Question

Prove the following by using the principle of mathematical induction for all n ∈ N: 10^{2n – 1} + 1 is divisible by 11.

Answer 1

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

P(n): 10^{2n – 1} + 1 is divisible by 11.

It can be observed that P(n) is true for n = 1 since P(1) = 10^{2.1 – 1} + 1 = 11, which is divisible by 11.

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

10^{2k – 1} + 1 is divisible by 11.

∴10^{2k – 1} + 1 = 11m, where m ∈ N … (1)

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

Consider

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.