Question

The sum of $n$ terms of the following series; $1^3+3^3+5^3+7^3+...$ 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 $n^2(2n^2-1)$

$S=[1^3+2^3+3^3+...+(2n{)}^3]-[2^3+4^3+...+(2n{)}^3]$
$=[1^3+2^3+3^3+...+(2n{)}^3]-2^3[1^3+2^3+3^3+...+(n{)}^3]$
$={\left[\frac{\left(2n\right)(2n+1)}{2}\right]}^2-8{\left[\frac{\left(n\right)(n+1)}{2}\right]}^2$
$=n^2{(2n+1)}^2-2n^2(n+1{)}^2$
$=n^2[4n^2+4n+1-2n^2-4n-2]$
$=n^2(2n^2-1)$