Question

If P(n) : 2ⁿ < n!
Then the smallest positive integer for which P(n) is true if

• A. 2
• B. 3
• C. 4
• D. 5

Answer 1

The Correct Option is C

If P(n) : \(2^n < n!\), then the smallest positive integer for which P(n) is true if

We test values of \(n\) starting from 1:

• n = 1: \(2^1 < 1!\) => \(2 < 1\) (False)
• n = 2: \(2^2 < 2!\) => \(4 < 2\) (False)
• n = 3: \(2^3 < 3!\) => \(8 < 6\) (False)
• n = 4: \(2^4 < 4!\) => \(16 < 24\) (True)

The smallest positive integer for which P(n) is true is 4.

Answer: (C) 4

Answer 2

We test small values of $ n $:

• For $ n = 1 $: $$ 2^1 = 2, \quad 1! = 1. \quad 2 > 1, \quad \text{so } P(1) \text{ is false.} $$
• For $ n = 2 $: $$ 2^2 = 4, \quad 2! = 2. \quad 4 > 2, \quad \text{so } P(2) \text{ is false.} $$
• For $ n = 3 $: $$ 2^3 = 8, \quad 3! = 6. \quad 8 > 6, \quad \text{so } P(3) \text{ is false.} $$
• For $ n = 4 $: $$ 2^4 = 16, \quad 4! = 24. \quad 16 < 24, \quad \text{so } P(4) \text{ is true.} $$

The smallest positive integer for which P(n) is true is 4.

Answer: (C) 4