Sets

Question

Prove the following by using the principle of mathematical induction for all

is divisible by 8.

Answer

Let is divisible by 8.**I. **For n = 1,

P(1) : is divisible by 8

is divisible by 8 81 - 17 is divisible by 8 64 is divisible by 8

which is true

∴ P(n) is true for n = 1**II. **Let the statement be true for n = m,

∴ is divisible by 8

...(i)**III. **For n = m + 1,

is divisible by 8

Now,

= [By (i)]

= 72k + 72m + 81 - 8m - 17 = 72k + 64m + 64 = 8(9k + 8m + 8),

= 8k' where k' = 9k + 8m +

∴ is divisible by 8.

P(m + 1) is true.

∴ P(m) is true P (m + 1) is true.

Hence, by the principle of mathematical induction, P(n) is true for all