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 = 23 m + 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