<s>[INST] Mathemathic induction proof I need to prove that $$\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4} + \cdots + \frac{1}{n(n+1)}=\frac{n}{n+1}$$
Here is what I tried:
\begin{align}
& \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4} + \cdots + \frac{1}{n(n+1)}+\frac{1}{n+1(n+2)} \\[10pt]
= {} & \frac{n}{n+1}+\frac{1}{n+1(n+2)} = \frac{(n+1)(n+2)+(n+1)}{n^2(n+2)}
\end{align}
I am stuck from here, I feel there must be someway to simplify the last part to get to
$$\frac{n+1}{n+2}$$
 [/INST] $$\frac{n}{n+1}+\frac{1}{(n+1)(n+2)}=\frac{n(n+2)+1}{(n+1)(n+2)}=\frac{(n+1)^2}{(n+1)(n+2)}$$
</s>