Question

Prove that xⁿ – yⁿ is divisible by x – y; when n is a + ve integer.

Answer 1

Let T(n) be the statement: 

xⁿ – yⁿ is divisible by x – y. 

Basic Step:

For n = 1, x¹ – y¹ = x – y is divisible by (x – y) 

⇒ T(1) is true 

Induction Step: 

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

Now, we prove T(k + 1) is true. 

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 ) Since x^k(x – y) is divisible by (x – y) and (x^k – y^k ) is divisible by (x – y) (By induction step, i.e., assuming T(k) is true), therefore,

x^k+1 – y^k+1 = x^k(x – y) + y(x^k – y^k) is divisible by (x – y) ⇒ T(k + 1) is true, whenever T(k) is true. 

⇒ T(n) holds for all positive integral values of n.