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 [UPSEAT 2002]

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

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
$S=\frac{a}{1-r}\Rightarrow r=1-\frac{a}{S}$
Now $S_n$ = Sum of n terms
$=a\left(\frac{1-r^n}{1-r}\right)=\frac{a}{1-r}(1-r^n)=S[1-{(1-\frac{a}{S})}^n].$