Using the principle of mathematical induction. Prove that `(x^(n)-y^(n))` is divisible by (x-y) for all ` n in N`.

Using the principle of mathematical induction. Prove that `(x^(n)-y^(n

1.	Using the principle of mathematical induction. Prove that `(x^(n)-y^(n))` is divisible by (x-y) for all ` n in N`.
Answer» Let the given statement be P(n). Then, `P(n): (x^(n)-y^(n))` is divisible by (x - y). When n =1, the given statement becomes: `(x^(1)-y^(1))` is divisible by (x-y), which is clearly true. ` :. ` P(1) is true. Let P(k) be true. Then, `P(k): (x^(k)-y^(k))` is divisible by (x - y). ...(i) Now, `(x^(k+1)-y^(y+1))` ` ={x^(k+1)-x^(k)y+x^(k)y-y^(k+1)}` [on adding and subtracting ` x^(k) y`] ` = x^(k) (x-y)+y(x^(k)-y^(k))`, which is divisible by (x - y) [using (i)]. Thus, P(k+1) is true, whenever P(k) is true. `:. `P(1) is true and P(k+1) is true, whenever P(k) is true. Hence, by the principle of mathematical induction, it follows that `(x^(n)-y^(n))` is divisible by (x-y) for all ` n in N`.