7 divides 2³ⁿ-1

Let the given statement be P(n), i.e.,

P(n): 2³ⁿ – 1 is divisible by 7.

It can be observed that P(n) is true for n = 1 since 2^3 × 1 –  1 = 8 – 1 = 7, which is divisible by 7.

Let P(k) be true for some positive integer k, i.e.,

P(k): 2^3k – 1 is divisible by 7.

∴2^3k – 1 = 7m; where m ∈ N … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

Therefore, 2³ⁿ – 1 is divisible by 7.

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

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.