# 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

${S}_{n}=\frac{{a}_{1}(1-{r}^{n})}{1-r},\text{}\text{}r\ne 1$,

where $n$
is the number of terms, ${a}_{1}$
is the first term and $r$
is the common ratio.

**Example 1:**

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

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

**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.

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

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

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

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.

^{th}term of an geometric series, that is, the sum of the first $n$ terms of an geometric sequence.

See also: sigma notation of a series and sum of the first $n$ terms of an arithmetic sequence

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