Proving 13+23+⋯+n3=(n(n+1)2)21^3+ 2^3 + \cdots + n^3 = \left(\frac{n(n+1)}{2}\right)^2 using induction

How can I prove that

13+23++n3=(n(n+1)2)2

for all nN? I am looking for a proof using mathematical induction.

Thanks

Answer

Let the induction hypothesis be
(13+23+33++n3)=(1+2+3++n)2
Now consider:
(1+2+3++n+(n+1))2
=(1+2+3++n)2+(n+1)2+2(n+1)(1+2+3++n)=(13+23+33++n3)+(n+1)2+2(n+1)(n(n+1)/2)=(13+23+33++n3)+(n+1)2+n(n+1)2=(13+23+33++n3)+(n+1)3
QED

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