# Prove the following using mathematical induction

$\sum_{i=1}^{n}i^3 = (\sum_{i=1}^{n}i)^2$

For n=1, it is obvious.

Assuming that

$\sum_{i=1}^{n}i^3 = (\sum_{i=1}^{n}i)^2$

We want to prove that

$\sum_{i=1}^{n+1}i^3 = (\sum_{i=1}^{n+1}i)^2$

Left side =

$\sum_{i=1}^{n}i^3 + (n+1)^3 = (\sum_{i=1}^{n}i)^2 + (n+1)^3 \text { by our assumption}$

Right side =

$(\sum_{i=1}^{n}i + (n+1))^2 = (\sum_{i=1}^{n}i )^2 + 2 (\sum_{i=1}^{n}i )(n+1) + (n+1)^2$

All we need to do is to prove

$(n+1)^3 = 2 (\sum_{i=1}^{n}i )(n+1) + (n+1)^2$

That is

$(n+1)^2 = 2 (\sum_{i=1}^{n}i ) + (n+1)$

which is true since

$2 (\sum_{i=1}^{n}i ) = n(n+1)$

The proof is done.

answered Oct 15, 2015 by (660 points)
edited Oct 15, 2015 by w1002