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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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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 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.

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.

  Is there an error in this question or solution?
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APPEARS IN

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