Mathematical Induction
Serial order wise

Ex 4.1, 20 - Prove: 102n-1 + 1 is divisble by 11 - Chapter 4 - Divisible

Ex 4.1, 20 - Chapter 4 Class 11 Mathematical Induction - Part 2
Ex 4.1, 20 - Chapter 4 Class 11 Mathematical Induction - Part 3 Ex 4.1, 20 - Chapter 4 Class 11 Mathematical Induction - Part 4

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Question20 Prove the following by using the principle of mathematical induction for all n ∈ N: 102n – 1 + 1 is divisible by 11. Introduction If a number is divisible by 11, 22 = 11 × 2 = 11 × 7 = 11 × 9 Any number divisible by 11 = 11 × Natural number Question20 Prove the following by using the principle of mathematical induction for all n ∈ N: 102n – 1 + 1 is divisible by 11. Let P(n): 102n – 1 + 1 = 11d where d ∈ N For n=1, L.H.S = 102.1 – 1 + 1 = 101 + 1 = 10 + 1 = 11 = 11 × 1 = R.H.S ∴P(n) is true for n = 1 Assume P(k) is true 102k – 1 + 1 = 11m, where m ∈ N We will prove that P(k + 1) is true. L.H.S = 102(k+1) – 1 + 1 = 102k+2 - 1 + 1 = 10( 2k - 1) + 2 + 1 = 10(2k - 1) .102 + 1 = (11m – 1) .102 + 1 = (11m – 1) × 100 + 1 = 100 × 11m – 100 + 1 = 100 × 11m – 99 = 100 × 11m – 9 × 11 = 11 (100m – 9) = 11r ,where r = (100m – 9) is some natural number ∴ P(k + 1) is true whenever P(k) is true. ∴ By the principle of mathematical induction, P(n) is true for n, where n is a natural number

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.