Which of them is not in A.P ?

Options

A. `1, - 1, - 3, - 5`
B. `-2, 2, -2, 2, -2`……..
C. `-10, -6, -2, 2,……..`
D. `2,4,6,8,10,……..`

Answer

To determine which of the given sequences is not in Arithmetic Progression (A.P.), we will analyze each sequence by calculating the common difference between consecutive terms.

### Step-by-Step Solution:

1. **Identify the sequences**: Let's denote the sequences as follows:
- Sequence A: 1, -1, -3, -5
- Sequence B: -2, 2, -2, 2
- Sequence C: -10, -6, -2, 2
- Sequence D: 2, 4, 6, 8, 10

2. **Check Sequence A**:
- Calculate the differences:
- d1 = -1 - 1 = -2
- d2 = -3 - (-1) = -2
- d3 = -5 - (-3) = -2
- Since the common difference (d) is the same (-2), Sequence A is in A.P.

3. **Check Sequence B**:
- Calculate the differences:
- d1 = 2 - (-2) = 4
- d2 = -2 - 2 = -4
- d3 = 2 - (-2) = 4
- The differences are not the same (4, -4, 4), so Sequence B is not in A.P.

4. **Check Sequence C**:
- Calculate the differences:
- d1 = -6 - (-10) = 4
- d2 = -2 - (-6) = 4
- d3 = 2 - (-2) = 4
- The common difference (d) is the same (4), so Sequence C is in A.P.

5. **Check Sequence D**:
- Calculate the differences:
- d1 = 4 - 2 = 2
- d2 = 6 - 4 = 2
- d3 = 8 - 6 = 2
- d4 = 10 - 8 = 2
- The common difference (d) is the same (2), so Sequence D is in A.P.

6. **Conclusion**:
- The only sequence that is not in A.P. is Sequence B: -2, 2, -2, 2.

### Final Answer:
Sequence B is not in A.P.