STATEMENT-1 : For ` n ne N, n gt 1 , 2^(n) gt 1 n ^((n-1)/(2))` and
STATEMENT-2 : A.M. of distinct positive number is greater then G.M.

Options

A. Statemant-1 is True , Statement-2 is True, Statement -2 is a correct explanation for Statement-1
B. Statemant-1 is True , Statement-2 is True, Statement -2 is NOT a correct explanation for Statement-1
C. Statement-1 is True, Stetement-2 is False.
D. Statement-1 is False, Statement-2 is True

Answer

To analyze the statements provided, we will evaluate each statement step by step.

### Step 1: Analyze Statement 1
**Statement 1:** For \( n \in \mathbb{N}, n > 1, \, 2^n > n^{(n-1)/2} \)

1. **Identify the natural numbers greater than 1:** The natural numbers greater than 1 are \( n = 2, 3, 4, \ldots \).
2. **Check the inequality for various values of \( n \):**
- For \( n = 2 \):
\[ 2^2 = 4 \quad \text{and} \quad 2^{(2-1)/2} = 2^{1/2} = \sqrt{2} \approx 1.41 \]
Thus, \( 4 > 1.41 \) is true.
- For \( n = 3 \):
\[ 2^3 = 8 \quad \text{and} \quad 3^{(3-1)/2} = 3^{1} = 3 \]
Thus, \( 8 > 3 \) is true.
- For \( n = 4 \):
\[ 2^4 = 16 \quad \text{and} \quad 4^{(4-1)/2} = 4^{3/2} = 8 \]
Thus, \( 16 > 8 \) is true.
- For \( n = 5 \):
\[ 2^5 = 32 \quad \text{and} \quad 5^{(5-1)/2} = 5^{2} = 25 \]
Thus, \( 32 > 25 \) is true.
- For \( n = 6 \):
\[ 2^6 = 64 \quad \text{and} \quad 6^{(6-1)/2} = 6^{5/2} = 6 \cdot \sqrt{6} \approx 14.7 \]
Thus, \( 64 > 14.7 \) is true.

We can observe that for \( n = 2, 3, 4, 5, 6 \), the inequality holds true.

3. **Conclusion for Statement 1:** The statement appears to be true for natural numbers greater than 1 based on the values checked.

### Step 2: Analyze Statement 2
**Statement 2:** The Arithmetic Mean (A.M.) of distinct positive numbers is greater than the Geometric Mean (G.M.).

1. **Define A.M. and G.M.:**
- For two distinct positive numbers \( a \) and \( b \):
\[ \text{A.M.} = \frac{a + b}{2} \]
\[ \text{G.M.} = \sqrt{ab} \]

2. **Apply the Arithmetic Mean-Geometric Mean Inequality:**
- According to the A.M.-G.M. inequality:
\[ \frac{a + b}{2} \geq \sqrt{ab} \]
- The equality holds when \( a = b \). Since we are considering distinct positive numbers, we have:
\[ \frac{a + b}{2} > \sqrt{ab} \]

3. **Conclusion for Statement 2:** This statement is true for any two distinct positive numbers.

### Final Conclusion
- **Statement 1:** True
- **Statement 2:** True