Induction method is used to prove a statement. Most commonly, it is used to prove a statement, involving, say n where n represents the set of all natural numbers.
Induction method involves two steps, One, that the statement is true for n=1 and say n=2. Two, we assume that it is true for n=k and prove that if it is true for n=k, then it is also true for n=k+1.
First Step - Now for 1/(1*2)+1/(2*3)+...+1/(n(n+1))=n/(n+1), we know for n=1, we have 1/(1*2)=1/2 and for n=2, we have 1/(1*2)+1/(2*3)=1/2+1/6=2/3=2/(2+1).
Hence, given statement is true for n=1 and n=2.
Second Step - Assume it is true for n=k, hence
1/(1*2)+1/(2*3)+...+1/(k(k+1))=k/(k+1)
Now let us test it for n=k+1 i.e.
1/(1*2)+1/(2*3)+...+1/(k(k+1))+1/((k+1)(k+2))
= k/(k+1)+1/((k+1)(k+2))
= (k(k+2)+1)/((k+1)(k+2))
= (k^2+2k+1)/((k+1)(k+2))
= (k+1)^2/((k+1)(k+2))
= (k+1)/(k+2)
Hence we see that the statement is true for n=k+1 if it is true for n=k.
Hence 1/(1*2)+1/(2*3)+...+1/(n(n+1))=n/(n+1)
