Question

By the principle of mathematical induction, prove the following:

aⁿ – bⁿ is divisible by a – b, for all n ∈ N.

Answer 1

Let P(n) denote the statement aⁿ – bⁿ is divisible by a – b.

Put n = 1. Then P(1) is the statement: a¹ – b¹ = a – b is divisible by a – b

∴ P(1) is true. Now assume that the statement be true for n = k

(i.e.,) assume P(k) be true, (i.e.,) a^k – b^k is divisible by (a – b) be true.

`=> (a^k - b^k)/(a-b)` = m (say) where m ∈ N

⇒ a^k – b^k = m(a – b)

⇒ a^k = b^k + m(a – b) ……. (1)

Now to prove P(k + 1) is true, (i.e.,) to prove: a^k+1 – b^k+1 is divisible by a – b

Consider a^k+1 – b^k+1 = a^k . a – b^k . b

= [b^k + m(a – b)] a – b^k . b [∵ a^k = bm + k(a – b)]

= b^k . a + am(a – b) – b^k . b

= b^k . a – b^k . b + am(a – b)

= b^k(a – b) + am(a – b)

= (a – b) (b^k + am) is divisible by (a – b)

∴ P(k + 1) is true.

By the principle of Mathematical induction. P(n) is true for all n ∈ N.

∴ aⁿ – bⁿ is divisible by a – b for n ∈ N.