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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.,
32k + 2 – 8k – 9 is divisible by 8.
∴32k + 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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