<s>[INST] How do I prove this using mathematical induction? \begin{align}
1\cdot3+2\cdot4+3\cdot5+...+n(n+2) = \frac{n(n+1)(2n+7)}{6}
\end{align}
Using the mathematical induction step I arrive at this :
\begin{align}
1\cdot3+2\cdot4+3\cdot5+...+n+1(n+3) = \frac{n+1(n+2)(2n+9)}{6}
\end{align}
And I don't see any other way to continue except to divide
\begin{align}
n+1(n+3)
\end{align}
into
\begin{align}
n(n+2)+something
\end{align}
and substitute it with the beginning of the fraction.
But that doesn't get me anywhere.
 [/INST] First prove the basis of the induction: If $n=1$, then 
$$1*3=^? \frac{(1)(1+1)(2(1)+7)}{6}=\frac{1\cdot 2\cdot 9}{6}=3$$
So efectivelly, $1*3=3$. Suppose by induction hypothesis that 
$$1*3+2*4+\dots + k*(k+2)=\frac{k(k+1)(2k+7)}{6}$$
Then we prove it for $n=k+1$:
$$1*3+2*4+\dots+ k(k+2)+(k+1)(k+3)=\frac{k(k+1)(2k+7)}{6}+(k+1)(k+3)$$
$$\frac{k(k+1)(2k+7)}{6}+(k+1)(k+3)=(k+1)\left( \frac{k(2k+7)}{6}+k+4\right)$$
$$(k+1)\left( \frac{k(2k+7)}{6}+k+3\right)=\frac{(k+1)}{6}\left( 2k^2+7k+6k+18\right)$$
$$\frac{(k+1)}{6}\left( 2k^2+7k+6k+18\right)=\frac{(k+1)(k+2)(2k+9)}{6}$$
$$\frac{(k+1)((k+1)+2)(2(k+1)+7)}{6}$$
That is exactly the equation for $k+1$. So it's done.
</s>