Proving $1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$ using induction How can I prove that

$$1^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2$$ 

for all $n \in \mathbb{N}$? I am looking for a proof using mathematical induction.
Thanks

Let the induction hypothesis be 
$$ (1^3+2^3+3^3+\cdots+n^3)=(1+2+3+\cdots+n)^2$$
Now consider:
$$ (1+2+3+\cdots+n + (n+1))^2 $$
$$\begin{align} 
& = \color{red}{(1+2+3+\cdots+n)^2} + (n+1)^2 + 2(n+1)\color{blue}{(1+2+3+\cdots+n)}\\
& = \color{red}{(1^3+2^3+3^3+\cdots+n^3)} + (n+1)^2 + 2(n+1)\color{blue}{(n(n+1)/2)}\\
& = (1^3+2^3+3^3+\cdots+n^3) + \color{teal}{(n+1)^2 + n(n+1)^2}\\
& = (1^3+2^3+3^3+\cdots+n^3) + \color{teal}{(n+1)^3} \end {align}$$
QED
