# Sum of the First $n$ Terms of a Geometric Sequence

If a sequence is geometric there are ways to find the sum of the first $n$ terms, denoted ${S}_{n}$, without actually adding all of the terms.

To find the sum of the first ${S}_{n}$ terms of a geometric sequence use the formula
${S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r},\text{}\text{}r\ne 1$,
where $n$ is the number of terms, ${a}_{1}$ is the first term and $r$ is the common ratio.

The sum of the first $n$ terms of a geometric sequence is called geometric series.

Example 1:

Find the sum of the first 8 terms of the geometric series if ${a}_{1}=1$ and $r=2$.

${S}_{8}=\frac{1\left(1-{2}^{8}\right)}{1-2}=255$

Example 2:

Find ${S}_{10}$ of the geometric sequence $24,12,6,\cdots$.

First, find $r$

$r=\frac{{r}_{2}}{{r}_{1}}=\frac{12}{24}=\frac{1}{2}$

Now, find the sum:

${S}_{10}=\frac{24\left(1-{\left(\frac{1}{2}\right)}^{10}\right)}{1-\frac{1}{2}}=\frac{3069}{64}$

Example 3:

Evaluate.

$\sum _{n=1}^{10}3{\left(-2\right)}^{n-1}$

(You are finding ${S}_{10}$ for the series $3-6+12-24+\cdots$, whose common ratio is $-2$.)

$\begin{array}{l}{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\\ {S}_{10}=\frac{3\left[1-{\left(-2\right)}^{10}\right]}{1-\left(-2\right)}=\frac{3\left(1-1024\right)}{3}=-1023\end{array}$

In order for an infinite geometric series to have a sum, the common ratio $r$ must be between $-1$ and $1$.  Then as $n$ increases, ${r}^{n}$ gets closer and closer to $0$.  To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, $S=\frac{{a}_{1}}{1-r}$, where ${a}_{1}$ is the first term and $r$ is the common ratio.

Example 4:

Find the sum of the infinite geometric sequence
$27,18,12,8,\cdots$.

First find $r$

$r=\frac{{a}_{2}}{{a}_{1}}=\frac{18}{27}=\frac{2}{3}$

Then find the sum:

$S=\frac{{a}_{1}}{1-r}$

$S=\frac{27}{1-\frac{2}{3}}=81$

Example 5:

Find the sum of the infinite geometric sequence
$8,12,18,27,\cdots$ if it exists.

First find $r$

$r=\frac{{a}_{2}}{{a}_{1}}=\frac{12}{8}=\frac{3}{2}$

Since $r=\frac{3}{2}$ is not less than one the series has no sum.

There is a formula to calculate the $n$th term of an geometric series, that is, the sum of the first $n$ terms of an geometric sequence.