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)
[Adding and subtracting y2k]
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