Q. 183.8( 5 Votes )

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

x^{2n} – y^{2n} is divisible by x + y.

Answer :

Let the given statement be P(n), as

P(n): x^{2n} – y^{2n} is divisible by (x + y).

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

P(1): x^{2} - y^{2} = (x - y)(x + y);

∴ It is true for n = 1.

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

P(k):x^{2k} - y^{2k} = m(x + y) where m ∈ N.

x^{2k} = y^{2k} + m(x + y) ………….(1)

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

P(k + 1):x^{2k + 2} - y^{2k + 2}

⇒ x^{2k}.x^{2} - y^{2k + 2}

⇒ [y^{2k} + m(x + y)]x^{2} - y^{2k + 2} From equation(1)

⇒ m(x + y)x^{2} + y^{2k}(x^{2} - y^{2})

⇒ m(x + y)x^{2} + y^{2k}(x - y)(x + y)

⇒ (x + y)[mx^{2} + y^{2k}(x - y)]

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

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

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