To Prove:
41n – 14n is divisible by 27
Let us prove this question by principle of mathematical induction (PMI) for all natural numbers
Let P(n):41n – 14n is divisible by 27
For n = 1 P(n) is true since 41n – 14n = 411 – 141 = 27
Which is multiple of 27
Assume P(k) is true for some positive integer k , ie,
= 41n – 14n is divisible by 27
∴ 41k – 14k = m x 27, where m ∈ N …(1)
We will now prove that P(k + 1) is true whenever P( k ) is true
Consider
[Adding and subtracting 14k]
is a natural number
Therefore 41k+1 - 14k+1 is divisible of 27
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.