Prove by the method of induction that xn-yn is divisible by x-y

Let P(n) be the statement: 

x ⁿ – yⁿ is divisible by x – y. 

 For n = 1, x ¹ – y¹ = x – y is divisible by (x – y)

 Thus, P(1) is true.

Assume that P(k) is true, i.e., for k∈N x ^k – y^k is divisible by (x – y)

Now, For P(k + 1)

 X^k+1  – y^k+1  = x^k . x – y^k y = x^k.x – x^k . y + x^k .y – y^k . y

(Adding and subtracting x^k.y) 

= x^k(x – y) + y (x^k – y^k )

As x^k(x – y) is divisible by (x – y) and (x^k – y^k ) is divisible by 

 (x – y),therefore, x^k+1 – y^k+1 = x^k(x – y) + y(x^k – y^k) is divisible by (x – y)

 P(k + 1) is true, whenever P(k) is true. By mathematical induction method, the statement P(n) is true for all natural number n.