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Geometric Sequences and Series

Determine if a sequence is geometric

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.

Consider these sequences.

Example

Try it.

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

ⓐ $4,8,16,32,64,128\text{,}\ \text{\ldots }$

ⓑ $-2,6,-12,36,-72,216\text{,}\ \text{\ldots }$

Solution

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

ⓐ

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

ⓑ

Find the ratio of the consecutive terms.	$\ \begin{array}{lllllllllll}-2, & & 6, & & -12, & & 36, & & -72, & & 216,\text{\ldots } \\ \\ \\ \frac{6}{-2} & & \frac{-12}{6} & & \frac{36}{-12} & & \frac{-72}{36} & & \frac{216}{-72} \\ -3 & & -2 & & -3 & & -2 & & -3 & & \end{array}$
	The sequence is not geometric. There is no common ratio.

ⓒ

	$\ 27,9,3,1,\frac{1}{3},\frac{1}{9}\text{,}\ \text{\ldots }$
Find the ratio of the consecutive terms.	$\ \begin{array}{lllll}\frac{9}{27} & \frac{3}{9} & \frac{1}{3} & \frac{\frac{1}{3}}{1} & \frac{\frac{1}{9}}{\frac{1}{3}} \\ \\ \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3}\end{array}$
	The sequence is geometric. The common ratio is $r=\frac{1}{3}.$

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

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Find the General Term (

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 ${a}_{1}$ and the common ratio is r. We will then look for a pattern.

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

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

\[{a}_{n}={a}_{1}{r}^{n-1}\]

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

Example

Try it.

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

Solution

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

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

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Find the Sum of the First

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

\[{S}_{n}={a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+...+{a}_{1}{r}^{n-1}\]

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

\[r{S}_{n}={a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+...+{a}_{1}{r}^{n}\]

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.

	$\begin{array}{lllll}\underset{\text{____________________________________________________}}{\begin{array}{lll}\ {S}_{n} & = & {a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+\ldots +{a}_{1}{r}^{n-1} \\ r{S}_{n} & = & \ {a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+\ldots +{a}_{1}{r}^{n-1}+{a}_{1}{r}^{n}\end{array}} \\ \\ {S}_{n}-r{S}_{n}\ =\ {a}_{1}\ {\text{-a}}_{1}{r}^{n}\end{array}$
We factor both sides.	${S}_{n}(1-r)={a}_{1}(1-{r}^{n})$
To obtain the formula for ${S}_{n},$ divide both sides by $(1-r).$	${S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r}$

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

Try it.

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

Solution

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

Knowing ${a}_{1}=7,r=2,$ and $n=20,$ use the sum formula.	$\ {S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r}$
Substitute in the values.	$\ {S}_{20}=\frac{7(1-{2}^{20})}{1-2}$
Simplify.	$\ {S}_{20}=7,340,025$

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.

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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 ${a}_{1}$ and common ratio is r and is written

\[{a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+\ldots +{a}_{1}{r}^{n-1}+\ldots\]

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

Let’s look at the infinite geometric series $3+6+12+24+48+96+\ldots .$ 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 ${a}_{1}=3$ and $r=2$

\[\begin{array}{lllllllllllll}{S}_{n} & = & \frac{{a}_{1}(1-{r}^{n})}{1-r} & & & \ {S}_{n} & = & \frac{{a}_{1}(1-{r}^{n})}{1-r} & & & \ {S}_{n} & = & \frac{{a}_{1}(1-{r}^{n})}{1-r} \\ {S}_{10} & = & \frac{3(1-{2}^{10})}{1-2} & & & \ {S}_{30} & = & \frac{3(1-{2}^{30})}{1-2} & & & \ {S}_{50} & = & \frac{3(1-{2}^{50})}{1-2} \\ {S}_{10} & = & 3,069 & & & \ {S}_{30} & = & 3,221,225,469 & & & \ {S}_{50} & \approx & 3.38\ \times \ {10}^{15}\end{array}\]

