{n})}^{n t} - 1)}{\frac{r}{n}}

Example 12.30

New parents decide to invest $100 per month in an annuity for their baby daughter. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at her eighteenth birthday?

Show solution

Solution

To find the Annuity formula, $A_{t} = \frac{P ({(1 + \frac{r}{n})}^{n t} - 1)}{\frac{r}{n}}, A_{t} = \frac{P ({(1 + \frac{r}{n})}^{n t} - 1)}{\frac{r}{n}},$ we need to identify P, r, n, and t.

$\begin{matrix} Identify P, the amount invested each month. & P = 100 \\ Identify r, the annual interest rate, in decimal form. & r = 0.05 \\ \begin{matrix} Identify n, the number of times the deposit \\ will be made and the interest compounded \\ each year. \end{matrix} & n = 12 \\ Identify t, the number of years. & t = 18 \\ \begin{matrix} Knowing P = 100, r = 0.05, n = 12 and \\ t = 18, use the sum formula. \end{matrix} & A_{t} = \frac{P ({(1 + \frac{r}{n})}^{n t} - 1)}{\frac{r}{n}} \\ Substitute in the values. & A_{t} = \frac{100 ({(1 + \frac{0.05}{12})}^{12 \cdot 18} - 1)}{\frac{0.05}{12}} \\ \begin{matrix} Use the calculator to evaluate. Be sure to \\ use parentheses as needed. \end{matrix} & A_{t} = 34,920.20 \\ The child will have $ 34,920.20 when she turns 18. \end{matrix} \begin{matrix} Identify \\ Identify & \begin{matrix} Identify \\ will be made and the interest compounded \\ each year. \end{matrix} & Identify & \begin{matrix} Knowing \\ t = 18, \end{matrix} & Substitute in the values. & \begin{matrix} Use the calculator to evaluate. Be sure to \\ use parentheses as needed. \end{matrix} \end{matrix}$

Try It 12.59

New grandparents decide to invest $200 per month in an annuity for their grandson. The account will pay 5% interest per year which is compounded monthly. How much will be in the child’s account at his twenty-first birthday?

Try It 12.60

Arturo just got his first full-time job after graduating from college at age 27. He decided to invest $200 per month in an IRA (an annuity). The interest on the annuity is 8%, which is compounded monthly. How much will be in the Arturo’s account when he retires at his sixty-seventh birthday?

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

Practice Makes Perfect

Determine if a Sequence is Geometric

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

127.

$3, 12, 48, 192, 768, 3072, … 3, 12, 48, 192, 768, 3072,$

128.

$2, 10, 50, 250, 1250, 6250, … 2, 10, 50, 250, 1250, 6250,$

129.

$48, 24, 12, 6, 3, \frac{3}{2}, … 48, 24, 12, 6, 3, \frac{3}{2},$

130.

$54, 18, 6, 2, \frac{2}{3}, \frac{2}{9}, … 54, 18, 6, 2, \frac{2}{3}, \frac{2}{9},$

131.

$−3, 6, −12, 24, −48, 96, … −3, 6, −12, 24, −48, 96,$

132.

$2, −6, 18, −54, 162, −486, … 2, −6, 18, −54, 162, −486,$

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

133.

$48, 24, 12, 6, 3, \frac{3}{2}, … 48, 24, 12, 6, 3, \frac{3}{2},$

134.

$12, 6, 0, −6, −12, −18, … 12, 6, 0, −6, −12, −18,$

135.

$−7, −2, 3, 8, 13, 18, … −7, −2, 3, 8, 13, 18,$

136.

$5, 9, 13, 17, 21, 25, … 5, 9, 13, 17, 21, 25,$

137.

$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64}, … \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \frac{1}{32}, \frac{1}{64},$

138.

$4, 8, 12, 24, 48, 96, … 4, 8, 12, 24, 48, 96,$

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

139.

$a_{1} = 4 a_{1} = 4$ and $r = 3 r = 3$

140.

$a_{1} = 9 a_{1} = 9$ and $r = 2 r = 2$

141.

$a_{1} = −4 a_{1} = −4$ and $r = −2 r = −2$

142.

$a_{1} = −5 a_{1} = −5$ and $r = −3 r = −3$

143.

$a_{1} = 27 a_{1} = 27$ and $r = \frac{1}{3} r = \frac{1}{3}$

144.

$a_{1} = 64 a_{1} = 64$ and $r = \frac{1}{4} r = \frac{1}{4}$

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.

145.

Find $a_{11} a_{11}$ given $a_{1} = 8 a_{1} = 8$ and $r = 3 . r = 3 .$

146.

Find $a_{13} a_{13}$ given $a_{1} = 7 a_{1} = 7$ and $r = 2 . r = 2 .$

147.

Find $a_{10} a_{10}$ given $a_{1} = −6 a_{1} = −6$ and $r = −2 . r = −2 .$

148.

Find $a_{15} a_{15}$ given $a_{1} = −4 a_{1} = −4$ and $r = −3 . r = −3 .$

149.

Find $a_{10} a_{10}$ given $a_{1} = 100,000 a_{1} = 100,000$ and $r = 0.1 . r = 0.1 .$

150.

Find $a_{8} a_{8}$ given $a_{1} = 1,000,000 a_{1} = 1,000,000$ and $r = 0.01 . r = 0.01 .$

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

151.

Find $a_{9} a_{9}$ of the sequence, $9, 18, 36, 72, 144, 288, … 9, 18, 36, 72, 144, 288,$

152.

Find $a_{12} a_{12}$ of the sequence, $5, 15, 45, 135, 405, 1215, … 5, 15, 45, 135, 405, 1215,$

153.

Find $a_{15} a_{15}$ of the sequence, $−486, 162, −54, 18, −6, 2, … −486, 162, −54, 18, −6, 2,$

154.

Find $a_{16} a_{16}$ of the sequence, $224, −112, 56, −28, 14, −7, … 224, −112, 56, −28, 14, −7,$

155.

Find $a_{10} a_{10}$ of the sequence, $1, 0.1, 0.01, 0.001, 0.0001, 0.00001, … 1, 0.1, 0.01, 0.001, 0.0001, 0.00001,$

156.

Find $a_{9} a_{9}$ of the sequence, $1000, 100, 10, 1, 0.1, 0.01, … 1000, 100, 10, 1, 0.1, 0.01,$

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.

157.

$8, 24, 72, 216, 648, 1944, … 8, 24, 72, 216, 648, 1944,$

158.

$7, 14, 28, 56, 112, 224, … 7, 14, 28, 56, 112, 224,$

159.

$−6, 12, −24, 48, −96, 192, … −6, 12, −24, 48, −96, 192,$

160.

$−4, 12, −36, 108, −324, 972, … −4, 12, −36, 108, −324, 972,$

161.

$81, 27, 9, 3, 1, \frac{1}{3}, … 81, 27, 9, 3, 1, \frac{1}{3},$

162.

$256, 64, 16, 4, 1, \frac{1}{4}, \frac{1}{16}, … 256, 64, 16, 4, 1, \frac{1}{4}, \frac{1}{16},$

In the following exercises, find the sum of the geometric sequence.

163.

$\sum_{i = 1}^{15} {(2)}^{i} \sum_{i = 1}^{15} {(2)}^{i}$

164.

$\sum_{i = 1}^{10} {(3)}^{i} \sum_{i = 1}^{10} {(3)}^{i}$

165.

$\sum_{i = 1}^{9} 4 {(2)}^{i} \sum_{i = 1}^{9} 4 {(2)}^{i}$

166.

$\sum_{i = 1}^{8} 5 {(3)}^{i} \sum_{i = 1}^{8} 5 {(3)}^{i}$

167.

$\sum_{i = 1}^{10} 9 {(\frac{1}{3})}^{i} \sum_{i = 1}^{10} 9 {(\frac{1}{3})}^{i}$

168.

$\sum_{i = 1}^{15} 4 {(\frac{1}{2})}^{i} \sum_{i = 1}^{15} 4 {(\frac{1}{2})}^{i}$

Find the Sum of an Infinite Geometric Series

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

169.

$1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} + \frac{1}{729} + \dots 1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \frac{1}{81} + \frac{1}{243} + \frac{1}{729} + \dots$

170.

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \dots 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} + \dots$

171.

$6 - 2 + \frac{2}{3} - \frac{2}{9} + \frac{2}{27} - \frac{2}{81} + \dots 6 - 2 + \frac{2}{3} - \frac{2}{9} + \frac{2}{27} - \frac{2}{81} + \dots$

172.

$−4 + 2 - 1 + \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \dots −4 + 2 - 1 + \frac{1}{2} - \frac{1}{4} + \frac{1}{8} - \dots$

173.

$6 + 12 + 24 + 48 + 96 + 192 + \dots 6 + 12 + 24 + 48 + 96 + 192 + \dots$

174.

