Question

Question 13 Prove the following by using the principle of mathematical induction for all n∈N: (1+ 3/1) (1+ 5/4) (1+ 7/9) ... (1+ (2n+1)/n^2) = (n+1)^2

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

(1 + 3/1)(1+5/4)(1+7/9) .......{1 + (2n+1)/n²} = (n+1)²

Let P(n): (1 + 3/1)(1+5/4)(1+7/9) .......{1 + (2n+1)/n²} = (n+1)²

step1:- for n = 1
P(1) : (1 + 3/1) = (1 + 3) = 4
= (1+1)² = 2² = 4
which is true.

step2:- for n = k
P(k): (1 + 3/1)(1 + 5/4)(1+7/9) ......{1 + (2k+1)/k²} = (k+1)²__________(1)

step3 :- for n = (k+1)
P(k+1): (1 + 3/1)(1 + 5/4)(1 + 7/9) .......{1 +(2k+3)/(k+1)²} = (k+2)²

from equation (1),
(1 +3/1)(1 + 5/4)(1 + 7/9) ........{1 + (2k+1)/k²} = (k+1)²
add ' {1 + (2k+3)/(k+1)²}' both sides,
(1 + 3/1)(1 + 5/4)(1 + 7/9) .......{1 + (2k+1)/k²} = (k+1)²{1 + (2k+3)/(k+1)²}
= (k+1)²{(k+1)² + (2k+3)}/(k+1)²
= (k²+2k+1 +2k+3}
= {k² + 4k + 4}
= (k+2)²
hence, P(k+1) is true when P(k) is true. from the mathematical induction , statement is true for all natural numbers.