Q8 of 30 Page 70

For any natural number n, xn – yn is divisible by x – y, where x integers with x ≠ y.

Given; P(n) = xn – yn is divisible by x – y, x integers with x ≠ y.

P(0) = x0 – y0 = 0; is divisible by x − y.


P(1) = x − y ; is divisible by x − y.


P(2) = x2 – y2


= (x +y)(x−y); is divisible by x−y.


P(3) = x3 – y3


= (x−y)(x2+xy+y2); is divisible by x−y.


Let P(k) = xk – yk is divisible by x – y;


⇒ xk – yk = a(x−y).


⇒ P(k+1) = xk+1 – yk+1


= xk(x−y) + y(xk−yk)


= xk(x−y) +y a(x−y); is divisible by x − y.


⇒ P(k+1) is true when P(k) is true.


∴ By Mathematical Induction P(n) xn – yn is divisible by x – y, where x integers with x ≠ y; is true for any natural number n.


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