  Question

By principle of mathematical Induction show that an​​​​​​-bn  is divisible by a+b when n is a positive integer.

Solution

Using the principle of mathematical induction, prove that (xn - yn) is divisible by (x - y)for all n ∈ N. Solution: Let the given statement be P(n). Then,  P(n): (xn - yn) is divisible by (x - y).  When n = 1, the given statement becomes: (x1 - y1) is divisible by (x - y), which is clearly true.  Therefore P(1) is true.  Let p(k) be true. Then,  P(k): xk - yk is divisible by (x-y).  Now, xk + 1 - yk + 1 = xk + 1 - xky - yk + 1                               [on adding and subtracting x)ky]  = xk(x - y) + y(xk - yk), which is divisible by (x - y) [using (i)]  ⇒ P(k + 1): xk + 1 - yk + 1is divisible by (x - y)  ⇒ P(k + 1) is true, whenever P(k) is true.  Thus, P(1) is true and P(k + 1) is true, whenever P(k) is true.  Hence, by the Principal of Mathematical Induction, P(n) is true for all n ∈ N.   Mathematics

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