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

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

\[\begin{array}{lllllllllllll}{S}_{n} & = & \frac{{a}_{1}(1-{r}^{n})}{1-r} & & & \ {S}_{n} & = & \frac{{a}_{1}(1-{r}^{n})}{1-r} & & & \ {S}_{n} & = & \frac{{a}_{1}(1-{r}^{n})}{1-r} \\ \\ \\ {S}_{10} & = & \frac{\frac{1}{2}(1-{(\frac{1}{2})}^{10})}{1-\frac{1}{2}} & & & \ {S}_{20} & = & \frac{\frac{1}{2}(1-{(\frac{1}{2})}^{20})}{1-\frac{1}{2}} & & & \ {S}_{30} & = & \frac{\frac{1}{2}(1-{(\frac{1}{2})}^{30})}{1-\frac{1}{2}} \\ \\ \\ {S}_{10} & \approx & .9990234375 & & & \ {S}_{20} & \approx & 0.9999990463 & & & \ {S}_{30} & \approx & 0.9999999991\end{array}\]\[\begin{array}{lll}{S}_{n} & = & \frac{{a}_{1}(1-{r}^{n})}{1-r} \\ {S}_{} & = & \frac{{a}_{1}(1-0)}{1-r} \\ {S}_{} & = & \frac{{a}_{1}}{1-r}\end{array}\]

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

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

Try it.

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?

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.

\[1000+1000(0.8)+1000{(0.8)}^{2}+\ldots\]

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

	$\begin{array}{lll}\ {S}_{} & = & \frac{{a}_{1}}{1-r}\end{array}$
Substitute in the values, ${a}_{1}=1,000$ and $r=0.8.$	$\begin{array}{lll}\ {S}_{} & = & \frac{1,000}{1-0.8}\end{array}$
Evaluate.	$\begin{array}{lll}\ {S}_{} & = & 5,000\end{array}$

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

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 $A=P{(1+\frac{r}{n})}^{nt}$ 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 $n=1.$ That means there is one deposit per year.

	$\ A=P{(1+\frac{r}{n})}^{nt}$
Let $n=1.$	$\ A=P{(1+\frac{r}{1})}^{1t}$
Simplify.	$\ A=P{(1+r)}^{t}$

After three years, the value of the annuity is

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

General Term (nth term) of a Geometric Sequence: The general term of a geometric sequence with first term ${a}_{1}$ and the common ratio r is
\[{a}_{n}={a}_{1}{r}^{n-1}\]
Sum of the First n Terms of a Geometric Series: The sum, ${S}_{n},$ of the n terms of a geometric sequence is
\[{S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r}\]
where ${a}_{1}$ is the first term and r is the common ratio.
Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is ${a}_{1}$ and common ratio is r and is written
\[{a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+\ldots +{a}_{1}{r}^{n-1}+\ldots\]
Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is ${a}_{1}$ and common ratio r,
$\begin{array}{l} \\ \\ \text{If}\ |r|<1,\ \text{the sum is} \\ \\ {S}_{}=\frac{{a}_{1}}{1-r} \\ \text{We say the series converges.} \\ \text{If}\ |r|\ge 1,\ \text{the infinite geometric series does not have a sum. We say the series diverges.}\end{array}$
Value of an Annuity with Interest Compounded $n$ 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
\[{A}_{t}=\frac{P({(1+\frac{r}{n})}^{nt}-1)}{\frac{r}{n}}\]

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.

Try it.

$3,12,48,192,768,3072\text{,}\ \text{\ldots }$

Solution

The sequence is geometric with common ratio $r=4.$

Try it.

$2,10,50,250,1250,6250\text{,}\ \text{\ldots }$

Try it.

$72,36,18,9,\frac{9}{2}\text{,}\frac{9}{4}\text{,}\ \text{\ldots }$

Solution

The sequence is geometric with common ratio $r=\frac{1}{2}.$

Try it.

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

Try it.

$-3,6,-12,24,-48,96\text{,}\ \text{\ldots }$

Solution

The sequence is geometric with a common ratio $r=-2.$

Try it.

$2,-6,18,-54,162,-486\text{,}\ \text{\ldots }$

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.

Try it.

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

Solution

The sequence is geometric with common ratio $r=\frac{1}{2}.$

Try it.

$12,6,0,-6,-12,-18\text{,}\ \text{\ldots }$

Try it.

$-7,-2,3,8,13,18\text{,}\ \text{\ldots }$

Solution

The sequence is arithmetic with common difference $d=5.$

Try it.

$5,9,13,17,21\text{,}25\text{,}\ \text{\ldots }$

Try it.

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

Solution

The sequence is geometric with common ratio $r=\frac{1}{2}.$

Try it.

$4,8,12,24,48,96\text{,}\ \text{\ldots }$

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

Try it.

