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Example 6 - Mathematical Induction

Last updated at May 29, 2023 by

Example 6 - Prove that 2.7n + 3.5n - 5 is divisible by 24 - Divisible

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Example 6 Prove that 2.7n + 3.5n 5 is divisible by 24, for all n N. Introduction If a number is divisible by 24, 48 = 24 2 72 = 24 3 96 = 24 4 Any number divisible by 24 = 24 Natural number Example 6 Prove that 2.7n + 3.5n 5 is divisible by 24, for all n N. Let P(n) : 2.7n + 3.5n 5 = 24d, when d N For n = 1, L.H.S = 2.71 + 3.51 5 = 2.7 + 3.5 5 = 14 + 15 5 = 24 = 24 1 = R.H.S , P(n) is true for n = 1 Assume P(k) is true 2.7k + 3.5k 5 = 24m, when m N We will prove that P(k + 1) is true. L.H.S = 2.7k+1 + 3.5k+1 5 = 2.7k . 71 + 3.5k . 51 5 = 7. (2.7k) + 5 . 3.5k 5 = 7 [24m 3.5k + 5] + 15.5k 5 = 7 24m (7 3). 5k + (7.5) + 15.5k 5 = 7 24m 21. 5k + 35 + 15.5k 5 = 7 24m 21. 5k + 15.5k + 35 5 = 7 24m 6.5k + 30 = 7 24m 6 (5k 5) (5k 5) is a multiple of 4 = 7 24m 6 (4p) = 7 24m 24p = 24 (7m p) = 24 r; where r = 7m p, 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

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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.