$5 + 15 + 45 + 135 + 405 + 1215 + \dots 5 + 15 + 45 + 135 + 405 + 1215 + \dots$

175.

$1,024 + 512 + 256 + 128 + 64 + 32 + \dots 1,024 + 512 + 256 + 128 + 64 + 32 + \dots$

176.

$6,561 + 2187 + 729 + 243 + 81 + 27 + \dots 6,561 + 2187 + 729 + 243 + 81 + 27 + \dots$

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

177.

$0. \bar{3} 0. \bar{3}$

178.

$0. \bar{6} 0. \bar{6}$

179.

$0. \bar{7} 0. \bar{7}$

180.

$0. \bar{2} 0. \bar{2}$

181.

$0. \overset{—}{45} 0. \overset{—}{45}$

182.

$0. \overset{—}{27} 0. \overset{—}{27}$

Apply Geometric Sequences and Series in the Real World

In the following exercises, solve the problem.

183.

Find the total effect on the economy of each government tax rebate to each household in order to stimulate the economy if each household will spend the indicated percent of the rebate in goods and services.

	Tax rebate to each household	Percent spent on goods and services
ⓐ	$$ 1,000 $ 1,000$	$85 % 85 %$
ⓑ	$$ 1,000 $ 1,000$	$75 % 75 %$
ⓒ	$$ 1,500 $ 1,500$	$90 % 90 %$
ⓓ	$$ 1,500 $ 1,500$	$80 % 80 %$

184.

New grandparents decide to invest $$ 100 $ 100$ per month in an annuity for their grandchild. The account will pay $6 % 6 %$ interest per year which is compounded monthly (12 times a year). How much will be in the child’s account at their twenty-first birthday?

185.

Berenice just got her first full-time job after graduating from college at age 30. She decided to invest $$ 500 $ 500$ per quarter in an IRA (an annuity). The interest on the annuity is $7 % 7 %$ which is compounded quarterly (4 times a year). How much will be in the Berenice’s account when she retires at age 65?

186.

Alice wants to purchase a home in about five years. She is depositing $$ 500 $ 500$ a month into an annuity that earns $5 % 5 %$ per year that is compounded monthly (12 times a year). How much will Alice have for her down payment in five years?

187.

Myra just got her first full-time job after graduating from college. She plans to get a master’s degree, and so is depositing $$ 2,500 $ 2,500$ a year from her year-end bonus into an annuity. The annuity pays $6.5 % 6.5 %$ per year and is compounded yearly. How much will she have saved in five years to pursue her master’s degree?

Writing Exercises

188.

In your own words, explain how to determine whether a sequence is geometric.

189.

In your own words, explain how to find the general term of a geometric sequence.

190.

In your own words, explain the difference between a geometric sequence and a geometric series.

191.

In your own words, explain how to determine if an infinite geometric series has a sum and how to find it.

Self Check

This figure shows seven rows and four columns. The first row is the header row and reads, “I can”, “Confidently”, “With some help”, and “No, I don’t get it. The first column reads, “Determine if a sequence is geometric”, “Find the general term (nth term) of a”, “Geometric sequence”, “Find the sum of an Infinite geometric series”, Use geometric sequences to solve applications”. The remaining columns are blank.

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

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

Learning Objectives

Be Prepared 12.3

Determine if a Sequence is Geometric

Geometric Sequence

Example 12.21

Solution

Try It 12.41

Try It 12.42

Example 12.22

Solution

Try It 12.43

Try It 12.44

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

General Term (nth term) of a Geometric Sequence

Example 12.23

Solution

Try It 12.45

Try It 12.46

Example 12.24

Solution

Try It 12.47

Try It 12.48

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

Sum of the First n Terms of a Geometric Series

Example 12.25

Solution

Try It 12.49

Try It 12.50

Example 12.26

Solution

Try It 12.51

Try It 12.52

Find the Sum of an Infinite Geometric Series

Infinite Geometric Series

Sum of an Infinite Geometric Series

Example 12.27

Solution

Try It 12.53

Try It 12.54

Example 12.28

Solution

Try It 12.55

Try It 12.56

Apply Geometric Sequences and Series in the Real World

Example 12.29

Solution

Try It 12.57

Try It 12.58

Value of an Annuity with Interest Compounded n n Times a Year

Example 12.30

Solution

Try It 12.59

Try It 12.60

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Practice Makes Perfect

Writing Exercises

Self Check

Value of an Annuity with Interest Compounded $n n$ Times a Year