${a}_{1}=4$ and $r=3$

Solution

$4,12,36,108,324$

Try it.

${a}_{1}=9$ and $r=2$

Try it.

${a}_{1}=-4$ and $r=-2$

Solution

$-4,8,-16,32,-64$

Try it.

${a}_{1}=-5$ and $r=-3$

Try it.

${a}_{1}=27$ and $r=\frac{1}{3}$

Solution

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

Try it.

${a}_{1}=64$ and $r=\frac{1}{4}$

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.

Try it.

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

Solution

$472,392$

Try it.

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

Try it.

Find ${a}_{10}$ given ${a}_{1}=-6$ and $r=-2.$

Solution

3,072

Try it.

Find ${a}_{15}$ given ${a}_{1}=-4$ and $r=-3.$

Try it.

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

Solution

$0.0001$

Try it.

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

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

Try it.

Find ${a}_{9}$ of the sequence, $9,18,36,72,144,288\text{,}\ \text{\ldots }$

Solution

${a}_{9}=2,304.$ The general term is ${a}_{n}=9{(2)}^{n-1}.$

Try it.

Find ${a}_{12}$ of the sequence, $5,15,45,135,405,1215\text{,}\ \text{\ldots }$

Try it.

Find ${a}_{15}$ of the sequence, $-486,162,-54,18,-6,2\text{,}\ \text{\ldots }$

Solution

${a}_{15}=-\frac{2}{19,683}.$ The general term is ${a}_{n}=-486{(-\frac{1}{3})}^{n-1}.$

Try it.

Find ${a}_{16}$ of the sequence, $224,-112,56,-28,14,-7\text{,}\ \text{\ldots }$

Try it.

Find ${a}_{10}$ of the sequence, $1,0.1,0.01,0.001,0.0001,0.00001\text{,}\ \text{\ldots }$

Solution

${a}_{10}=0.000000001.$ The general term is ${a}_{n}={(0.1)}^{n-1}.$

Try it.

Find ${a}_{9}$ of the sequence, $1000,100,10,1,0.1,0.01\text{,}\ \text{\ldots }$

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.

Try it.

$8,24,72,216,648,1944\text{,}\ \text{\ldots }$

Solution

$57,395,624$

Try it.

$7,14,28,56,112,224\text{,}\ \text{\ldots }$

Try it.

$-6,12,-24,48,-96,192\text{,}\ \text{\ldots }$

Solution

$-65,538$

Try it.

$-4,12,-36,108,-324,972\text{,}\ \text{\ldots }$

Try it.

$81,27,9,3,1,\frac{1}{3}\text{,}\ \text{\ldots }$

Solution

$\frac{7,174,453}{59,049}\approx 121.5$

Try it.

$256,64,16,4,1,\frac{1}{4},\frac{1}{16}\text{,}\ \text{\ldots }$

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

Try it.

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

Solution

$65,534$

Try it.

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

Try it.

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

Solution

$4088$

Try it.

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

Try it.

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

Solution

$\frac{29,524}{6561}\approx 4.5$

Try it.

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

Find the Sum of an Infinite Geometric Series

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

Try it.

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

Solution

$\frac{3}{2}$

Try it.

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

Try it.

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

Solution

$\frac{9}{2}$

Try it.

$-4+2-1+\frac{1}{2}-\frac{1}{4}+\frac{1}{8}-\ldots$

Try it.

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

Solution

no sum as $r\ge 1$

Try it.

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

Try it.

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

Solution

2,048

Try it.

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

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

Try it.

$0.\overset{-}{3}$

Solution

$\frac{1}{3}$

Try it.

$0.\overset{-}{6}$

Try it.

$0.\overset{-}{7}$

Solution

$\frac{7}{9}$

Try it.

$0.\overset{-}{2}$

Try it.

$0.\overset{—}{45}$

Solution

$\frac{5}{11}$

Try it.

$0.\overset{—}{27}$

Apply Geometric Sequences and Series in the Real World

In the following exercises, solve the problem.

Try it.

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	Total Effect on the economy
ⓐ	$\text{\$}1,000$	$85\text{\%}$
ⓑ	$\text{\$}1,000$	$75\text{\%}$
ⓒ	$\text{\$}1,500$	$90\text{\%}$
ⓓ	$\text{\$}1,500$	$80\text{\%}$

Solution

Try it.

New grandparents decide to invest $\text{\$}100$ per month in an annuity for their grandchild. The account will pay $6\text{\%}$ 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?

