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
$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}, r \neq 1$ ,
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 $r = 2$ .

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

Example 2:

Find $S_{10}$ of the geometric sequence $24, 12, 6, \dots$ .

First, find $r$ .

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

Now, find the sum:

$S_{10} = \frac{24 (1 - {(\frac{1}{2})}^{10})}{1 - \frac{1}{2}} = \frac{3069}{64}$

Example 3:

Evaluate.

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

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

$\begin{array}{l} S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{10} = \frac{3 [1 - {(- 2)}^{10}]}{1 - (- 2)} = \frac{3 (1 - 1024)}{3} = - 1023 \end{array}$

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
$27, 18, 12, 8, \dots$ .

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
$8, 12, 18, 27, \dots$ 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.

Sum of the First n Terms of a Geometric Sequence

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