So I took the equality for $n = k$ and tried to use it to prove the equality for $n = k+1$. I was unable to however get to the answer.
PS - I don't know how to type an equation here so I have attached a screenshot of the problem. Sorry for the inconvenience
Prove using mathematical induction: $$ \frac{3}{1\cdot2} + \frac{4}{2\cdot3}+\frac{5}{3\cdot4}+....+\frac{n+1}{(n-1)n}+\frac{n+2}{n(n+1)} = 1+ \frac1 2 + \frac13 +....+\frac1{n} +\frac n {n+1} $$
Hint:
Induction step comes to proving that: $$\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n-1}+\frac{1}{n}+\frac{n}{n+1}\right)+\frac{n+3}{\left(n+1\right)\left(n+2\right)}=\cdots$$$$=1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n-1}+\frac{1}{n}+\frac{1}{n+1}+\frac{n+1}{n+2}$$
or equivalently: $$\frac{n}{n+1}+\frac{n+3}{\left(n+1\right)\left(n+2\right)}=\cdots=\frac{1}{n+1}+\frac{n+1}{n+2}$$Can you do that yourself?