Try it.

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

Solution

$\text{\$}295,581.88$

Try it.

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

Try it.

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

Solution

$\text{\$}14,234.10$

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: $\frac{24}{32}.$
If you missed this problem, review .

ഉത്തരം വെളിപ്പെടുത്തുക

$\frac{3}{4}$
Evaluate: ⓐ ${3}^{4}$ ⓑ ${(\frac{1}{2})}^{4}.$
If you missed this problem, review .

ഉത്തരം വെളിപ്പെടുത്തുക

ⓐ 81; ⓑ $\frac{1}{16}$
If $f(x)=4\cdot {3}^{x},$ find ⓐ $f(1)$ ⓑ $f(2)$ ⓒ $f(3).$
If you missed this problem, review .

ഉത്തരം വെളിപ്പെടുത്തുക

ⓐ 12; ⓑ 36; ⓒ 108

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

ⓐ $4,8,16,32,64,128\text{,}\ \text{\ldots }$

ⓑ $-2,6,-12,36,-72,216\text{,}\ \text{\ldots }$

ഉത്തരം വെളിപ്പെടുത്തുക

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

ⓐ

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

ⓑ

Find the ratio of the consecutive terms.	$\ \begin{array}{lllllllllll}-2, & & 6, & & -12, & & 36, & & -72, & & 216,\text{\ldots } \\ \\ \\ \frac{6}{-2} & & \frac{-12}{6} & & \frac{36}{-12} & & \frac{-72}{36} & & \frac{216}{-72} \\ -3 & & -2 & & -3 & & -2 & & -3 & & \end{array}$
	The sequence is not geometric. There is no common ratio.

ⓒ

	$\ 27,9,3,1,\frac{1}{3},\frac{1}{9}\text{,}\ \text{\ldots }$
Find the ratio of the consecutive terms.	$\ \begin{array}{lllll}\frac{9}{27} & \frac{3}{9} & \frac{1}{3} & \frac{\frac{1}{3}}{1} & \frac{\frac{1}{9}}{\frac{1}{3}} \\ \\ \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3}\end{array}$
	The sequence is geometric. The common ratio is $r=\frac{1}{3}.$

Determine if each sequence is geometric. If so indicate the common ratio.
ⓐ $7,21,63,189,567,1,701\text{,}\ \text{\ldots }$
ⓑ $64,16,4,1,\frac{1}{4},\frac{1}{16}\text{,}\ \text{\ldots }$
ⓒ $2,4,12,48,240,1,440\text{,}\ \text{\ldots }$

ഉത്തരം വെളിപ്പെടുത്തുക

ⓐ The sequence is geometric with common ratio $r=3.$ ⓑ The sequence is geometric with common ratio $r=\frac{1}{4}.$ ⓒ The sequence is not geometric. There is no common ratio.
Determine if each sequence is geometric. If so indicate the common ratio.
ⓐ $-150,-30,-15,-5,-\frac{5}{2},0\text{,}\ \text{\ldots }$
ⓑ $5,10,20,40,80,160\text{,}\ \text{\ldots }$
ⓒ $8,4,2,1,\frac{1}{2},\frac{1}{4}\text{,}\ \text{\ldots }$

ഉത്തരം വെളിപ്പെടുത്തുക

ⓐ The sequence is not geometric. There is no common ratio. ⓑ The sequence is geometric with common ratio $r=2.$ ⓒ The sequence is geometric with common ratio $r=\frac{1}{2}.$
Write the first five terms of the sequence where the first term is 3 and the common ratio is $r=-2.$

ഉത്തരം വെളിപ്പെടുത്തുക

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.
\[\begin{array}{lllllllllllll}{a}_{1} & & & {a}_{2} & & & {a}_{3} & & & {a}_{4} & & & {a}_{5} \\ 3 & & & 3\cdot (-2) & & & -6\cdot (-2) & & & 12\cdot (-2) & & & -24\cdot (-2) \\ & & & -6 & & & 12 & & & -24 & & & 48\end{array}\]
The sequence is $3,-6,12,-24,48\text{,}\ \text{\ldots }$
Write the first five terms of the sequence where the first term is 7 and the common ratio is $r=-3.$

ഉത്തരം വെളിപ്പെടുത്തുക

$7,-21,63,-189,567$
Write the first five terms of the sequence where the first term is 6 and the common ratio is $r=-4.$

