Prove the following using the principle of mathematical induction for all n ∈ N

41ⁿ – 14ⁿ is a multiple of 27.

Let the given statement be P(n), as

P(n):41ⁿ - 14ⁿ is divisible by 27.

First, we check if it is true for n = 1,

P(1):41¹ - 14¹ = 27;

∴ It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

P(k):41^k - 14^k = 27m where m ∈ N.

41^k = 14^k + 27m ………….(1)

We shall prove that P(k + 1)is true,

P(k + 1):41^{k + 1} - 14^{k + 1}

⇒ 41^k.41 - 14^{k + 1}

⇒ (14^k + 27m)41 - 14^{k + 1} From equation(1)

⇒ 27.41m + 14^k(41 - 14)

⇒ 27.41m + 14^k.27

⇒ 27(41m + 14^k)

We proved that P(k + 1) is true.

Hence by principle of mathematical induction it is true for all n ∈ N.