Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: 102n – 1 + 1 is Divisible by 11

Question

Prove the following by using the principle of mathematical induction for all n ∈ N: 10²ⁿ^{– 1}+ 1 is divisible by 11

Solution

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

P(n): 10²ⁿ^{– 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.,

10²^k^{– 1}+ 1 is divisible by 11.

∴10²^k^{– 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.

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