Write of the following are A.P.s ? If they form an A.P., find the common difference and write three more terms .
`sqrt(3),sqrt(6),sqrt(9),sqrt(12),…..`

Answer

To determine if the sequence \( \sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \ldots \) forms an Arithmetic Progression (A.P.), we need to check if the difference between consecutive terms is constant.

### Step 1: Identify the terms
The given terms are:
- \( a_1 = \sqrt{3} \)
- \( a_2 = \sqrt{6} \)
- \( a_3 = \sqrt{9} \)
- \( a_4 = \sqrt{12} \)

### Step 2: Calculate the differences
Now, we will calculate the differences between consecutive terms:
1. **Difference between the second and first term:**
\[ d_1 = a_2 - a_1 = \sqrt{6} - \sqrt{3} \]
2. **Difference between the third and second term:**
\[ d_2 = a_3 - a_2 = \sqrt{9} - \sqrt{6} \]

### Step 3: Simplify the differences
- For \( d_1 \):
\[ d_1 = \sqrt{6} - \sqrt{3} \]
To simplify, we can rationalize:
\[ d_1 = \frac{(\sqrt{6} - \sqrt{3})(\sqrt{6} + \sqrt{3})}{\sqrt{6} + \sqrt{3}} = \frac{6 - 3}{\sqrt{6} + \sqrt{3}} = \frac{3}{\sqrt{6} + \sqrt{3}} \]

- For \( d_2 \):
\[ d_2 = \sqrt{9} - \sqrt{6} = 3 - \sqrt{6} \]

### Step 4: Compare the differences
Now, we need to check if \( d_1 \) is equal to \( d_2 \):
\[ \sqrt{6} - \sqrt{3} \neq 3 - \sqrt{6} \]

Since \( d_1 \) is not equal to \( d_2 \), the sequence does not have a constant difference.

### Conclusion
Since the differences between consecutive terms are not equal, the sequence \( \sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, \ldots \) does not form an A.P. Therefore, we cannot find a common difference or write three more terms.