Question

If $S$ is the sum of infinite terms of a G.P., whose first term is $a,$ then the sum of the first $n$ terms is ___.

• A. $S{(1-\frac{a}{S})}^n$
• B. $S[1-{(1-\frac{a}{S})}^n]$
• C. $a[1-{(1-\frac{a}{S})}^n]$
• D. $S[1-{(1+\frac{a}{S})}^n]$

Answer 1

The correct option is B

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

Let $r$ be the common ratio of the G.P.
Then,
$S=\text{Sum of infinite terms}=\frac{a}{1-r}\Rightarrow r=1-\frac{a}{S}$
Now, $S_n$ = Sum upto n terms
$=a\left(\frac{1-r^n}{1-r}\right)=\frac{a}{1-r}(1-r^n)=S[1-{(1-\frac{a}{S})}^n].$