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

Answer

To determine if the sequence \(3, 3+\sqrt{2}, 3+2\sqrt{2}, 3+3\sqrt{2}\) forms an arithmetic progression (A.P.), we need to check if the differences between consecutive terms are equal.

### Step 1: Identify the terms
The terms of the sequence are:
- First term (\(a_1\)): \(3\)
- Second term (\(a_2\)): \(3 + \sqrt{2}\)
- Third term (\(a_3\)): \(3 + 2\sqrt{2}\)
- Fourth term (\(a_4\)): \(3 + 3\sqrt{2}\)

### Step 2: Calculate the difference between the first two terms
To find the common difference, we calculate:
\[ d_1 = a_2 - a_1 = (3 + \sqrt{2}) - 3 = \sqrt{2} \]

### Step 3: Calculate the difference between the second and third terms
Next, we calculate:
\[ d_2 = a_3 - a_2 = (3 + 2\sqrt{2}) - (3 + \sqrt{2}) = 2\sqrt{2} - \sqrt{2} = \sqrt{2} \]

### Step 4: Calculate the difference between the third and fourth terms
Now, we calculate:
\[ d_3 = a_4 - a_3 = (3 + 3\sqrt{2}) - (3 + 2\sqrt{2}) = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \]

### Step 5: Check if the differences are equal
Since \(d_1 = d_2 = d_3 = \sqrt{2}\), the differences are equal. Therefore, the sequence forms an arithmetic progression.

### Step 6: Find the common difference
The common difference \(d\) is:
\[ d = \sqrt{2} \]

### Step 7: Write three more terms of the A.P.
To find the next three terms, we keep adding the common difference \(d\):
- Fifth term (\(a_5\)):
\[ a_5 = a_4 + d = (3 + 3\sqrt{2}) + \sqrt{2} = 3 + 4\sqrt{2} \]
- Sixth term (\(a_6\)):
\[ a_6 = a_5 + d = (3 + 4\sqrt{2}) + \sqrt{2} = 3 + 5\sqrt{2} \]
- Seventh term (\(a_7\)):
\[ a_7 = a_6 + d = (3 + 5\sqrt{2}) + \sqrt{2} = 3 + 6\sqrt{2} \]

### Final Answer
The sequence is an A.P. with a common difference of \(\sqrt{2}\). The next three terms are:
\[ 3 + 4\sqrt{2}, \quad 3 + 5\sqrt{2}, \quad 3 + 6\sqrt{2} \]