# Thread: Prove by induction? I think...

1. ## Prove by induction? I think...

I am very lost on this problem and am not sure where to begin.. Any help would be much appreciated!

Prove the Following:

$sum_{i=0}^n i^3$ = $sum_{i=0}^n (i)^2$

2. $\sum_{i=0}^2 i^3=0^3+1^3+2^3=1+8=9$

$\sum_{i=0}^2 i^2=0^2+1^2+2^2=1+4=5$

am I missing something? seems like disproof by counter example

3. Originally Posted by vexiked
Prove the Following:
$sum_{i=0}^n i^3$ = $sum_{i=0}^n (i)^2$
Surely you mean $\sum\limits_{k = 1}^n {k^3 } = \left( {\sum\limits_{k = 1}^n k } \right)^2 ?$

Yes do it by induction. But simplify the right hand sum first.
$\sum\limits_{k = 1}^n k = \frac{{n(n + 1)}}{2}$ so that $\left( {\sum\limits_{k = 1}^n k } \right)^2 = \left( {\frac{{n(n + 1)}}{2}} \right)^2 = \frac{{n^2 (n + 1)^2 }}{4}$.