Prove that $(\sum_{i=1}^n i)^2$ = $\sum_{i=1}^n i^3$ by induction Prove that: $(\sum_{i=1}^n i)^2$ = $\sum_{i=1}^n i^3$
I can use the fact that $\sum_{i=1}^n i$ = n(n+1)/2 after the inductive hypothesis is invoked.
I'm not sure where to start, I would usually break down one side but there isn't usually two sums, so I'm not sure.

$$\Big(\sum_{i=1}^{n+1}i\Big)^2=\Big(\sum_{i=1}^ni+(n+1)\Big)^2=\Big(\sum_{i=1}^ni\Big)^2+2(n+1)\sum_{i=1}^ni+(n+1)^2=$$ $$\sum_{i=1}^ni^3+n(n+1)^2+(n+1)^2=\sum_{i=1}^ni^3+(n+1)^3=\sum_{i=1}^{n+1}i^3$$
