2^2n – 1 is divisible by 3.

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