1/1.2 + 1/2.3 = 1/3.4 +.........+ 1/n(n+1 )= n/n+1 using principle of mathematical induction

Let P(n) be the statement that 1/1.2 + 1/2.3 = 1/3.4 +.........+ 1/n(n + 1 ) =  n /( n + 1) 

So P(1) is true as 1/2 = 1/(1+1)

Let P(k) be true , hence 1/1.2 + 1/2.3 = 1/3.4 +.........+ 1/k(k + 1 ) =  k /( k + 1)  (1)

So P(k+1) = P(k) + $\frac{1}{(k+1).(k+2)}$ [using (1)] , we have
= $\frac{k}{k+1} +\frac{1}{(k+1)(k+2)} =\frac{1}{k+1}(k+\frac{1}{k+2}) =\frac{1}{k+1}\left(\frac{k^2+2k+1}{k+2}\right) =\frac{(k+1{)}^2}{(k+1)(k+2)}=\frac{k+1}{k+2}$
= $\frac{(k+1)}{\left[\right(k+1)+1]}$

Hence P(k+1) is also true .

So P(n) is also true .