Geometric Sequences & Series

Learning Outcomes

Find the common ratio for a geometric sequence.
List the terms of a geometric sequence.
Use a recursive formula for a geometric sequence.
Use an explicit formula for a geometric sequence.
Use the formula for the sum of the ﬁrst n terms of a geometric series.
Use the formula for the sum of an inﬁnite geometric series.
Solve annuity problems.

Many jobs offer an annual cost-of-living increase to keep salaries consistent with inflation. Suppose, for example, a recent college graduate finds a position as a sales manager earning an annual salary of $26,000. He is promised a 2% cost of living increase each year. His annual salary in any given year can be found by multiplying his salary from the previous year by 102%. His salary will be $26,520 after one year; $27,050.40 after two years; $27,591.41 after three years; and so on. When a salary increases by a constant rate each year, the salary grows by a constant factor. In this section we will review sequences that grow in this way.

Terms of Geometric Sequences

Finding Common Ratios

The yearly salary values described form a geometric sequence because they change by a constant factor each year. Each term of a geometric sequence increases or decreases by a constant factor called the common ratio. The sequence below is an example of a geometric sequence because each term increases by a constant factor of 6. Multiplying any term of the sequence by the common ratio 6 generates the subsequent term.

A sequence , {1, 6, 36, 216, 1296, ...} that shows all the numbers have a common ratio of 6.

A General Note: Definition of a Geometric Sequence

A geometric sequence is one in which any term divided by the previous term is a constant. This constant is called the common ratio of the sequence. The common ratio can be found by dividing any term in the sequence by the previous term. If ${a}_{1}$ is the initial term of a geometric sequence and $r$ is the common ratio, the sequence will be

$\left\{{a}_{1}, {a}_{1}r,{a}_{1}{r}^{2},{a}_{1}{r}^{3},...\right\}$ .

How To: Given a set of numbers, determine if they represent a geometric sequence.

Divide each term by the previous term.
Compare the quotients. If they are the same, a common ratio exists and the sequence is geometric.

Example: Finding Common Ratios

Is the sequence geometric? If so, find the common ratio.

$1,2,4,8,16,\dots$
$48,12,4,2,\dots$

Show Solution

Divide each term by the previous term to determine whether a common ratio exists.

$\begin{align}&\frac{2}{1}=2 && \frac{4}{2}=2 && \frac{8}{4}=2 && \frac{16}{8}=2 \end{align}$
The sequence is geometric because there is a common ratio. The common ratio is 2.
$\begin{align}&\frac{12}{48}=\frac{1}{4} && \frac{4}{12}=\frac{1}{3} && \frac{2}{4}=\frac{1}{2} \end{align}$
The sequence is not geometric because there is not a common ratio.

Analysis of the Solution

The graph of each sequence is shown in Figure 1. It seems from the graphs that both (a) and (b) appear have the form of the graph of an exponential function in this viewing window. However, we know that (a) is geometric and so this interpretation holds, but (b) is not.

Graph of two sequences where graph (a) is geometric and graph (b) is exponential.

Q & A

If you are told that a sequence is geometric, do you have to divide every term by the previous term to find the common ratio?

No. If you know that the sequence is geometric, you can choose any one term in the sequence and divide it by the previous term to find the common ratio.

Try It

Is the sequence geometric? If so, find the common ratio.
$5,10,15,20,\dots$

Show Solution

The sequence is not geometric because $\dfrac{10}{5}\ne \dfrac{15}{10}$ .

Try It

Is the sequence geometric? If so, find the common ratio.

$100,20,4,\dfrac{4}{5},\dots$

Show Solution

The sequence is geometric. The common ratio is $\dfrac{1}{5}$ .

Writing Terms of Geometric Sequences

Now that we can identify a geometric sequence, we will learn how to find the terms of a geometric sequence if we are given the first term and the common ratio. The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly. For instance, if the first term of a geometric sequence is ${a}_{1}=-2$ and the common ratio is $r=4$ , we can find subsequent terms by multiplying $-2\cdot 4$ to get $-8$ then multiplying the result $-8\cdot 4$ to get $-32$ and so on.

