Using the principle of mathematical induction, prove that (2³ⁿ – 1) is divisible by 7 for all n ∈ N.

Let P(n) = 2³ⁿ – 1,

For n = 1,

P (1) = 2³⁽¹⁾ -1

= 8-1

= 7

Which is divisible by 7.

Hence P (1) is true.

Consider P(k) to be true.

i.e. 2^3(k) -1 is divisible by 7.

⇒ 2^3(k) -1 = 7λ for some λ ∈ N

We have to prove that P(k+1) is true i.e. 2^3(k+1) -1 is divisible by 7.

Now,

2^3(k+1) -1 = 2^3k × 2³ – 1

= (7λ +1) 2³ – 1

= 56λ + 8 – 1

= 7(8λ +1)

Which is divisible by 7.

∴ P(k+1) is true when P(k) is true.

Hence by Mathematical induction P(n) is true for all n ∈ N.