KVPY Sample Paper KVPY Stream-SX Model Paper-11

The sum of the following series \[1+6+\frac{9\,({{1}^{2}}+{{2}^{2}}+{{3}^{3}})}{7}+\frac{12\,({{1}^{2}}+{{2}^{2}}+{{3}^{2}}+{{4}^{2}})}{9}\] \[+\frac{15\,({{1}^{2}}+{{2}^{2}}+...+{{5}^{2}})}{11}+...\] up to 15 terms, is:

A) 7820

B) 7830

C) 7520

D) 7510

Correct Answer: A

Solution :

\[{{T}_{n}}=\frac{(3+(n-1)\times 3)({{1}^{2}}+{{2}^{2}}+...+{{n}^{2}})}{(2n+1)}\]

\[{{T}_{n}}=\frac{3.\frac{{{n}^{2}}(n+1)(2n+1)}{6}}{2n+1}+\frac{{{n}^{2}}(n+1)}{2}\]

\[{{S}_{15}}=\,\frac{1}{2}\sum\limits_{n\,\,=\,\,1}^{15}{({{n}^{3}}+{{n}^{2}})}\]

\[=\,\frac{1}{2}\left[ {{\left( \frac{15(15+1)}{2} \right)}^{2}}+\frac{15\times 16\times 31}{6} \right]\]

= 7820.