Question

If $S_r$ denotes the sum of the infinite geometric series whose first term is $r$ and common ratio is $\frac{1}{1+r}$, where $r\in N$, then the value of $\sum _{r=1}^{10}{S}_r^2$ is

• A. $505$
• B. $385$
• C. $384$
• D. $659$

Answer 1

The correct option is A $505$

From given condition, we get
$S_r=\frac{r}{1-\frac{1}{r+1}}\Rightarrow S_r=r+1$

Now, $\sum _{r=1}^{10}{S}_r^2$
$=\sum _{r=1}^{10}(r+1{)}^2=2^2+3^2+\dots +{11}^2=1^2+2^2+3^2+\dots +{11}^2-1^2=\frac{11\left(12\right)\left(23\right)}{6}-1=505$