Question

Prove the following by using the principle of mathematical induction for all n ∈ N: x²ⁿ – y²ⁿ is divisible by x + y.

Answer 1

Let the given statement be P(n), i.e.,

P(n): x²ⁿ – y²ⁿ is divisible by x + y.

It can be observed that P(n) is true for n = 1.

This is so because x^{2 × 1} – y^{2 × 1} = x² – y² = (x + y) (x – y) is divisible by (x + y).

Let P(k) be true for some positive integer k, i.e.,

x^2k – y^2k is divisible by x + y.

∴x^2k – y^2k = m (x + y), where m ∈ N … (1)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Consider

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.