Prove by the method of induction, for all n ∈ N. 1/1.3 + 1/3.5 + 1/5.7 + ....+ 1/((2n-1)(2n+1)) = n/(2n+1)

Question

Prove by the method of induction, for all n ∈ N.

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

Answer 1

Let P(n) =

 1/1.3 + 1/3.5 + 1/5.7 + ....+ \(\frac{1}{(2n-1)(2n+1)}=\frac{n}{2n+1}\) , for all n ∈ N.

Step I:

Put n = 1

L.H.S.= \(\frac1{1.3}=\frac13\)

R.H.S. = \(\frac{1}{2(1)+1}=\frac13\)

∴ L.H.S. = R.H.S.

∴ P(n) is true for n = 1.

Step II: Let us assume that P(n) is true for n = k.

∴ 1/1.3 + 1/3.5 + 1/5.7 + ....+ \(\frac{1}{(2k-1)(2k+1)}=\frac{k}{2k+1}\)......(i)

Step III: We have to prove that P(n) is true for n = k + 1,

i.e., to prove that

∴ P(n) is true for n = k + 1.

Step IV: From all the steps above, by the principle of mathematical induction, P(n) is true for all n ∈ N.

\(\therefore\) 1/1.3 + 1/3.5 + 1/5.7 + ....+ \(\frac{1}{(2n-1)(2n+1)}=\frac{n}{2n+1}\), for all n ∈ N.