M2 Mathematical Induction

6) Let P(n) be the statement "8n + 2 x 7n - 1 is divisible by 7", then:

When n = 1, 8n + 2 x 7n - 1 = 21 which is divisible by 7, hence P(1) is true.

Suppose that P(k) is true for some positive integer k, i.e.

8k + 2 x 7k - 1 = 7M where M is some natural no.

When n = k + 1:

8k+1 + 2 x 7k+1 - 1 = 8 x 8k + 14 x 7k - 1

= 8 (8k + 2 x 7k - 1) - 2 x 7k + 7

= 8 (7M) - 2 x 7k + 7

which is also divisible by 7.

Hence P(k + 1) is also true and by the principle of MI, P(n) is true for all positive integers n.

7) Let P(n) be the statement "32n+2 - 8n - 1 is divisible by 64", then:

When n = 1, 32n+2 - 8n - 1 = 64 which is divisible by 64, hence P(1) is true.

Suppose that P(k) is true for some positive integer k, i.e.

32k+2 - 8k - 1 = 64M where M is some natural no.

When n = k + 1:

32k+4 - 8k - 9 = 9 x 32k+2 - 8k - 9

= 9 (32k+2 - 8k - 1) + 64k

= 9 (64M) + 64k

which is also divisible by 64.

Hence P(k + 1) is also true and by the principle of MI, P(n) is true for all positive integers n.

Ching 炒錯題牙
7) Let P(n) be the statement "32n+2 - 8n - 9