Question

If first term of a decreasing infinite G.P. is 1 and sum is $S$, then sum of squares of its terms is-

• A. $S^2$
• B. $1/S^2$
• C. $S^2/(2S-1)$
• D. $S^2/(2S+1)$

Answer 1

Answer: (C)

First term of infinite G.P. $=1$ and sum $=5$
Let common ratio is $r$
So $S=\frac{1}{1-r}$
Let sum of squares of its terms is $S^{\prime }$.
$\therefore S^{\prime }=1+r^2+r^4\dots \dots \infty$
$\therefore S^{\prime }=\frac{1}{1-r^2}=\frac{1}{(1+r)(1-r)}=\frac{S}{\left(1+\frac{S-1}{S}\right)}\left[\because S=\frac{1}{1-r}\right]$
$\therefore S^{\prime }=\frac{S}{\left(\frac{2S-1}{S}\right)}=\frac{S^2}{2S-1}$