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