$\begin{align}&{a}_{1}=-2 \\ &{a}_{2}=\left(-2\cdot 4\right)=-8 \\ &{a}_{3}=\left(-8\cdot 4\right)=-32 \\ &{a}_{4}=\left(-32\cdot 4\right)=-128 \end{align}$

The first four terms are $\left\{-2,-8,-32,-128\right\}$ .

How To: Given the first term and the common factor, find the first four terms of a geometric sequence.

Multiply the initial term, ${a}_{1}$ , by the common ratio to find the next term, ${a}_{2}$ .
Repeat the process, using ${a}_{n}={a}_{2}$ to find ${a}_{3}$ and then ${a}_{3}$ to find ${a}_{4,}$ until all four terms have been identified.
Write the terms separated by commons within brackets.

Example: Writing the Terms of a Geometric Sequence

List the first four terms of the geometric sequence with ${a}_{1}=5$ and $r=-2$ .

Show Solution

Multiply ${a}_{1}$ by $-2$ to find ${a}_{2}$ . Repeat the process, using ${a}_{2}$ to find ${a}_{3}$ , and so on.

$\begin{align}&{a}_{1}=5 \\ &{a}_{2}=-2{a}_{1}=-10 \\ &{a}_{3}=-2{a}_{2}=20 \\ &{a}_{4}=-2{a}_{3}=-40 \end{align}$

The first four terms are $\left\{5,-10,20,-40\right\}$ .

Try It

List the first five terms of the geometric sequence with ${a}_{1}=18$ and $r=\dfrac{1}{3}$ .

Show Solution

$\left\{18,6,2,\dfrac{2}{3},\dfrac{2}{9}\right\}$

Explicit Formulas for Geometric Sequences

Using Explicit Formulas for Geometric Sequences

Because a geometric sequence is an exponential function whose domain is the set of positive integers, and the common ratio is the base of the function, we can write explicit formulas that allow us to find particular terms.

${a}_{n}={a}_{1}{r}^{n - 1}$

Let’s take a look at the sequence $\left\{18\text{, }36\text{, }72\text{, }144\text{, }288\text{, }...\right\}$ . This is a geometric sequence with a common ratio of 2 and an exponential function with a base of 2. An explicit formula for this sequence is

${a}_{n}=18\cdot {2}^{n - 1}$

Graph of the geometric sequence.

A General Note: Explicit Formula for a Geometric Sequence

The nth term of a geometric sequence is given by the explicit formula:

${a}_{n}={a}_{1}{r}^{n - 1}$

Example: Writing Terms of Geometric Sequences Using the Explicit Formula

Given a geometric sequence with ${a}_{1}=3$ and ${a}_{4}=24$ , find ${a}_{2}$ .

Show Solution

The sequence can be written in terms of the initial term and the common ratio $r$ .

$3,3r,3{r}^{2},3{r}^{3},\dots$

Find the common ratio using the given fourth term.

$\begin{align}&{a}_{n}={a}_{1}{r}^{n - 1} \\ &{a}_{4}=3{r}^{3} && \text{Write the fourth term of sequence in terms of }{a}_{1}\text{ and }r \\ &24=3{r}^{3} && \text{Substitute }24\text{ for }{a}_{4} \\ &8={r}^{3} && \text{Divide} \\ &r=2 && \text{Solve for the common ratio} \end{align}$

Find the second term by multiplying the first term by the common ratio.

$\begin{align}{a}_{2} & =2{a}_{1} \\ & =2\left(3\right) \\ & =6 \end{align}$

Analysis of the Solution

The common ratio is multiplied by the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by multiplying the first term by the common ratio nine times or by multiplying by the common ratio raised to the ninth power.

Try It

Given a geometric sequence with ${a}_{2}=4$ and ${a}_{3}=32$ , find ${a}_{6}$ .

Show Solution

${a}_{6}=16\text{,}384$

Example: Writing an Explicit Formula for the nth Term of a Geometric Sequence

Write an explicit formula for the $n\text{th}$ term of the following geometric sequence.

