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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