Question

If $S_n$ denotes the sum of $n$ terms of a G.P. whose first term and common ratio are $a$ and $r(r\ne 1)$ respectively, then $S_1+S_2+S_3+...+S_n$ is-

• A. $\frac{na}{1-r}$
• B. $\frac{na}{1-r}-\frac{ar(1-r^n)}{(1-r{)}^2}$
• C. $\frac{a}{1-rn}$
• D. $\frac{a}{1-r^n}-\frac{r(1-r^n)}{(1-r{)}^2}$

Answer 1

Answer: (B)

$\frac{na}{1-r}-\frac{ar(1-r^n)}{(1-r{)}^2}$

Since for first term $a$ and the common ratio $r(r\ne 1)$
$S_n=\frac{a(1-r^n)}{(1-r)}$
$\Rightarrow S_1=\frac{a(1-r)}{(1-r)}=a$
$\Rightarrow S_2=\frac{a(1-r^2)}{(1-r)}=a(1+r)$
and so on $...$
Now, $S_1+S_2+...+S_n=a+a(1+r)+...+\frac{a(1-r^n)}{(1-r)}$
$S_1+S_2+...+S_n=a[1+(1+r)+\frac{(1-r^3)}{1-r}...+\frac{(1-r^n)}{(1-r)}]$
$\Rightarrow S_1+S_2+...+S_n=a[\frac{n}{(1-r)}-\frac{r\left({1-r}^n\right)}{{(1-r)}^2}]$
Hence, option 'B' is correct.