$\left\{2,10,50,250,\dots\right\}$

Show Solution

The first term is 2. The common ratio can be found by dividing the second term by the first term.

$\dfrac{10}{2}=5$

The common ratio is 5. Substitute the common ratio and the first term of the sequence into the formula.

$\begin{align}&{a}_{n}={a}_{1}{r}^{\left(n - 1\right)} \\ &{a}_{n}=2\cdot {5}^{n - 1} \end{align}$

The graph of this sequence shows an exponential pattern.

Graph of the geometric sequence.

Try It

Write an explicit formula for the following geometric sequence.

$\left\{-1,3,-9,27,\dots\right\}$

Show Solution

${a}_{n}=-{\left(-3\right)}^{n - 1}$

In real-world scenarios involving arithmetic sequences, we may need to use an initial term of ${a}_{0}$ instead of ${a}_{1}$ . In these problems we can alter the explicit formula slightly by using the following formula:

${a}_{n}={a}_{0}{r}^{n}$

Example: Solving Application Problems with Geometric Sequences

In 2013, the number of students in a small school is 284. It is estimated that the student population will increase by 4% each year.

Write a formula for the student population.
Estimate the student population in 2020.

Show Solution

The situation can be modeled by a geometric sequence with an initial term of 284. The student population will be 104% of the prior year, so the common ratio is 1.04.Let $P$ be the student population and $n$ be the number of years after 2013. Using the explicit formula for a geometric sequence we get
${P}_{n} =284\cdot {1.04}^{n}$
We can find the number of years since 2013 by subtracting.
$2020 - 2013=7$
We are looking for the population after 7 years. We can substitute 7 for $n$ to estimate the population in 2020.
${P}_{7}=284\cdot {1.04}^{7}\approx 374$
The student population will be about 374 in 2020.

Try It

A business starts a new website. Initially the number of hits is 293 due to the curiosity factor. The business estimates the number of hits will increase by 2.6% per week.

Write a formula for the number of hits.
Estimate the number of hits in 5 weeks.

Show Solution

${P}_{n} = 293\cdot 1.026{a}^{n}$
The number of hits will be about 333.

The following video provides a short lesson on some of the topics covered in this lesson.

A couple decides to start a college fund for their daughter. They plan to invest $50 in the fund each month. The fund pays 6% annual interest, compounded monthly. How much money will they have saved when their daughter is ready to start college in 6 years? In this section we will learn how to answer this question. To do so we need to consider the amount of money invested and the amount of interest earned.

Geometric Series

Just as the sum of the terms of an arithmetic sequence is called an arithmetic series, the sum of the terms in a geometric sequence is called a geometric series. Recall that a geometric sequence is a sequence in which the ratio of any two consecutive terms is the common ratio, $r$ . We can write the sum of the first $n$ terms of a geometric series as

${S}_{n}={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n - 1}$ .

Just as with arithmetic series, we can do some algebraic manipulation to derive a formula for the sum of the first $n$ terms of a geometric series. We will begin by multiplying both sides of the equation by $r$ .

$r{S}_{n}={a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}$

Next, we subtract this equation from the original equation.

$\begin{align}{S}_{n}&={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n - 1} \\ -r{S}_{n}&=-\left({a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}\right) \\ \hline \left(1-r\right){S}_{n}&={a}_{1}-{a}_{1}{r}^{n}\end{align}$

Notice that when we subtract, all but the first term of the top equation and the last term of the bottom equation cancel out. To obtain a formula for ${S}_{n}$ , factor $a_1$ on the right hand side and divide both sides by $\left(1-r\right)$ .

${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\text{ r}\ne \text{1}$

A General Note: Formula for the Sum of the First n Terms of a Geometric Series

A geometric series is the sum of the terms in a geometric sequence. The formula for the sum of the first $n$ terms of a geometric sequence is represented as

${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\text{ r}\ne \text{1}$

How To: Given a geometric series, find the sum of the first n terms.

Identify ${a}_{1},r,\text{and}n$ .
Substitute values for ${a}_{1},r$ , and $n$ into the formula ${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$ .
Simplify to find ${S}_{n}$ .

