<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] When $n=1$, it is trival. 
Suppose it is true for $n=k$, ie. $\sum \limits_i^k i(i+1)=\frac{k(k+1)(k+2)}{3}$. 
Consider $n=k+1$. 
$$\begin{align}
\sum \limits_i^{k+1} i(i+1)=&(k+1)(k+2)+\sum \limits_i^{k} i(i+1) \\
=&(k+1)(k+2)+\frac{k(k+1)(k+2)}{3} \\
=&\frac{(k^2+4k+3)(k+2)}{3} \\
=&\frac{(k+1)(k+1+1)(k+1+2)}{3}.
\end{align}
$$
Done!
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