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

10^{2n – 1} + 1 is divisible by 11.

Let the given statement be P(n), as

is divisible by 11.

First, we check if it is true for n = 1,

∴ It is true for n = 1.

Now we assume that it is true for some positive integer k, such that

P(k):10^{2k - 1} + 1 = 11m where m ∈ N

10^{2k - 1} = 11m - 1 ………….(1)

We shall prove that P(k + 1)is true,

P(k + 1):10^{2k + 1} + 1

⇒ 10^{2k - 1}.10² + 1

⇒ (11m - 1).100 + 1 From equation(1)

⇒ 1100m - 100 + 1

⇒ 1100m - 99

⇒ 11(100m - 9)

We proved that P(k + 1) is true.

Hence by principle of mathematical induction it is true for all n ∈ N.