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

(23n – 1) is a multiple of 7

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To Prove:

23n – 1 is a multiple of 7

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

23n – 1 is a multiple of 7

Let P(n): 23n – 1 which is a multiple of 7

For n = 1 P(n) is true since 23 – 1 = 8 - 1 = 7, which is multiple of 7

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

= 23k – 1 = 7m, where m ∈ N …(1)

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

Consider,

[Adding and subtracting 23]

= 7 x r , where r = 2m + 1 is a natural number

Therefore 23n - 1 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

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