Question

Sum of n terms of the following series $1^2+3^3+5^3+7^3+\dots$ is

• A. $n^2(2n^2-1)$
• B. $n^3(n-1)$
• C. $n^3+8n+4$
• D. $2n^4+3n^2$

Answer 1

The correct option is A $n^2(2n^2-1)$

The nth term of given series $t_n=(2n-1{)}^3$
Sum of series $=\sum t_n=\sum (2n-1{)}^3$
$\sum \left\{\begin{array}{c}(8n^3-3.4n^2+3.2n-1)\end{array}\right\}$
$=2n^2(n+1{)}^2-2n(n+1\left)\right(2n+1)+3n(n+1)-n$
$=2n^4+4n^3+2n^2-2n[2n^2+3n+1]+3n^2+3n-n$
$=2n^4+4n^3+2n^2-4n^3-6n^2-2n+3n^2+3n-n$
$=2n^4-n^2$
$=n^2(2n^2-1)$