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

Explanation

Let `P(n) : x^(n) - y^(n)` is divisible by `x-y`, where x and y are any integers such that `x ne y , n in N`.
For `n = 1`
`P(1) = x^(1) - y^(1) = x-y`, which is divisible by `(x-y)`.
So, `P(1)` is true.
Let `P(n)` be true for some `n = k`
Then `x^(k) - y^(k)` is divisible by `(x-y)`.
or `x^(k) - y^(k) = m(x-y), m in N"......"(1)`
Now, `x^(k+1) - y^(k+1)`
`= x^(k) xx x - x^(k) xx y + x^(k ) xx y- y^(k) xx y`
`= x^(k) (x-y) + y(x^(k)-y^(k))`
`= x^(k) (x-y) + ym(x-y) , [Using (i)]` `= (x-y) [x^(k)+ym]` , which is divisible by `(x-y)`
Thus, `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.