Answer
See below.
Work Step by Step
Proofs using mathematical induction consist of two steps:
1) The base case: here we prove that the statement holds for the first natural number.
2) The inductive step: assume the statement is true for some arbitrary natural number $n$ larger than the first natural number. Then we prove that then the statement also holds for $n + 1$.
Hence, here we have:
1) For $n=1: \frac{1}{1(2)}=\frac{1}{1+1}$.
2) Assume for $n=k: \frac{1}{1(2)}+ \frac{1}{2(3)}++ \frac{1}{k(k+1)}= \frac{k}{k+1}$. Then for $n=k+1$:
$ \frac{1}{1(2)}+ \frac{1}{2(3)}++ \frac{1}{k(k+1)}+\frac{1}{(k+1)(k+2)}= \frac{k}{k+1}+\frac{1}{(k+1)(k+2)}=\frac{k+1}{(k+2)}$
Thus we proved what we wanted to.