I am very lost on this problem and am not sure where to begin.. Any help would be much appreciated!
Prove the Following:
$\displaystyle sum_{i=0}^n i^3$ = $\displaystyle sum_{i=0}^n (i)^2$
Surely you mean $\displaystyle \sum\limits_{k = 1}^n {k^3 } = \left( {\sum\limits_{k = 1}^n k } \right)^2 ?$
Yes do it by induction. But simplify the right hand sum first.
$\displaystyle \sum\limits_{k = 1}^n k = \frac{{n(n + 1)}}{2}$ so that $\displaystyle \left( {\sum\limits_{k = 1}^n k } \right)^2 = \left( {\frac{{n(n + 1)}}{2}} \right)^2 = \frac{{n^2 (n + 1)^2 }}{4}$.
Now do your induction.