Example: Finding the First n Terms of a Geometric Series

Use the formula to find the indicated partial sum of each geometric series.

${S}_{11}$ for the series $8 + -4 + 2 + \dots$
$\sum\limits _{k=1}^6 3\cdot {2}^{k}$

Show Solution

${a}_{1}=8$ , and we are given that $n=11$ .We can find $r$ by dividing the second term of the series by the first.
$r=\dfrac{-4}{8}=-\frac{1}{2}$
Substitute values for ${a}_{1}, r, \text{and} n$ into the formula and simplify.
$\begin{align}\\ &{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r} \\[1mm] &{S}_{11}=\dfrac{8\left(1-{\left(-\frac{1}{2}\right)}^{11}\right)}{1-\left(-\frac{1}{2}\right)}\approx 5.336 \\ \text{ } \end{align}$
Find ${a}_{1}$ by substituting $k=1$ into the given explicit formula.
${a}_{1}=3\cdot {2}^{1}=6$
We can see from the given explicit formula that $r=2$ . The upper limit of summation is 6, so $n=6$ .Substitute values for ${a}_{1},r$ , and $n$ into the formula, and simplify.
$\begin{align}\\ &{S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r} \\[1mm] &{S}_{6}=\frac{6\left(1-{2}^{6}\right)}{1 - 2}=378 \end{align}$

Try It

Use the formula to find the indicated partial sum of each geometric series.
${S}_{20}$ for the series $1\text{,}000 + 500 + 250 + \dots$

Show Solution

$\approx 2,000.00$

Use the formula to determine the sum $\sum\limits _{k=1}^{8}{3}^{k}$

Show Solution

Example: Solving an Application Problem with a Geometric Series

At a new job, an employee’s starting salary is $26,750. He receives a 1.6% annual raise. Find his total earnings at the end of 5 years.

Show Solution

The problem can be represented by a geometric series with ${a}_{1}=26,750$ ; $n=5$ ; and $r=1.016$ . Substitute values for ${a}_{1}$ , $r$ , and $n$ into the formula and simplify to find the total amount earned at the end of 5 years.

$\begin{align}{S}_{n}&=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r} \\ {S}_{5}&=\dfrac{26\text{,}750\left(1-{1.016}^{5}\right)}{1 - 1.016}\approx 138\text{,}099.03 \end{align}$

He will have earned a total of $138,099.03 by the end of 5 years.

Try It

At a new job, an employee’s starting salary is $32,100. She receives a 2% annual raise. How much will she have earned by the end of 8 years?

Show Solution

Using the Formula for the Sum of an Infinite Geometric Series

Thus far, we have looked only at finite series. Sometimes, however, we are interested in the sum of the terms of an infinite sequence rather than the sum of only the first n terms. An infinite series is the sum of the terms of an infinite sequence. An example of an infinite series is $2+4+6+8+\dots$ .

This series can also be written in summation notation as $\sum\limits _{k=1}^{\infty} 2k$ , where the upper limit of summation is infinity. Because the terms are not tending to zero, the sum of the series increases without bound as we add more terms. Therefore, the sum of this infinite series is not defined. When the sum is not a real number, we say the series diverges.

Determining Whether the Sum of an Infinite Geometric Series is Defined

If the terms of an infinite geometric series approach 0, the sum of an infinite geometric series can be defined. The terms in this series approach 0:

$1+0.2+0.04+0.008+0.0016+\dots$

The common ratio is $r=0.2$ . As n gets large, the values of of $r^n$ get very small and approach 0. Each successive term affects the sum less than the preceding term. As each succeeding term gets closer to 0, the sum of the terms approaches a finite value. The terms of any infinite geometric series with [latex]-1

DETERMINING WHETHER THE SUM OF AN INFINITE GEOMETRIC SERIES IS DEFINED

The sum of an infinite series is defined if the series is geometric and [latex]-1

How To: Given the first several terms of an infinite series, determine if the sum of the series exists.

Find the ratio of the second term to the first term.
Find the ratio of the third term to the second term.
Continue this process to ensure the ratio of a term to the preceding term is constant throughout. If so, the series is geometric.
If a common ratio, r, was found in step 3, check to see if [latex]-1

Example: Determining Whether the Sum of an Infinite Series is Defined

Determine whether the sum of each infinite series is defined.

