Write of the following are A.P.s ? If they form an A.P., find the common difference and write three more terms .
2,4,8,16

Answer

To determine whether the sequence 2, 4, 8, 16 is an Arithmetic Progression (A.P.), we need to check if the difference between consecutive terms is constant.

### Step-by-Step Solution:

1. **Identify the Terms**: The given terms are:
- First term (a1) = 2
- Second term (a2) = 4
- Third term (a3) = 8
- Fourth term (a4) = 16

2. **Calculate the Differences**:
- Difference between the second and first term:
\[ d_1 = a2 - a1 = 4 - 2 = 2 \]
- Difference between the third and second term:
\[ d_2 = a3 - a2 = 8 - 4 = 4 \]
- Difference between the fourth and third term:
\[ d_3 = a4 - a3 = 16 - 8 = 8 \]

3. **Check for Constant Differences**:
- We have:
- \(d_1 = 2\)
- \(d_2 = 4\)
- \(d_3 = 8\)
- Since \(d_1\), \(d_2\), and \(d_3\) are not equal, the sequence does not have a constant difference.

4. **Conclusion**:
- Since the differences between consecutive terms are not the same, the sequence 2, 4, 8, 16 does not form an Arithmetic Progression (A.P.).

5. **Finding Next Terms**:
- Although the sequence does not form an A.P., we can still write the next terms based on the pattern observed:
- The terms seem to be doubling each time:
- 2, 4, 8, 16, ...
- Next term: \(16 \times 2 = 32\)
- Next term: \(32 \times 2 = 64\)
- Next term: \(64 \times 2 = 128\)
- Therefore, the next three terms are 32, 64, and 128.

### Final Answer:
- The sequence 2, 4, 8, 16 is **not an A.P.**
- The next three terms based on the pattern are **32, 64, and 128**.