ഉത്തരം വെളിപ്പെടുത്തുക

$6,-24,96,-384,1536$

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

ഉത്തരം വെളിപ്പെടുത്തുക

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

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

ഉത്തരം വെളിപ്പെടുത്തുക

$\frac{1}{6,561}$
Find the twelfth term of a sequence where the first term is 256 and the common ratio is $r=\frac{1}{4}.$

ഉത്തരം വെളിപ്പെടുത്തുക

$\frac{1}{16,384}$

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

ഉത്തരം വെളിപ്പെടുത്തുക

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

	$\ 3,6,12,24,48,96\text{,}\ \text{\ldots }$
The first term is three.	$\ {a}_{1}=3$
Find the common ratio.	$\ \begin{array}{lllll}\frac{6}{3} & \frac{12}{6} & \frac{24}{12} & \frac{48}{24} & \frac{96}{48} \\ 2 & 2 & 2 & 2 & 2\end{array}$
	The common ratio is $r=2.$
To find the twelfth term, ${a}_{12},$ use the formula with ${a}_{1}=3\ \text{and}\ r=2.$	$\ {a}_{n}={a}_{1}{r}^{n-1}$
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 ${a}_{1}=3\ \text{and}\ r=2.$	$\ {a}_{n}=3{(2)}^{n-1}$

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

ഉത്തരം വെളിപ്പെടുത്തുക

${a}_{9}=39,366.$ The general term is ${a}_{n}=6{(3)}^{n-1}.$
Find the eleventh term of the sequence 7, 14, 28, 56, 112, 224, … Then find the general term for the sequence.

ഉത്തരം വെളിപ്പെടുത്തുക

${a}_{11}=7,168.$ The general term is ${a}_{n}=7{(2)}^{n-1}.$

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

ഉത്തരം വെളിപ്പെടുത്തുക

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

Knowing ${a}_{1}=7,r=2,$ and $n=20,$ use the sum formula.	$\ {S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r}$
Substitute in the values.	$\ {S}_{20}=\frac{7(1-{2}^{20})}{1-2}$
Simplify.	$\ {S}_{20}=7,340,025$

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

ഉത്തരം വെളിപ്പെടുത്തുക

$3,145,725$
Find the sum of the first 20 terms of the geometric sequence 6, 18, 54, 162, 486, 1,458, …

ഉത്തരം വെളിപ്പെടുത്തുക

$10,460,353,200$
Find the sum: $\sum _{i=1}^{15}2{(3)}^{i}.$

ഉത്തരം വെളിപ്പെടുത്തുക

To find the sum, we will use the formula ${S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r},$ which requires ${a}_{1}$ 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 ${a}_{1}$.

Find the common ratio.

Knowing ${a}_{1}=6,$$r=3,$ and $n=15,$
use the sum formula.

Substitute in the values.
Simplify.
Find the sum: $\sum _{i=1}^{15}6{(2)}^{i}.$

ഉത്തരം വെളിപ്പെടുത്തുക

$393,204$
Find the sum: $\sum _{i=1}^{10}5{(2)}^{i}.$

ഉത്തരം വെളിപ്പെടുത്തുക

$10,230$

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

ഉത്തരം വെളിപ്പെടുത്തുക

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

Find the common ratio.	$\ r=\frac{18}{54}\ r=\frac{6}{18}\ldots$
	$\ r=\frac{1}{3}\ r=\frac{1}{3}\ \|r\|<1$
Identify ${a}_{1}.$	$\ {a}_{1}=54$
Knowing ${a}_{1}=54,r=\frac{1}{3},$ use the sum formula.	$\ {S}_{}=\frac{{a}_{1}}{1-r}$
Substitute in the values.	$\ {S}_{}=\frac{54}{1-\frac{1}{3}}$
Simplify.	$\ {S}_{}=81$

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

ഉത്തരം വെളിപ്പെടുത്തുക

96
Find the sum of the infinite geometric series $64+16+4+1+\frac{1}{4}+\frac{1}{16}+\ldots$

ഉത്തരം വെളിപ്പെടുത്തുക

$\frac{256}{3}$

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

ഉത്തരം വെളിപ്പെടുത്തുക

Rewrite the $0.\overset{-}{5}$ showing the repeating five.	$\ 0.5555555555555\text{\ldots }$
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.	$\ r=\frac{0.05}{0.5}\ r=\frac{0.005}{0.05}\ldots$
	$\ r=0.1\ r=0.1\ \|r\|<1$
Identify ${a}_{1}.$	$\ {a}_{1}=0.5$
Knowing ${a}_{1}=0.5,r=0.1,$ use the sum formula.	$\ {S}_{}=\frac{{a}_{1}}{1-r}$
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.	$0.\overset{-}{5}=\frac{5}{9}$

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

ഉത്തരം വെളിപ്പെടുത്തുക

$\frac{4}{9}$
Write the repeating decimal $0.\overset{-}{8}$ as a fraction.

