# Thread: Am I on the right path to this induction problem?

1. ## Am I on the right path to this induction problem?

Professor gave us this problem to work at home after lecture ended. I believe I am right. Then i just have to manipulate algebra at the end.
http://imgur.com/gallery/wC11kX1

2. ## Re: Am I on the right path to this induction problem?

Originally Posted by math951
Professor gave us this problem to work at home after lecture ended. I believe I am right. Then i just have to manipulate algebra at the end.
http://imgur.com/gallery/wC11kX1
So suppose that $\sum\limits_{k = 1}^N {{k^3}} = {\left( {\frac{{N(N + 1)}}{2}} \right)^2}$ is true

Then we look at $\sum\limits_{k = 1}^{N+1} {{k^3}}$
\begin{align*}\sum\limits_{k = 1}^{N + 1} {{k^3}} &= \sum\limits_{k = 1}^N {{k^3}} + {(N + 1)^3}\\ &= {\left( {\frac{{N(N + 1)}}{2}} \right)^2} + {(N + 1)^3}\\ &= \left( {\frac{{{N^2}{{(N + 1)}^2}}}{4}} \right) + {(N + 1)^3}\\ &={(N + 1)^2}\left\{ {\left( {\frac{{{N^2}}}{4}} \right) + (N + 1)} \right\}\\ &= {(N + 1)^2}\left( {\frac{{{N^2} + 4N + 4}}{4}} \right)\\ &= {(N + 1)^2}\left( {\frac{{{{(N + 2)}^2}}}{4}} \right)\\ &= {\left( {\frac{{(N + 1)(N + 2)}}{2}} \right)^2}\end{align*}