Question

If P(n): 2n < n!, n ∈ N, then P(n) is true for all n ≥ ______.

Answer 1

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.