Question

Find the sum of first $10$ terms of following series
$S=\frac{3\left(1\right)}{1}+\frac{5(1^3+2^3)}{1^2+2^2}+\frac{7(1^3+2^3+3^3)}{1^2+2^2+3^2}+....$

• A. $440$
• B. $450$
• C. $660$
• D. $220$

Answer 1

The correct option is C $660$

First we shall find the $r^{\text{th}}$ term of the given series

$T_r=\frac{(2r+1){\left[\frac{r(r+1)}{2}\right]}^2}{\frac{r(r+1)(2r+1)}{6}}=\frac{3r(r+1)}{2}$

$T_r=\frac{3}{2}[r^2+r]$

$S_n=\sum T_r=\frac{3}{2}[\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}]$

$S_{10}=\frac{3}{2}[\frac{10\times 11\times 21}{6}+\frac{10\times 11}{2}]$

$S_{10}=\frac{3}{2}[385+55]=660$