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 kN x k – yk is divisible by (x – y)
Now, For P(k + 1)
Xk+1 – yk+1 = xk . x – yk y = xk.x – xk . y + xk .y – yk . y
(Adding and subtracting xk.y)
= xk(x – y) + y (xk – yk )
As xk(x – y) is divisible by (x – y) and (xk – yk ) is divisible by
(x – y),therefore, xk+1 – yk+1 = xk(x – y) + y(xk – yk) is divisible by (x – y)
P(k + 1) is true,...
