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