Statement -1: `1.3.5 . . .(2n-1)len^(n)"for all "ninN` Statement -2: `GMleAM`

Options

A. Statement -1 is true, Statement -2 is True, Statement -2 is a correct explanation for Statement for Statement -1.
B. Statement -1 is true, Statement -2 is True, Statement -2 is not a correct explanation for Statement for Statement -1.
C. Statement -1 is true, Statement -2 is False.
D. Statement -1 is False, Statement -2 is True.

Explanation

To solve the given problem, we need to analyze the two statements and prove their validity step by step.

### Step-by-Step Solution:

1. **Understanding the Statements**:
- **Statement 1**: The product of the first n odd numbers (1, 3, 5, ..., (2n-1)) is less than or equal to \( n^n \).
- **Statement 2**: The geometric mean (GM) of these odd numbers is less than or equal to the arithmetic mean (AM).

2. **Finding the Product of Odd Numbers**:
- The product of the first n odd numbers can be expressed as:
\[ P = 1 \times 3 \times 5 \times \ldots \times (2n - 1) \]

3. **Calculating the Geometric Mean (GM)**:
- The GM of these n odd numbers is given by:
\[ GM = \sqrt[n]{P} \]

4. **Calculating the Arithmetic Mean (AM)**:
- The sum of the first n odd numbers is:
\[ S = 1 + 3 + 5 + \ldots + (2n - 1) = n^2 \]
- Therefore, the AM is:
\[ AM = \frac{S}{n} = \frac{n^2}{n} = n \]

5. **Applying the GM ≤ AM Inequality**:
- According to Statement 2, we have:
\[ GM \leq AM \implies \sqrt[n]{P} \leq n \]
- Raising both sides to the power of n gives:
\[ P \leq n^n \]

6. **Conclusion**:
- We have shown that the product of the first n odd numbers is indeed less than or equal to \( n^n \), which confirms Statement 1.
- Since Statement 2 (GM ≤ AM) is a valid inequality, both statements are true.

### Final Answer:
Both Statement 1 and Statement 2 are true, and Statement 2 serves as a correct explanation for Statement 1.