Question

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

Answer 1

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): x^k - y^k is divisible by (x-y).

Now, x^{k + 1} - y^{k + 1} = x^{k + 1} - x^ky - y^{k + 1}

[on adding and subtracting x)^ky]

= x^k(x - y) + y(x^k - y^k), which is divisible by (x - y) [using (i)]

⇒ P(k + 1): x^{k + 1} - y^{k + 1}is 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.