Wednesday, April 27, 2016

algebra precalculus - Prove by induction sumni=1i3=fracn2(n+1)24 for nge1

Prove the following statement S(n) for n1:



ni=1i3=n2(n+1)24



To prove the basis, I substitute 1 for n in S(n):




1i=113=1=12(2)24



Great. For the inductive step, I assume S(n) to be true and prove S(n+1):



n+1i=1i3=(n+1)2(n+2)24



Considering the sum on the left side:



n+1i=1i3=ni=1i3+(n+1)3




I make use of S(n) by substituting its right side for ni=1i3:



n+1i=1i3=n2(n+1)24+(n+1)3



This is where I get a little lost. I think I expand the equation to be



=(n4+2n3+n2)4+(n+1)3



but I'm not totally confident about that. Can anyone provide some guidance?

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