# Sum of a Geometric Progression

The sum of a geometric progression is given by the formula
$S_{n} = a\left ( \frac{r^{n}- 1}{r -1} \right )$ if r > 1

and
$S_{n} = a\left ( \frac{1 -r^{n}}{1 -r} \right )$ if r < 1

where, first term = a
common ratio = r
sum of nth terms = $S_{n}$
Proof of sum of a G.P. Let $S_{n}$ denote the sum of 'n' terms of the G.P. with first term 'a' and common ratio 'r'
$S_{n} = a + ar + ar^{2} + ar^{3} + ...+ ar^{n - 2} + ar^{n - 1}$ --------(1)
Multiplying both sides by r
$r. S_{n} = ar + ar^{2} + ar^{3} + ...+ ar^{n - 1} + ar^{n}$ --------(2)
Subtract equation (2) from (1)
$S_{n} - r. S_{n} = a - ar^{n}$
∴ $S_{n}(1- r) = a(1 - r^{n}$)

$S_{n} = a\left ( \frac{1 -r^{n}}{1 -r} \right ) r\neq 1$
OR
$S_{n} = a\left ( \frac{r^{n}- 1}{r -1} \right )$

## Examples on Sum of a Geometric Progression

1) Find the sum of seven terms of G.P. 3,6,12,...
Solution :
Here first term = a = 3
common ratio = r = 2
n = 7
Since r > 1
$S_{n} = a\left ( \frac{r^{n}- 1}{r -1} \right )$

$S_{7} = 3\left ( \frac{2^{7}- 1}{2 -1} \right )$

$S_{7} = 3\left ( \frac{128 - 1}{2 -1} \right )$
$S_{7} = 3 \times$ 127
$S_{7}$ = 381

2) Find the sum of 10 terms of the G.P. 1, 1/2, 1/4,1/8, ....
Here first term = a = 1
common ratio = r = 1/2
n = 10
Since r < 1
$S_{n} = a\left ( \frac{1 -r^{n}}{1 -r} \right )$

$S_{10} = 1\left ( \frac{1 -\left (\frac{1}{2} \right )^{10}}{1 -\frac{1}{2}} \right )$

$S_{10} = 1\left ( \frac{1 -\left (0.5 \right )^{10}}{1 -0.5} \right )$

$S_{10} = 1\left ( \frac{1 -0.0009765}{0.5} \right )$

$S_{10} = 1\left ( \frac{0.9990}{0.5} \right )$

$S_{10}$ = 1.9980

3) Find the sum of 8 terms of the G.P. 1, 3,9, 27, ....
Solution :
Here first term = a = 1
common ratio = r = 3
n = 8
Since r > 1
$S_{n} = a\left ( \frac{r^{n}- 1}{r -1} \right )$

$S_{8} = 1\left ( \frac{3^{8}- 1}{3 -1} \right )$

$S_{8} = 1\left ( \frac{6561 - 1}{2} \right )$
$S_{8} = \left ( \frac{6560}{2} \right )$
$S_{8}$ = 3280