$$1^3+2^3+\cdots+n^3=\left[\frac{n(n+1)}2\right]^2$$
so far I have..
$$1^3+2^3+\cdots+k^3+(k+1)^3=\left[\frac{(k+1)(k+2)}2\right]^2$$
then..
$$\left[\frac{k(k+1)}2\right]^2+(k+1)^3=\left[\frac{(k+1)(k+2)}2\right]^2$$
where do I go from here so that the rhs equals the lhs?
Inductive step: $$\begin{align*}\underbrace{1^3+2^3+\ldots+n^3}_{=\left(\dfrac{n(n+1)}{2}\right)^2}+(n+1)^3&=\left(\dfrac{n(n+1)}{2}\right)^2+(n+1)^3=(n+1)^2\cdot\left(\frac{n^2}{4}+(n+1)\right)=\\&=(n+1)^2\cdot\left(\frac{n^2+4n+4}{4}\right)=(n+1)^2\cdot\left(\frac{(n+2)^2}{2^2}\right)=\\&=\left(\frac{(n+1)(n+2)}{2}\right)^2\end{align*}$$