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

$S_{n} = a + ar + ar^{2} + ar^{3} + ...+ ar^{n - 2} + ar^{n - 1}$ --------(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

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

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, ....

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

From sum of a Geometric Progression to Home

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