Using the principle of mathematical induction, prove each of the following for all n ϵ N:

(2³ⁿ – 1) is a multiple of 7

To Prove:

, which is multiple of 7

Let us prove this question by principle of mathematical induction (PMI) for all natural numbers

is multiple of 7

Let P(n): , which is multiple of 7

For n = 1 P(n) is true since , which is multiple of 7

Assume P(k) is true for some positive integer k , ie,

= , where m ∈ N …(1)

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

Consider ,

[Adding and subtracting ]

[ Using 1 ]

, where r = is a natural number

Therefore is multiple of 7

Therefore, P (k + 1) is true whenever P(k) is true

By the principle of mathematical induction, P(n) is true for all natural numbers ie, N

Hence proved