Question

x^n-y^n is divisible by x-y

Answer 1

Step-by-step explanation:

Let P(n) : xn – yn is divisible by x – y, where x and y are any integers with x≠y.

Now, P(l): x1 -y1 = x-y, which is divisible by (x-y)

Hence, P(l) is true.

Let us assume that, P(n) is true for some natural number n = k.

P(k): xk -yk is divisible by (x – y)

or   xk-yk = m(x-y),m ∈ N …(i)

Now, we have to prove that P(k + 1) is true.

P(k+l):xk+l-yk+l

= xk-x-xk-y + xk-y-yky

= xk(x-y) +y(xk-yk)

= xk(x – y) + ym(x – y)  (using (i))

= (x -y) [xk+ym], which is divisible by (x-y)

Hence, P(k + 1) is true whenever P(k) is true.

So, by the principle of mathematical induction P(n) is true for any natural number n.