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Question
If P(n): 2n < n!, n ∈ N, then P(n) is true for all n ≥ ______.
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Solution
If P(n): 2n < n!, n ∈ N, then P(n) is true for all n ≥ 4.
Explanation:
Given that P(n): 2n < n!, ∀ n ∈ N
For n = 1
2 < 1 .....(Not true)
For n = 2
2 × 2 < 2!
⇒ 4 < 2 ....(Not true)
For n = 3
2 × 3 < 3!
⇒ 6 < 3.2.1
⇒ 6 < 6 ....(Not true)
For n = 4
2 × 4 < 4!
⇒ 8 < 4.3.2.1
⇒ 8 < 24 .......(True)
For n = 5
2 × 5 < 5!
⇒ 10 < 5.4.3.2.1
⇒ 10 < 120 ......(True)
So, P(n) is the true for n ≥ 4.
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