Example 6 - Mathematical Induction

Last updated at April 16, 2024 by

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

Example 6 - Chapter 4 Class 11 Mathematical Induction - Part 2

Example 6 - Chapter 4 Class 11 Mathematical Induction - Part 3
Example 6 - Chapter 4 Class 11 Mathematical Induction - Part 4

Transcript

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

Go Ad-free
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 14 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.