Prove $1^3 + 2^3 + \ldots n^3 = \frac{n^2(n + 1)^2}{4}$ with mathematical induction

Question

Hello guys I'm trying to prove this equation with mathematical induction method.

So for n = 1 I know that :

1 = 1

Now I know that it works for some k

I wanna prove it for k + 1:

Since :

I can rewrite it as :

So I'm stuck here.. I don't know what should I do.. Should I first square (k + 1)^2 or rewrite (k + 1)^3 ??'
Either way I get weird result and don't know how to proceed with this next step to prove mathematical
induction. I mean I can multiple k^2 with the first parenthesis but I think I should leave it that way and do smth other than that. I'll appreciate any help. Thanks 🙂

Answer 1

You are almost there! Notice that \begin{align*} \frac{k^{2}(k+1)^{2}}{4} + (k+1)^{3} & = \frac{k^{2}(k+1)^{2}+4(k+1)^{3}}{4}\\\\ & = \frac{(k^{2}+4(k+1))(k+1)^{2}}{4}\\\\ & = \frac{(k^{2} + 4k + 4)(k+1)^{2}}{4}\\\\ & = \frac{(k+2)^{2}(k+1)^{2}}{4}\\\\ & = \frac{(k+1)^{2}(k+2)^{2}}{4}\\\\ & = \frac{(k+1)^{2}((k+1)+1)^{2}}{4} \end{align*} and we are done.

Hopefully this helps!