<s>[INST] 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
 [/INST] Hint: $(1+2+...+n+(n+1))^2 - (1+2+...+n)^2 = 2(1+2+...+n)(n+1) + (n+1)^2 = (n+1)(2(1+2+...+n) + (n+1)) = (n+1)(n(n+1) + n+1) = (n+1)(n+1)^2 = (n+1)^3$
We use here the (slightly easier to prove) result $\sum_{k=1}^{n} k = n(n+1)/2$
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