$12+8+4+/dots$
$\dfrac{3}{4}+\dfrac{1}{2}+\dfrac{1}{3}+\dots$
$\sum\limits _{k=1}^{\infty}{27}\cdot\left(\dfrac{1}{3}\right)^k$
$\sum\limits _{k=1}^{\infty}{5k}$

Show Solution

The ratio of the second term to the first is $\frac{2}{3}$ , which is not the same as the ratio of the third term to the second, $\frac{1}{2}$ . The series is not geometric.
The ratio of the second term to the first is the same as the ratio of the third term to the second. The series is geometric with a common ratio of $\frac{2}{3}$ . The sum of the infinite series is defined.
The given formula is exponential with a base of $\frac{1}{3}$ ; the series is geometric with a common ratio of $\frac{1}{3}$ . The sum of the infinite series is defined.
The given formula is not exponential. The series is arithmetic, not geometric and so cannot yield a finite sum.

try it

Determine whether the sum of the infinite series is defined.

$\dfrac{1}{3}+\dfrac{1}{2}+\dfrac{3}{4}+\dfrac{9}{8}+\cdots$
$24+(-12)+6+(-3)+\dots$
$\sum\limits _{k=1}^{\infty} 15\cdot(-0.3)^k$

Show Solution

The series is geometric, but $r=\dfrac{3}{2}>1$ . The sum is not defined.
The series is geometric with $r=-\dfrac{1}{2}$ . The sum is defined.
The series is geometric with $r=-0.3$ . The sum is defined.

Finding Sums of Infinite Series

When the sum of an infinite geometric series exists, we can calculate the sum. The formula for the sum of an infinite series is related to the formula for the sum of the first n terms of a geometric series.

${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$

We will examine an infinite series with $r=\frac{1}{2}$ . What happens to $r^n$ as n increases?

$\begin{align} &{\left(\frac{1}{2}\right)}^{2} = \frac{1}{4} \\&{\left(\frac{1}{2}\right)}^{3} = \frac{1}{8} \\&{\left(\frac{1}{2}\right)}^{4} = \frac{1}{16} \end{align}$

The value of $r^n$ decreases rapidly. What happens for greater values of n?

$\begin{align} &{\left(\frac{1}{2}\right)}^{10} = \frac{1}{1\text{,}024} \\&{\left(\frac{1}{2}\right)}^{20} = \frac{1}{1\text{,}048\text{,}576} \\&{\left(\frac{1}{2}\right)}^{30} = \frac{1}{1\text{,}073\text{,}741\text{,}824} \end{align}$

As n gets large, $r^n$ gets very small. We say that as n increases without bound, $r^n$ approaches 0. As $r^n$ approaches 0, $1-r^n$ approaches 1. When this happens the numerator approaches $a_1$ . This gives us the formula for the sum of an infinite geometric series.

A General Note: FORMULA FOR THE SUM OF AN INFINITE GEOMETRIC SERIES

The formula for the sum of an infinite geometric series with [latex]-1 $S=\dfrac{{a}_{1}}{1-r}$

How To: Given an infinite geometric series, find its sum.

Identify $a_1$ and r.
Confirm that [latex]-1
Substitute values for $a_1$ and r into the formula, $S=\dfrac{{a}_{1}}{1-r}$ .
Simplify to find S.

Example: Finding the Sum of an Infinite Geometric Series

Find the sum, if it exists, for the following:

$10+9+8+7+\dots$
$248.6+99.44+39.776+\dots$
$\sum\limits _{k=1}^{\infty}4\text{,}374\cdot\left(-\dfrac{1}{3}\right)^{k-1}$
$\sum\limits _{k=1}^{\infty}\dfrac{1}{9}\cdot\left(\dfrac{4}{3}\right)^{k}$

