Q. 183.8( 5 Votes )

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

x2n – y2n is divisible by x + y.

Answer :

Let the given statement be P(n), as

P(n): x2n – y2n is divisible by (x + y).

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

P(1): x2 - y2 = (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):x2k - y2k = m(x + y) where m N.

x2k = y2k + m(x + y) ………….(1)

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

P(k + 1):x2k + 2 - y2k + 2

x2k.x2 - y2k + 2

[y2k + m(x + y)]x2 - y2k + 2 From equation(1)

m(x + y)x2 + y2k(x2 - y2)

m(x + y)x2 + y2k(x - y)(x + y)

(x + y)[mx2 + y2k(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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