<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] If  $S_r= 1^r+2^r+...+n^r=f(n)$  and 
$\sigma =n(n+1)$  &  $\sigma'=2n+1$
$S_1=\frac{1}{2}\sigma$
$S_2=\frac{1}{6}\sigma\sigma'$
$S_3=\frac{1}{4}\sigma^2$
$S_4=\frac{1}{30}\sigma\sigma'(3\sigma-1)$
$S_5=\frac{1}{12}\sigma^2(2\sigma-1)$
$S_6=\frac{1}{42}\sigma\sigma'(3\sigma^2-3\sigma+1)$
$S_7=\frac{1}{24}\sigma^2(3\sigma^2-4\sigma+2)$
$S_8=\frac{1}{90}\sigma\sigma'(5\sigma^3-10\sigma^2+9\sigma-3)$
$S_9=\frac{1}{20}\sigma^2(2\sigma^3-5\sigma^2+6\sigma-3)$
$S_{10}=\frac{1}{66}\sigma\sigma'(3\sigma^4-10\sigma^3+17\sigma^2-15\sigma+5)$
proof of this is based on the theorem 
if r is a positive integer, $s_r$ can be expressed as a polynomial in $n$ of which the highest term in $\frac{n^{r+1}}{r+1}$
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