Write of the following are A.P.s ? If they form an A.P., find the common difference and write three more terms .
`sqrt(2),sqrt(3),sqrt(9),sqrt(12)…..`

Answer

To determine whether the sequence \( \sqrt{2}, \sqrt{3}, \sqrt{9}, \sqrt{12} \) forms an Arithmetic Progression (A.P.), we need to check if the differences between consecutive terms are equal.

### Step-by-Step Solution:

1. **Identify the Terms**:
The given terms are:
- First term \( a_1 = \sqrt{2} \)
- Second term \( a_2 = \sqrt{3} \)
- Third term \( a_3 = \sqrt{9} \)
- Fourth term \( a_4 = \sqrt{12} \)

2. **Calculate the Differences**:
- Find the difference between the second and first terms:
\[ d_1 = a_2 - a_1 = \sqrt{3} - \sqrt{2} \]
- Find the difference between the third and second terms:
\[ d_2 = a_3 - a_2 = \sqrt{9} - \sqrt{3} = 3 - \sqrt{3} \]
- Find the difference between the fourth and third terms:
\[ d_3 = a_4 - a_3 = \sqrt{12} - \sqrt{9} = 2\sqrt{3} - 3 \]

3. **Compare the Differences**:
- We need to check if \( d_1 = d_2 = d_3 \).
- Since \( d_1 = \sqrt{3} - \sqrt{2} \), \( d_2 = 3 - \sqrt{3} \), and \( d_3 = 2\sqrt{3} - 3 \) are not equal, we conclude that the differences are not the same.

4. **Conclusion**:
Since the differences between the consecutive terms are not equal, the sequence \( \sqrt{2}, \sqrt{3}, \sqrt{9}, \sqrt{12} \) does not form an Arithmetic Progression (A.P.).

### Final Answer:
No, the sequence does not form an A.P.

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