For all n ≥ 1. prove that 1/1.3 + 1/3.5+1/5.7+....+ 1/[(2n-1)(2n+1)] = n/(2n+1)

Question

For all n ≥ 1. prove that

\(\frac{1}{1.3}+ \frac{1}{3.5}+ \frac{1}{5.7} + .....+ \frac{1}{(2n-1)(2n+1)} = \frac{n}{(2n+1)}\)

Answer 1

\(\frac{1}{1.3}+ \frac{1}{3.5}+ \frac{1}{5.7} + .....+ \frac{1}{(2n-1)(2n+1)}\)

Put n = 1

p(1) = \(\frac{1}{3} = \frac{1}{(2+1)} = \frac{1}{3}\) which is true.

Assuming that true of p(k)

p(k) : \(\frac{1}{1.3}+ \frac{1}{3.5}+ \frac{1}{5.7} + .....+ \frac{1}{(2n-1)(2n+1)}\) = \(\frac{k}{(2k + 1)}\)

Let p(k+1):

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