Proving that $1\cdot 2+2\cdot 3+\cdots+n\left( n+1 \right) =\frac { n\left( n+1 \right) \left( n+2 \right) }{ 3 } $ by induction 
Prove that 
  $$1\cdot 2+2\cdot 3+\cdots+n\left( n+1 \right) =\frac { n\left( n+1 \right) \left( n+2 \right)  }{ 3 }. $$

I can get to $1/3(k+1)(k+2) + (k+1)(k+2)$ but then finishing off  and rearranging the problem to $1/3(k+2)(k+3)$ is where I always get stuck.

Here is an easy way of going about the inductive step:
\begin{align}
\sum_{i=1}^{k+1}i(i+1)&= \sum_{i=1}^ki(i+1)+(k+1)(k+2)\\[1em]
&= \frac{k(k+1)(k+2)}{3}+(k+1)(k+2)\\[1em]
&= \frac{k(k+1)(k+2)+3(k+1)(k+2)}{3}\\[1em]
&= \frac{(k+1)(k+2)(k+3)}{3}.
\end{align}
See if you can determine how each line was obtained (where the inductive hypothesis was applied, etc.).
