Question

Question 14 Prove the following by using the principle of mathematical induction for all n∈N: (1+ 1/1) (1+ 1/2) (1+ 1/3) ... (1+ (1/n)) = (n+1)

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

(1 + 1/1)(1 + 1/2)(1 + 1/3)......(1 + 1/n) = (n+1)
Let P(n): (1+1/1)(1+1/2)(1+1/3)....(1+1/n) = (n+1)

step1 :- for n = 1
P(1): (1+1/1) = 1+1 = 2
it's true.

step2:- for n = k
P(k): (1+1/1)(1+1/2)(1+1/3) ......(1+1/k) = (k+1) __________(1)

step3:- for n = (k+1)
P(k+1): (1 + 1/1)(1+1/2)(1+1/3).....(1+1/(k+1))= (k+1+1) = (k+2)
from eqn (1)
(1+1/1)(1+1/2)(1+1/3).. ..(1+1/k) = (k+1)
multiply both sides, { 1 + 1/(k+1)}
(1 + 1/1)(1+1/2)(1+1/3).....{1+1/(k+1)} = (k+1){1+1/(k+1)}
= (k+1) (k+2)/(k+1)
= (k+2)

hence, P(k+1) is true when p(k) is true . form the principle of mathematical induction , statement is true for all natural numbers .