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 is

• A. $S\left(1-\frac{a}{S}\right)^n$
• B. $S\left[1-\left(1-\frac{a}{S}\right)\right]^n$
• C. $a\left[1-\left(1-\frac{a}{S}\right)^n\right]$
• D. none of these

Answer 1

Answer: (B)

$S\left[1-\left(1-\frac{a}{S}\right)\right]^n$

Answer 2

If $r$ is the common ratio of the $G.P$. then $S= \frac{a}{1-r}$ $\Rightarrow 1-r=\frac{a}{S}$ $\Rightarrow r=1-\frac{a}{S}$ Now $S_{n} = \frac{a\left(1-r^{n}\right)}{1-r}$ $= \frac{a}{\frac{a}{S}}\left[1-\left(1-\frac{a}{S}\right)^{n}\right]$ $=S \left[1-\left(1 - \frac{a}{S}\right)^{n} \right]$