Can Induction Prove the Equality of Cubic and Square Sums?

[Trentonx]

Homework Statement

Prove by induction \sum\limits_{i=0}^n i^3 = (\sum\limits_{i=0}^N i)^2

Homework Equations

The Attempt at a Solution

So I used N=1 and found that indeed, 1^3 = (1)^2

Then I assumed it was valid up to some limit k, and tried to find it for k+1
(1^3+2^3+...+k^3+(k+1)^3)=(1+2+...+k+k+1)^2
(9+...+2k^3+3k^2+1)=(4+...+2k)^2
Right here I can see a problem, since the RHS will have a k^2 term, and the LHS will have k^3. Where did I go wrong? Are they supposed to be equal?
Thanks for any help.

 

[pongo38]

Do you know the expression Sum of i 1 to n is n*(n+1)/2 ? There's a pattern there. When you sqaure it, maybe you will find a k^3