Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: X2n – Y2n is Divisible by X + Y.

Question

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

Solution

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 x²^×¹ – y²^×¹ = x² – y² = (x + y) (x – y) is divisible by (x + y).

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

x²^k – y²^k is divisible by x + y.

∴x²^k – y²^k = 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.

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Principle of Mathematical Induction

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