<s>[INST] 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.
 [/INST] Let $ 2 = \frac{1 \cdot 2 \cdot 3}{3} $ be the basis. The inductive step consists of simple distribution of multiplication:
$$
\begin{align}&\frac{n(n+1)(n+2)}{3} + (n+1)(n+2) \\&= \tfrac{1}{3}n^3+2n^2+\tfrac{11}{3}n+2 \\&= \frac{(n + 1)\Big((n+1)+1\Big)\Big((n + 1)+2\Big)}{3}\end{align}
$$
</s>