Assertion (A): For any natural number `n,(n)^(2)` is of the form `2q` or `2q+2`.
Reason (R):Square of every odd number is odd.

Options

A. Both A and R are true and R is the correct explanation for A .
B. Both A and Rare true and R is not correct explanation for A.
C. A is true but R is false.
D. A is false but R is true.

Answer

To solve the question, we need to analyze both the assertion (A) and the reason (R) provided.

### Step 1: Analyze the Assertion (A)
Assertion (A) states that for any natural number \( n \), \( n^2 \) is of the form \( 2q \) or \( 2q + 2 \).

1. **Understanding the Forms**:
- The expression \( 2q \) represents even numbers.
- The expression \( 2q + 2 \) also represents even numbers since \( 2q + 2 = 2(q + 1) \), which is still an even number.

2. **Checking for Even and Odd \( n \)**:
- If \( n \) is even, say \( n = 2k \) for some integer \( k \):
\[ n^2 = (2k)^2 = 4k^2 = 2(2k^2) \]
This is of the form \( 2q \).
- If \( n \) is odd, say \( n = 2k + 1 \):
\[ n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 \]
This is of the form \( 2q + 1 \), which is odd.

3. **Conclusion for Assertion**:
- The assertion is false because \( n^2 \) can be odd when \( n \) is odd, thus it cannot be expressed solely in the forms \( 2q \) or \( 2q + 2 \).

### Step 2: Analyze the Reason (R)
Reason (R) states that the square of every odd number is odd.

1. **Understanding Odd Squares**:
- If \( n \) is odd, \( n = 2k + 1 \) for some integer \( k \).
- We already calculated \( n^2 \):
\[ n^2 = (2k + 1)^2 = 4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 \]
- This shows that \( n^2 \) is indeed odd.

2. **Conclusion for Reason**:
- The reason is true because the square of any odd number is always odd.

### Final Conclusion
- Assertion (A) is false.
- Reason (R) is true.

### Answer
The correct answer is: Assertion (A) is false and Reason (R) is true.