First slide

Geometric progression

Question

If s is the sum to infinity of a G.P., whose first term is a, then the sum of the first n terms of the G.P. is

Moderate

A

$s [1 - {(1 - \frac{a}{s})}^{n}]$
B

$s {(1 - \frac{a}{s})}^{n}$
C

$a [1 - {(1 - \frac{a}{s})}^{n}]$
D

$\frac{a}{s} [1 - {(1 - \frac{a}{s})}^{n}]$

Solution

$\begin{array}{l} s = \frac{a}{1 - r} \Rightarrow 1 - r = \frac{a}{s} \Rightarrow r = 1 - \frac{a}{s} \\ S_{n} = \frac{a (1 - r^{n})}{1 - r} = \frac{a}{a / s} [1 - {(1 - \frac{a}{s})}^{n}] \\ = s [1 - {(1 - \frac{a}{s})}^{n}] \end{array}$

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