Prove the following by using the principle of mathematical induction for all nN: x 2 n y 2 n is divisible by x + y.

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

P(n): x2ny2n is divisible by x + y.

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

This is so because x2 × 1y2 × 1 = x2y2 = (x + y) (xy) is divisible by (x + y).

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

x2ky2k is divisible by x + y.

x2ky2k = m (x + y), where mN … (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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