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

We can try by induction, that is


*

*base case: $1^3=1^2$

*induction step: assume $1^3 + 2^3 +...+ n^3 = (1 + 2 +...+ n)^2=\frac{n^2(n+1)^2}{4}$ then


$$1^3 + 2^3 +...+ n^3+(n+1)^3=\frac{n^2(n+1)^2}{4}+(n+1)^3=\\=(n+1)^2\left(\frac{n^2}{4}+(n+1)\right)=(n+1)^2\frac{n^2+4n+4}{4}=\frac{(n+1)^2(n+2)^2}{4}$$
