Question

Value of $S_1{S}_n+S_2{S}_{n-1}+...+S_n{S}_1$ is

• A. $2n(n+1)$
• B. $n$
• C. $\frac{n}{3}(2n+3{)}^2$
• D. $\frac{n}{3}(2n^2+12n+13)$

Answer 1

The correct option is D $\frac{n}{3}(2n^2+12n+13)$

Sum of infinit terms,
$S_n=\frac{a}{1-r}=\frac{n}{1-\frac{n+1}{2n+1}}=2n+1$
$S_1{S}_n+S_2{S}_{n-1}+...+S_n{S}_1$
$={\sum }_{k=1}^n{S}_k{S}_{n-k+1}$
$={\sum }_{k=1}(2k+1)(2n+3-2k)$
$=\frac{n}{3}(2n^2+12n+13)$