ഉത്തരം വെളിപ്പെടുത്തുക

$\frac{8}{9}$

ഉത്തരം വെളിപ്പെടുത്തുക

\[1000+1000(0.8)+1000{(0.8)}^{2}+\ldots\]

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

	$\begin{array}{lll}\ {S}_{} & = & \frac{{a}_{1}}{1-r}\end{array}$
Substitute in the values, ${a}_{1}=1,000$ and $r=0.8.$	$\begin{array}{lll}\ {S}_{} & = & \frac{1,000}{1-0.8}\end{array}$
Evaluate.	$\begin{array}{lll}\ {S}_{} & = & 5,000\end{array}$

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

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?

ഉത്തരം വെളിപ്പെടുത്തുക

$\text{\$}10,000$
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?

ഉത്തരം വെളിപ്പെടുത്തുക

$\text{\$}3,333.33$

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?

ഉത്തരം വെളിപ്പെടുത്തുക

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

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

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?

ഉത്തരം വെളിപ്പെടുത്തുക

$\text{\$}88,868.36$
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?

ഉത്തരം വെളിപ്പെടുത്തുക

$\text{\$}698,201.57$
$3,12,48,192,768,3072\text{,}\ \text{\ldots }$

ഉത്തരം വെളിപ്പെടുത്തുക

The sequence is geometric with common ratio $r=4.$
$2,10,50,250,1250,6250\text{,}\ \text{\ldots }$
$72,36,18,9,\frac{9}{2}\text{,}\frac{9}{4}\text{,}\ \text{\ldots }$

ഉത്തരം വെളിപ്പെടുത്തുക

The sequence is geometric with common ratio $r=\frac{1}{2}.$
$54,18,6,2,\frac{2}{3},\frac{2}{9}\text{,}\ \text{\ldots }$
$-3,6,-12,24,-48,96\text{,}\ \text{\ldots }$

ഉത്തരം വെളിപ്പെടുത്തുക

The sequence is geometric with a common ratio $r=-2.$
$2,-6,18,-54,162,-486\text{,}\ \text{\ldots }$
$48,24,12,6,3,\frac{3}{2}\text{,}\ \text{\ldots }$

ഉത്തരം വെളിപ്പെടുത്തുക

The sequence is geometric with common ratio $r=\frac{1}{2}.$

Symbols used here

\sum_{k=1}^{n} a_k

summation
Add a_k for k = 1 up to n.

P(A),\ P(A \mid B)

probability, conditional probability
Chance of A; chance of A given that B happened.

imaginary unit
i² = −1.

\approx

approximately equal
Equal to the precision shown, not exactly.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\neq

not equal
The two sides are different.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Geometric Sequences and Series

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

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

നീ സ്വയം ശ്രമിക്ക്.

Parts of this page are adapted from OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

കൂടുതല്‍ Algebra

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	\(\ 27,9,3,1,\frac{1}{3},\frac{1}{9}\text{,}\ \text{\ldots }\)
Find the ratio of the consecutive terms.	\(\ \begin{array}{lllll}\frac{9}{27} & \frac{3}{9} & \frac{1}{3} & \frac{\frac{1}{3}}{1} & \frac{\frac{1}{9}}{\frac{1}{3}} \\ \\ \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & \frac{1}{3}\end{array}\)
	The sequence is geometric. The common ratio is \(r=\frac{1}{3}.\)

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

	\(\begin{array}{lllll}\underset{\text{____________________________________________________}}{\begin{array}{lll}\ {S}_{n} & = & {a}_{1}+{a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+\ldots +{a}_{1}{r}^{n-1} \\ r{S}_{n} & = & \ {a}_{1}r+{a}_{1}{r}^{2}+{a}_{1}{r}^{3}+\ldots +{a}_{1}{r}^{n-1}+{a}_{1}{r}^{n}\end{array}} \\ \\ {S}_{n}-r{S}_{n}\ =\ {a}_{1}\ {\text{-a}}_{1}{r}^{n}\end{array}\)
We factor both sides.	\({S}_{n}(1-r)={a}_{1}(1-{r}^{n})\)
To obtain the formula for \({S}_{n},\) divide both sides by \((1-r).\)	\({S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r}\)

