Assertion (A) : Common difference of the A.P.: `-5,-1,3,7,`……. Is 4.
Reason (R ) : Common difference of the A.P.a,a+d, `a+2d`, …., is given by d `=2^(nd)` term `-1^(st)` term.

Options

A. Both A and R are true and R is the correct explanation for A.
B. Both A and R are 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-by-Step Solution:

1. **Identify the terms of the A.P.**: The given A.P. is `-5, -1, 3, 7, ...`.

2. **Calculate the common difference (d)**:
- The common difference \( d \) of an arithmetic progression (A.P.) is calculated as the difference between any two consecutive terms.
- Here, we can find \( d \) by subtracting the first term from the second term:
\[ d = \text{second term} - \text{first term} = (-1) - (-5) \]
- Simplifying this gives:
\[ d = -1 + 5 = 4 \]

3. **Verify Assertion (A)**:
- The assertion states that the common difference of the A.P. is 4. From our calculation, we found that \( d = 4 \).
- Therefore, Assertion (A) is **True**.

4. **Understand the Reason (R)**:
- The reason states that the common difference of the A.P. can be given by the formula:
\[ d = a_n - a_1 \]
- Here, \( a_n \) is the \( n \)-th term and \( a_1 \) is the first term. However, this formula is not correctly stated for finding the common difference. The correct formula for the common difference is:
\[ d = a_2 - a_1 \]
- Since the reason is not correctly explaining the assertion, we conclude that Reason (R) is **False**.

5. **Conclusion**:
- Since Assertion (A) is True and Reason (R) is False, we can conclude that the assertion is correct, but the reason does not correctly explain it.

### Final Answer:
- Assertion (A) is True.
- Reason (R) is False.