Show Solution

There is not a constant ratio; the series is not geometric.
There is a constant ratio; the series is geometric. $a_1=248.6$ and $r=\dfrac{99.44}{248.6}=0.4$ , so the sum exists. Substitute $a_1=248.6$ and $r=0.4$ into the formula and simplify to find the sum.
$\begin{align} \\ &S=\frac{a_1}{1-r} \\[1.5mm] &S=\frac{248.6}{1-0.4}=\frac{1243}{3} \\ \text{ }\end{align}$
The formula is exponential, so the series is geometric with $r=-\frac{1}{3}$ . Find $a_1$ by substituting $k=1$ into the given explicit formula.
$\begin{align} \\ a_1=4\text{,}374\cdot\left(-\frac{1}{3}\right)^{1-1}=4\text{,}374 \\ \text{ }\end{align}$
Substitute $4\text{,}374$ and $r=-\frac{1}{3}$ into the formula, and simplify to find the sum.
$\begin{align}\\&S=\frac{a_1}{1-r} \\[1.5mm] &S=\frac{4\text{,}374}{1-\left(-\frac{1}{3}\right)}=3\text{,}280.5 \\ \text{ }\end{align}$
The formula is exponential, so the series is geometric, but $r>1$ . The sum does not exist.

Example: Finding an Equivalent Fraction for a Repeating Decimal

Find the equivalent fraction for the repeating decimal $0.\overline{3}$ .

Show Solution

We notice the repeating decimal $0.\overline{3}=0.333\dots$ . so we can rewrite the repeating decimal as a sum of terms.

$0.\overline{3}=0.3+0.03+0.003+\dots$

Looking for a pattern, we rewrite the sum, noticing that we see the first term multiplied to 0.1 in the second term, and the second term multiplied to 0.1 in the third term.

$\begin{align}0.\overline{3}&=0.3+0.3\cdot(0.1)+0.3\cdot(0.01)+0.3\cdot(0.001)+\dots \\ &=0.3+0.3\cdot(0.1)+0.3\cdot(0.1)^2+0.3\cdot(0.1)^3+\dots$

Notice the pattern; we multiply each consecutive term by a common ratio of 0.1 starting with the first term of 0.3. So, substituting into our formula for an infinite geometric sum, we have

$S=\dfrac{a_1}{1-r} =\dfrac{0.3}{1-0.1} =\dfrac{0.3}{0.9} =\dfrac{1}{3}$

try it

Find the sum if it exists.

$2+\dfrac{2}{3}+\dfrac{2}{9}+\dots$
$\sum\limits _{k=1}^{\infty}{0.76k+1}$
$\sum\limits _{k=1}^{\infty}\left(-\dfrac{3}{8}\right)^k$

Solution/reveal-answer]

3
The series is arithmetic. The sum does not exist.
$-\dfrac{3}{11}$

Annuities

At the beginning of the section, we looked at a problem in which a couple invested a set amount of money each month into a college fund for six years. An annuity is an investment in which the purchaser makes a sequence of periodic, equal payments. To find the amount of an annuity, we need to find the sum of all the payments and the interest earned. In the example the couple invests $50 each month. This is the value of the initial deposit. The account paid 6% annual interest, compounded monthly. To find the interest rate per payment period, we need to divide the 6% annual percentage interest (APR) rate by 12. So the monthly interest rate is 0.5%. We can multiply the amount in the account each month by 100.5% to find the value of the account after interest has been added.

We can find the value of the annuity right after the last deposit by using a geometric series with ${a}_{1}=50$ and $r=100.5\%=1.005$ . After the first deposit, the value of the annuity will be $50. Let us see if we can determine the amount in the college fund and the interest earned.

We can find the value of the annuity after $n$ deposits using the formula for the sum of the first $n$ terms of a geometric series. In 6 years, there are 72 months, so $n=72$ . We can substitute ${a}_{1}=50, r=1.005,$ and $n=72$ into the formula, and simplify to find the value of the annuity after 6 years.

${S}_{72}=\dfrac{50\left(1-{1.005}^{72}\right)}{1 - 1.005}\approx 4\text{,}320.44$

After the last deposit, the couple will have a total of $4,320.44 in the account. Notice, the couple made 72 payments of $50 each for a total of $72\left(50\right) = $3,600$ . This means that because of the annuity, the couple earned $720.44 interest in their college fund.

