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

{(41)ⁿ – (14)ⁿ} is divisible by 27.

To Prove:

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

Let P(n):

For n = 1 P(n) is true since

which is multiple of 27

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 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.