Question

Prove the following by using the principle of mathematical induction for all n ∈ N:

Answer 1

Let the given statement be P(n), i.e.,

It can be noted that P(n) is true for n = 1 since .

Let P(k) be true for some positive integer k, i.e.,

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

Consider

Hence,

Thus, P(k + 1) is true whenever P(k) is true.

Hence, by the principle of mathematical induction, statement P(n) is true for all natural numbers i.e., n.