# $S_{1}, S_{2},.....S_{n}$ are sums of infinite geometric series with first term $1,2,3,...n$ and common ratio $\large\frac{1}{2}$, $\large\frac{1}{3}$,...$\large\frac{1}{n+1}$ respectively. Find $\sum_{r=1}^{n}\;S_{r}$.

$(a)\;\large\frac{n}{3}(n+2)\qquad(b)\;\large\frac{n}{2}(n+3)\qquad(c)\;\large\frac{n^2+1}{3}\qquad(d)\;\large\frac{n(2n+1)}{6}$

Answer : (c) $\;\large\frac{n}{2}(n+3)$
Explanation : $For\;series\;S_{r}$
First term = r
Common ratio = $\;\large\frac{1}{r+1}$
$\sum_{r=1}^{n}\;S_{r}=\sum_{r=1}^{n}\;r+1$
$=\large\frac{n(n+1)}{2}+n$
$=\large\frac{n}{2}\;(n+3)\;.$