For all n ≥ 1, prove that 1/1.2 + 1/2.3 + 1/3.4 + .......... + 1/[n(n+1)] = n/(n+1)

Question

For all n ≥ 1, prove that

\(\frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4}\) + ......... + \(\frac{1}{n(n+1)} = \frac{n}{n+1}\)

Answer 1

Let p(n): \(\frac{1}{1.2} + \frac{1}{2.3} + \frac{1}{3.4}\) + ......... + \(\frac{1}{n(n+1)} = \frac{n}{n+1}\)

put n =1

p(1) = \(\frac{1}{2}\) = \(\frac{1}{(1+1)} = \frac{1}{2}\) Which is true.

Assuming that true for p(k)

Hence by using the principle of mathematical induction true for all n ∈ N.