Question

By using principle of mathematical induction for every natural number, (ab)ⁿ = ______.

Options

• aⁿbⁿ

• aⁿb

• abⁿ

• 1

Answer 1

By using principle of mathematical induction for every natural number, (ab)ⁿ = aⁿbⁿ.

Explanation:

Let P(n) be the given statement,

i.e. P(n) : (ab)ⁿ = aⁿbⁿ

We note that P(n) is true for n = 1, since (ab)¹ = a¹ b¹

Let P(k) be true, i.e. (ab)^k = a^k b^k ...(i)

We shall now prove that P(k + 1) is true whenever P(k) is true.

Now, we have (ab)^{k + 1}

= (ab)^k (ab)

= (a^k b^k) (ab) ...[by using (i)]

= (a^k . a¹) (b^k . b¹) = a^{k + 1} . b^{k + 1}

Therefore, P(k + 1) is also true whenever P(k) is true. Hence, by principle of mathematical induction, P(n) is true for all n ∈ N.