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Question
By the principle of mathematical induction, prove the following:
2n > n, for all n ∈ N.
Solution
Let P(n) denote the statement 2n > n for all n ∈ N
i.e., P(n): 2n > n for n ≥ 1
Put n = 1, P(1): 21 > 1 which is true.
Assume that P(k) is true for n = k
i.e., 2k > k for k ≥ 1
To prove P(k + 1) is true.
i.e., to prove 2k+1 > k + 1 for k ≥ 1
Since 2k > k
Multiply both sides by 2
2 . 2k > 2k
2k+1 > k + k
i.e., 2k+1 > k + 1 (∵ k ≥ 1)
∴ P(k + 1) is true whenever P(k) is true.
∴ By principal of mathematical induction P(n) is true for all n ∈ N.
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