Find the common difference of the A.P. `-5, -1, 3, 7`……… .

Answer

To find the common difference of the arithmetic progression (A.P.) given by the sequence \(-5, -1, 3, 7\), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Terms**:
The first few terms of the A.P. are:
- First term (\(a_1\)) = \(-5\)
- Second term (\(a_2\)) = \(-1\)
- Third term (\(a_3\)) = \(3\)
- Fourth term (\(a_4\)) = \(7\)

2. **Use the Formula for Common Difference**:
The common difference (\(d\)) in an A.P. is calculated as the difference between any two consecutive terms. We can use the first two terms to find \(d\):
\[ d = a_2 - a_1 \]

3. **Calculate the Common Difference**:
Substituting the values of \(a_1\) and \(a_2\):
\[ d = (-1) - (-5) \]
Simplifying this gives:
\[ d = -1 + 5 = 4 \]

4. **Verify with Other Terms**:
We can verify the common difference using the next pair of terms:
\[ d = a_3 - a_2 = 3 - (-1) = 3 + 1 = 4 \]
And also:
\[ d = a_4 - a_3 = 7 - 3 = 4 \]

5. **Conclusion**:
The common difference of the A.P. \(-5, -1, 3, 7\) is \(4\).