Ex 4.1, 22

Transcript

Ex 4.1,22 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 Ex 4.1,22 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