Ex 4.1, 22 - Prove 32n+2 - 8n -9 is divisble by 8 - Chapter 4 - Divisible

Ex 4.1, 22 - Chapter 4 Class 11 Mathematical Induction - Part 2

Ex 4.1, 22 - Chapter 4 Class 11 Mathematical Induction - Part 3
Ex 4.1, 22 - Chapter 4 Class 11 Mathematical Induction - Part 4

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Question22 Prove the following by using the principle of mathematical induction for all n N: 32n + 2 8n 9 is divisible by 8. Introduction If a number is divisible by 8, 16 = 8 2 24 = 8 3 64 = 8 8 Any number divisible by 8 = 8 Natural number Question22 Prove the following by using the principle of mathematical induction for all n N: 32n + 2 8n 9 is divisible by 8. Let P(n): 32n + 2 8n 9 =8d where d N i.e. d is a natural number For n = 1, L.H.S = 32 1 + 2 8 1 9 = 32+2 8 9 = 34 17 = 81 17 = 64 = 8 8 = R.H.S P(n) is true for n = 1 Assume P(k) is true 32k + 2 8k 9 = 8m; where m N We will prove that P(k + 1) is true. L.H.S = 32(k+1)+2 8(k+1) 9 = 32k+2 + 2 8k 8 - 9 = 32k+2. 32 8k 8 - 9 = 9 (32k+2) 8k 17 = 9 (8k + 9 + 8m) 8k 17 = 9 8k + 9 9 + 9 8m 8k 17 = 9 8k + 81 + 9 8m 8k 17 = 9 8k 8k + 81 17 + 9 8m = 9 8k 8k + 64 + 9 8m = 8k (9 1) + 64 + 9 8m = 8k 8 + 64 + 9 8m = 8k 8 + 8 8 + 9 8m = 8 (8k + 8 + 9m ) = 8r, where r =(9m + 8k + 8) is a 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

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo