How can I prove that

13+23+⋯+n3=(n(n+1)2)2

for all n∈N? 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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