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Example 4 - Prove that 7n - 3n is divisible by 4 - Chapter 4 - Examples

Example 4 - Chapter 4 Class 11 Mathematical Induction - Part 2
Example 4 - Chapter 4 Class 11 Mathematical Induction - Part 3
Example 4 - Chapter 4 Class 11 Mathematical Induction - Part 4

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Example 4 For every positive integer n, prove that 7n – 3n is divisible by 4 Introduction If a number is divisible by 4, 8 = 4 × 2 16 = 4 × 4 32 = 4 × 8 Any number divisible by 4 = 4 × Natural number Example 4 For every positive integer n, prove that 7n – 3n is divisible by 4. Let P(n) : 7n – 3n = 4d ,where d ∈ N For n = 1, L.H.S = 71 – 31 = 7 – 3 = 4 = 4 × 1 = R.H.S ∴ P(n) is true for n = 1 Assume P(k) is true P(k) : 7k – 3k = 4m, where m ∈ N We will prove that P(k + 1) is true. L.H.S = 7(k+1) – 3(k+1) = 7k .71 – 3k.31 = 7 . 7k – 3 . 3k = 7.(4m + 3k) – (3).3k = 7 × 4m + 7 × 3k – (3).3k = 7 × 4m + 3k (7 – 3) = 7 × 4m + 3k (4) = 4(7m + 3k) = 4r where r = (7m + 3k) 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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.