Prove the Following by Using the Principle of Mathematical Induction for All N ∈ N: 32n + 2 – 8n – 9 is Divisible by 8.

Question

Prove the following by using the principle of mathematical induction for all n ∈ N: 3²ⁿ^{+ 2} – 8n– 9 is divisible by 8.

Solution

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

P(n): 3²ⁿ^{+ 2} – 8n – 9 is divisible by 8.

It can be observed that P(n) is true for n = 1 since 3²^×^{1 + 2} – 8 × 1 – 9 = 64, which is divisible by 8.

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

3²^k^{+ 2} – 8k – 9 is divisible by 8.

∴3²^k^{+ 2} – 8k – 9 = 8m; 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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Principle of Mathematical Induction

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