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

Prove the following by using the principle of mathematical induction for all n ∈ Nx2n – y2n is divisible by x y.

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

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

x2k – y2k is divisible by x y.

x2k – y2k = m (y), where m ∈ N … (1)

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


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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NCERT Class 11 Mathematics Textbook
Chapter 4 Principle of Mathematical Induction
Q 21 | Page 95
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