Using the principle of mathematical induction, prove each of the following for all n ϵ N:  (x^(2n) – y^(2n)) is divisible by (x + y).

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Using the principle of mathematical induction, prove each of the following for all n ϵ N:

(x2n – y2n) is divisible by (x + y).

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To Prove:

x2n – y2n is divisible by x + y

Let us prove this question by principle of mathematical induction (PMI) for all natural numbers

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

For n = 1 P(n) is true since x2n – y2n = x2 - y2 = (x + y) x (x - y)

Which is divisible by x + y

Assume P(k) is true for some positive integer k , ie,

= x2k – y2k is divisible by x + y

Let x2k – y2k = m x (x + y), where m ϵ N.....(1)

We will now prove that P(k + 1) is true whenever P( k ) is true

Consider,

= x2(k + 1) - y2(k + 1)

which is factor of (x + y)

Therefore, P (k + 1) is true whenever P(k) is true

By the principle of mathematical induction, P(n) is true for all natural numbers ie, N

Hence proved