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Using the principle of mathematical induction, prove each of the following for all n ϵ N: {(41)n – (14)n } is divisible by 27.

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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.

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