Q:

Prove by induction $\sum_{k=1}^n k^3 =( \sum_{k=1}^n k )^2 $

Can anyone show me how to prove this example by induction? I can't figure it out. $$\sum_{k=1}^n k^3 =\left( \sum_{k=1}^n k \right)^2  $$ 

A:

Hint You can first pose $S_n = \sum_{k=1}^n k $ and apply the identity $(a+b)^2 = a^2 + 2ab + b^2$ with $S_{n+1}^2 = (S_n + (n+1))^2$. Using the recurrence hypothesis, you will find the result. 

