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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