Question

Prove statement by using the Principle of Mathematical Induction for all n ∈ N, that:

2²ⁿ – 1 is divisible by 3.

Answer 1

Let the statement P(n) given as

P(n): 2²ⁿ – 1 is divisible by 3, for every natural number n.

We observe that P(1) is true.

Since 2² – 1 = 4 – 1 = 3

1 is divisible by 3.

Assume that P(n) is true for some natural number k.

i.e., P(k) : 2^2k – 1 is divisible by 3.

i.e., 2^2k – 1 = 3q, where q ∈ N,

Now, to prove that P(k + 1) is true,

We have P(k + 1) : `2^(2(k + 1)) - 1`

= `2^(2k + 2)  –  1`

= 2^2k . 2² – 1

= 2^2k . 4 – 1

= 3.2^2k + (2^2k – 1)

= 3.2^2k + 3q

= 3(2^2k + q) = 3m, where m ∈ N,

Thus P(k + 1) is true, whenever P(k) is true.

Hence, by the Principle of Mathematical Induction P(n) is true for all natural numbers n.