Assertion- Reason Type Question:
STATEMENT -1: `1*3*5* ….*(2n-1) gt n^n, n in N` because
STATEMENT -2: the sum of the first n natural numbers is equal to `n^2`.

Options

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

Answer

To solve the assertion-reason type question, we need to analyze both statements provided:

**Statement 1:** \(1 \times 3 \times 5 \times \ldots \times (2n - 1) > n^n\) for \(n \in \mathbb{N}\)

**Statement 2:** The sum of the first \(n\) natural numbers is equal to \(n^2\).

### Step-by-Step Solution:

**Step 1: Analyze Statement 1**

We need to check if the product of the first \(n\) odd numbers is greater than \(n^n\).

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

This can also be represented using the double factorial:
\[ P(n) = \frac{(2n)!}{2^n \cdot n!} \]

To check if \(P(n) > n^n\), we can test specific values of \(n\).

**Step 2: Test for \(n = 1\)**

For \(n = 1\):
\[ P(1) = 1 \]
\[ 1^1 = 1 \]
Here, \(1 \not> 1\).

**Step 3: Test for \(n = 2\)**

For \(n = 2\):
\[ P(2) = 1 \times 3 = 3 \]
\[ 2^2 = 4 \]
Here, \(3 \not> 4\).

**Step 4: Test for \(n = 3\)**

For \(n = 3\):
\[ P(3) = 1 \times 3 \times 5 = 15 \]
\[ 3^3 = 27 \]
Here, \(15 \not> 27\).

**Step 5: Conclusion for Statement 1**

From the tests, we can see that \(P(n) \leq n^n\) for the values checked. Thus, Statement 1 is false.

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**Step 6: Analyze Statement 2**

The sum of the first \(n\) natural numbers is given by the formula:
\[ S(n) = \frac{n(n + 1)}{2} \]

We need to check if this is equal to \(n^2\).

**Step 7: Test for \(n = 1\)**

For \(n = 1\):
\[ S(1) = \frac{1(1 + 1)}{2} = 1 \]
\[ 1^2 = 1 \]
This is true.

**Step 8: Test for \(n = 2\)**

For \(n = 2\):
\[ S(2) = \frac{2(2 + 1)}{2} = 3 \]
\[ 2^2 = 4 \]
This is false.

**Step 9: Conclusion for Statement 2**

Since \(S(n) \neq n^2\) for \(n = 2\), Statement 2 is also false.

### Final Conclusion

Both Statement 1 and Statement 2 are false. Therefore, the answer is that neither statement is true.

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