Question

The inequality \( n!>2^{n-1} \) is true for

• A. for all \( n \in \mathbb{N} \)
• B. for all \( n>1 \)
• C. for all \( n>2 \)
• D. None of the above

Hint:

To verify inequalities involving factorials, check small values of \( n \) and observe the growth rates of the functions.

Answer 1

The Correct Option is C

Step 1: Analyze the inequality.
We are given the inequality \( n!>2^{n-1} \). We need to determine for which values of \( n \) this inequality holds true.

Step 2: Check small values of \( n \).
- For \( n = 1 \): \[ 1! = 1 \quad \text{and} \quad 2^{1-1} = 1 \] So, the inequality does not hold for \( n = 1 \). - For \( n = 2 \): \[ 2! = 2 \quad \text{and} \quad 2^{2-1} = 2 \] So, the inequality holds for \( n = 2 \). - For \( n = 3 \): \[ 3! = 6 \quad \text{and} \quad 2^{3-1} = 4 \] So, the inequality holds for \( n = 3 \). - For \( n = 4 \): \[ 4! = 24 \quad \text{and} \quad 2^{4-1} = 8 \] So, the inequality holds for \( n = 4 \).

Step 3: General observation.
As \( n \) increases, \( n! \) grows much faster than \( 2^{n-1} \). Hence, the inequality holds for \( n>2 \).

Step 4: Conclusion.
Thus, the inequality \( n!>2^{n-1} \) holds true for \( n>2 \), corresponding to option (C).