Question

Question 17 Prove the following by using the principle of mathematical induction for all n∈N: 1/3.5 + 1/5.7 + 1/7.9 + ... + 1/(2n+1)(2n+3) = n/3(2n+3)

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

1/3.5 + 1/5.7 + 1/7.9 +....1/(2n+1)(2n+3) = n/3(2n+3)

Let p(n):1/3.5 + 1/5.7+1/7.9+...1/(2n+1)(2n+3) = n/3(2n+3)

step1:- for n = 1
P(1):1/3.5 = 1/15 = 1/3(2×1+3)
It is true.

step2 :- for n= k
p(k):1/3.5 + 1/5.7 + 1/7.9 +..
..1/(2k+1)(2k+3) = k/3(2k+3) ----(1)

step3:- for n = k + 1
[ 1/3.5 + 1/5.7 + 1/7.9 + ...1/(2k+1)(2k+3)] + 1/{2(k+1)+1}{2(k+1)+3}
= k/3(2k +3) + 1/(2k+3)(2k+5)
= { k(2k+5)+3}/3(2k+3)(2k+5)
= (2k² + 5k +3)/3(2k+3)(2k+5)
= (k+1)(2k+3)/3(2k+3)(2k+5)
= (k+1)/3[2(k+1)+3]
So, p(k+1) is true when p(k) is true , from the principle of mathematical induction , statement is true for all natural numbers.