Question

By the principle of mathematical induction, prove the following:

3²ⁿ – 1 is divisible by 8, for all n ∈ N.

Answer 1

Let P(n) denote the statement 3²ⁿ – 1 is divisible by 8 for all n ∈ N

Put n = 1

P(1) is the statement 3²⁽¹⁾ – 1 = 3² – 1 = 9 – 1 = 8, which is divisible by 8

∴ P(1) is true.

Assume that P(k) is true for n = k.

i.e., 3^2k – 1 is divisible by 8 to be true.

Let 3^2k – 1 = 8m

To prove P(k + 1) is true.

i.e., to prove `3^(2(k+1)) - 1` is divisible by 8

Consider `3^(2(k+1)) - 1` = 3^2k+2 – 1

= 3^2k . 3² – 1

= 3^2k (9) – 1

= 3^2k (8 + 1) – 1

= 3^2k × 8 + 3^2k × 1 – 1

= 3^2k (8) + 3^2k – 1

= 3^2k (8) + 8m (∵ 3^2k – 1 = 8m)

= 8(3^2k + m), which is divisible by 8.

∴ P(k + 1) is true wherever P(k) is true.

∴ By principle of Mathematical Induction, P(n) is true for all n ∈ N.