Answer
$\sum\limits_{i =1}^{n}i^{3}=[\frac{n(n+1)}{2}]^{2}$
Work Step by Step
We need to prove $\sum\limits_{i =1}^{n}i^{3}=[\frac{n(n+1)}{2}]^{2}$
Consider $n=1,k,k+1$
$\sum\limits_{i =1}^{1}i^{3}=[\frac{1(1+1)}{2}]^{2}=1$
$\sum\limits_{i =1}^{k}i^{3}=[\frac{k(k+1)}{2}]^{2}$
$\sum\limits_{i =1}^{k+1}(i)^{3}=[\frac{k(k+1)}{2}]^{2}+(k+1)^{3}$
$\sum\limits_{i =1}^{k+1}(i)^{3}=\frac{1}{4}(k+1)^{2}(k+2)^{2}$
$\sum\limits_{i =1}^{k+1}i^{3}=[\frac{(k+1)((k+1)+1)}
{2}]^{2}$
Hence, $\sum\limits_{i =1}^{n}i^{3}=[\frac{n(n+1)}{2}]^{2}$
