Question

Question 20 Prove the following by using the principle of mathematical induction for all n∈N: 10^(2n – 1) + 1 is divisible by 11.

Class X1 - Maths -Principle of Mathematical Induction Page 95

Answer 1

10^(2n-1) +1,is divisible by 11.
Let P(n):10^(2n-1)+1

step 1:- for n= 1
P(1): 10^(2-1)+1 =11, is divisible by 11.

step2:- for n= k
P(k):10^(2k-1)+1, is divisible by 11.
Let , 11L = 10^(2k-1)+1

step3:- for n= k+1
P(k+1):10^{2(k+1)-1}+1
= 10^{(2k-1)+2} +1
= 10^(2k-1).10² + 1
= (11L -1)100 + 1
= 100×11L -99
= 11( 100L - 9)
It's clear that it's divisible by 11.
P(k+1) is true when p(k) is true.hence, from the principle of mathematical induction, statement is true for all natural numbers.