Which of the following are AP's ? If they form an A. P. , find the common difference d and write three more terms.
`3,3 + sqrt2 , 3 + 2 sqrt2 , 3 + 3 sqrt2,….`

Answer

To determine if the given sequence \(3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}, \ldots\) forms an arithmetic progression (AP), we will follow these steps:

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

### Step 2: Calculate the common difference \(d\)
To check if the sequence is an AP, we need to find the common difference \(d\) between consecutive terms.

1. Calculate \(d_1 = a_2 - a_1\):
\[ d_1 = (3 + \sqrt{2}) - 3 = \sqrt{2} \]

2. Calculate \(d_2 = a_3 - a_2\):
\[ d_2 = (3 + 2\sqrt{2}) - (3 + \sqrt{2}) = 2\sqrt{2} - \sqrt{2} = \sqrt{2} \]

3. Calculate \(d_3 = a_4 - a_3\):
\[ d_3 = (3 + 3\sqrt{2}) - (3 + 2\sqrt{2}) = 3\sqrt{2} - 2\sqrt{2} = \sqrt{2} \]

Since \(d_1 = d_2 = d_3 = \sqrt{2}\), the common difference \(d\) is the same for all terms.

### Step 3: Confirm it is an AP
Since the common difference is constant, the sequence is indeed an arithmetic progression.

### Step 4: Write three more terms
Now that we know the sequence is an AP with the first term \(a = 3\) and common difference \(d = \sqrt{2}\), we can find the next three terms.

1. The 5th term \(a_5\):
\[ a_5 = a + 4d = 3 + 4\sqrt{2} \]

2. The 6th term \(a_6\):
\[ a_6 = a + 5d = 3 + 5\sqrt{2} \]

3. The 7th term \(a_7\):
\[ a_7 = a + 6d = 3 + 6\sqrt{2} \]

### Final Answer
The next three terms are:
- \(3 + 4\sqrt{2}\)
- \(3 + 5\sqrt{2}\)
- \(3 + 6\sqrt{2}\)