How To: Given an initial deposit and an interest rate, find the value of an annuity.

Determine ${a}_{1}$ , the value of the initial deposit.
Determine $n$ , the number of deposits.
Determine $r$ .
1. Divide the annual interest rate by the number of times per year that interest is compounded.
2. Add 1 to this amount to find $r$ .
Substitute values for ${a}_{1},r,$ and $n$
into the formula for the sum of the first $n$ terms of a geometric series, ${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}$ .
Simplify to find ${S}_{n}$ , the value of the annuity after $n$ deposits.

Example: Solving an Annuity Problem

A deposit of $100 is placed into a college fund at the beginning of every month for 10 years. The fund earns 9% annual interest, compounded monthly, and paid at the end of the month. How much is in the account right after the last deposit?

Show Solution

The value of the initial deposit is $100, so ${a}_{1}=100$ . A total of 120 monthly deposits are made in the 10 years, so $n=120$ . To find $r$ , divide the annual interest rate by 12 to find the monthly interest rate and add 1 to represent the new monthly deposit.

$r=1+\dfrac{0.09}{12}=1.0075$

Substitute ${a}_{1}=100,r=1.0075,$ and $n=120$ into the formula for the sum of the first $n$ terms of a geometric series, and simplify to find the value of the annuity.

${S}_{120}=\dfrac{100\left(1-{1.0075}^{120}\right)}{1 - 1.0075}\approx 19\text{,}351.43$

So the account has $19,351.43 after the last deposit is made.

Try It

At the beginning of each month, $200 is deposited into a retirement fund. The fund earns 6% annual interest, compounded monthly, and paid into the account at the end of the month. How much is in the account if deposits are made for 10 years?

Show Solution

Key Equations

recursive formula for $nth$ term of a geometric sequence	${a}_{n}=r{a}_{n - 1},n\ge 2$
explicit formula for $nth$ term of a geometric sequence	${a}_{n}={a}_{1}{r}^{n - 1}$
sum of the first $n$ terms of a geometric series	${S}_{n}=\dfrac{{a}_{1}\left(1-{r}^{n}\right)}{1-r} , r\ne 1$
sum of an infinite geometric series with [latex]-1	${S}_{n}=\dfrac{{a}_{1}}{1-r}$

Key Concepts

A geometric sequence is a sequence in which the ratio between any two consecutive terms is a constant.
The constant ratio between two consecutive terms is called the common ratio.
The common ratio can be found by dividing any term in the sequence by the previous term.
The terms of a geometric sequence can be found by beginning with the first term and multiplying by the common ratio repeatedly.
A recursive formula for a geometric sequence with common ratio $r$ is given by ${a}_{n}=r{a}_{n - 1}$ for $n\ge 2$ .
As with any recursive formula, the initial term of the sequence must be given.
An explicit formula for a geometric sequence with common ratio $r$ is given by ${a}_{n}={a}_{1}{r}^{n - 1}$ .
In application problems, we sometimes alter the explicit formula slightly to ${a}_{n}={a}_{0}{r}^{n}$ .

The sum of the terms in a sequence is called a series.
A common notation for series is called summation notation, which uses the Greek letter sigma to represent the sum.
The sum of the terms in a geometric sequence is called a geometric series.
The sum of the first $n$ terms of a geometric series can be found using a formula.
The sum of an infinite series exists if the series is geometric with [latex]-1
If the sum of an infinite series exists, it can be found using a formula.
An annuity is an account into which the investor makes a series of regularly scheduled payments. The value of an annuity can be found using geometric series.

Glossary

annuity an investment in which the purchaser makes a sequence of periodic, equal payments

common ratio the ratio between any two consecutive terms in a geometric sequence

diverge a series is said to diverge if the sum is not a real number

geometric sequence a sequence in which the ratio of a term to a previous term is a constant

geometric series the sum of the terms in a geometric sequence

infinite series the sum of the terms in an infinite sequence

series the sum of the terms in a sequence

summation notation a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series

Enrichment 3 – Sequences & Series