Knowing \({a}_{1}=7,r=2,\) and \(n=20,\) use the sum formula.	\(\ {S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r}\)
Substitute in the values.	\(\ {S}_{20}=\frac{7(1-{2}^{20})}{1-2}\)
Simplify.	\(\ {S}_{20}=7,340,025\)

	\(\begin{array}{lll}\ {S}_{} & = & \frac{{a}_{1}}{1-r}\end{array}\)
Substitute in the values, \({a}_{1}=1,000\) and \(r=0.8.\)	\(\begin{array}{lll}\ {S}_{} & = & \frac{1,000}{1-0.8}\end{array}\)
Evaluate.	\(\begin{array}{lll}\ {S}_{} & = & 5,000\end{array}\)

Find the ratio of the consecutive terms.	\(\ \begin{array}{lllllllllll}4, & & 8, & & 16, & & 32, & & 64, & & 128,\text{\ldots } \\ \\ \\ \frac{8}{4} & & \frac{16}{8} & & \frac{32}{16} & & \frac{64}{32} & & \frac{128}{64} & & \\ 2 & & 2 & & 2 & & 2 & & 2\end{array}\)
	The sequence is geometric. The common ratio is \(r=2.\)

Find the ratio of the consecutive terms.	\(\ \begin{array}{lllllllllll}-2, & & 6, & & -12, & & 36, & & -72, & & 216,\text{\ldots } \\ \\ \\ \frac{6}{-2} & & \frac{-12}{6} & & \frac{36}{-12} & & \frac{-72}{36} & & \frac{216}{-72} \\ -3 & & -2 & & -3 & & -2 & & -3 & & \end{array}\)
	The sequence is not geometric. There is no common ratio.

	\(\ A=P{(1+\frac{r}{n})}^{nt}\)
Let \(n=1.\)	\(\ A=P{(1+\frac{r}{1})}^{1t}\)
Simplify.	\(\ A=P{(1+r)}^{t}\)

	\(\ 3,6,12,24,48,96\text{,}\ \text{\ldots }\)
The first term is three.	\(\ {a}_{1}=3\)
Find the common ratio.	\(\ \begin{array}{lllll}\frac{6}{3} & \frac{12}{6} & \frac{24}{12} & \frac{48}{24} & \frac{96}{48} \\ 2 & 2 & 2 & 2 & 2\end{array}\)
	The common ratio is \(r=2.\)
To find the twelfth term, \({a}_{12},\) use the formula with \({a}_{1}=3\ \text{and}\ r=2.\)	\(\ {a}_{n}={a}_{1}{r}^{n-1}\)
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 \({a}_{1}=3\ \text{and}\ r=2.\)	\(\ {a}_{n}=3{(2)}^{n-1}\)


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

Find the common ratio.	\(\ r=\frac{18}{54}\ r=\frac{6}{18}\ldots\)
	\(\ r=\frac{1}{3}\ r=\frac{1}{3}\ \|r\|<1\)
Identify \({a}_{1}.\)	\(\ {a}_{1}=54\)
Knowing \({a}_{1}=54,r=\frac{1}{3},\) use the sum formula.	\(\ {S}_{}=\frac{{a}_{1}}{1-r}\)
Substitute in the values.	\(\ {S}_{}=\frac{54}{1-\frac{1}{3}}\)
Simplify.	\(\ {S}_{}=81\)

Rewrite the \(0.\overset{-}{5}\) showing the repeating five.	\(\ 0.5555555555555\text{\ldots }\)
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.	\(\ r=\frac{0.05}{0.5}\ r=\frac{0.005}{0.05}\ldots\)
	\(\ r=0.1\ r=0.1\ \|r\|<1\)
Identify \({a}_{1}.\)	\(\ {a}_{1}=0.5\)
Knowing \({a}_{1}=0.5,r=0.1,\) use the sum formula.	\(\ {S}_{}=\frac{{a}_{1}}{1-r}\)
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.	\(0.\overset{-}{5}=\frac{5}{9}\)

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