Prove the following by the principle of mathematical induction: `1/(1. 2)+1/(2. 3)+1/(3. 4)++1/(n(n+1))=n/(n+1)`

Explanation

Let P (n) : `(1)/(1.2) +(1)/(2.3)+(1)/(3.4)`
`+…….+ (1)/(n.(n+1))=(n)/(n+1)`
For n=1
`L.H.S. =(1)/(1.2)=(1)/(2)`
`R.H.S. =(1)/(1+1) =(1)/(2)`
`L.H.S. + R.H.S.`
Therefore , P (n) is true for n=1
Let P (n) is true for n=K
`P (K) : (1)/(1.2)+(1)/(2.3)+(1)/(3.4) +.....+(1)/(K(K+1)) =(K)/(K+1)`
`" Adding "(1)/((K+1)(K+2))` on both sides
`(1)/(1.2)+(1)/(2.3)+(1)/(3.4)+.......+(1)/(K(K+1)) +(1)/((K+1)(K+2))`
`=(K)/(K+1) +(1)/((K+1)(K+2))`
`(K(K+2)+2)/((K+1)(K+2))`
`(K^(2)+2K+1)/((K+1)(K+2))`
`((K+1)^(2))/((K+1)(K+2))=(K+1)/((K+1)+1)`
`rArr` P (n) is also true for n =K+1
Hence by the principle of mathematical induction given statement is true for all natural numbers 'n'