Prove the following statement S(n) for n\ge1:
\sum_{i=1}^ni^3=\frac{n^2(n+1)^2}{4}
To prove the basis, I substitute 1 for n in S(n):
\sum_{i=1}^11^3=1=\frac{1^2(2)^2}{4}
Great. For the inductive step, I assume S(n) to be true and prove S(n+1):
\sum_{i=1}^{n+1}i^3=\frac{(n+1)^2(n+2)^2}{4}
Considering the sum on the left side:
\sum_{i=1}^{n+1}i^3=\sum_{i=1}^ni^3+(n+1)^3
I make use of S(n) by substituting its right side for \sum_{i=1}^ni^3:
\sum_{i=1}^{n+1}i^3=\frac{n^2(n+1)^2}{4}+(n+1)^3
This is where I get a little lost. I think I expand the equation to be
=\frac{(n^4+2n^3+n^2)}{4}+(n+1)^3
but I'm not totally confident about that. Can anyone provide some guidance?
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