Which of the following are AP's ? If they form an A. P. , find the common difference d and write three more terms.
`sqrt2, sqrt8, sqrt18, sqrt32,…`

Answer

To determine if the sequence \( \sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots \) forms an arithmetic progression (AP), we will follow these steps:

### Step 1: Simplify the terms
We first simplify each term in the sequence:
- The first term is already simplified:
\[ a_1 = \sqrt{2} \]
- The second term:
\[ a_2 = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2} \]
- The third term:
\[ a_3 = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} \]
- The fourth term:
\[ a_4 = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} \]

### Step 2: Identify the common difference
Next, we calculate the common difference \( d \) between consecutive terms:
- The difference between the second and first term:
\[ d = a_2 - a_1 = 2\sqrt{2} - \sqrt{2} = (2 - 1)\sqrt{2} = \sqrt{2} \]
- The difference between the third and second term:
\[ d = a_3 - a_2 = 3\sqrt{2} - 2\sqrt{2} = (3 - 2)\sqrt{2} = \sqrt{2} \]
- The difference between the fourth and third term:
\[ d = a_4 - a_3 = 4\sqrt{2} - 3\sqrt{2} = (4 - 3)\sqrt{2} = \sqrt{2} \]

Since the common difference \( d \) is the same for all pairs of consecutive terms, we conclude that the sequence is indeed an AP.

### Step 3: Find the next three terms
To find the next three terms of the AP, we use the formula for the \( n \)-th term of an AP:
\[ a_n = a_1 + (n-1)d \]
- The fifth term \( a_5 \):
\[ a_5 = a_1 + 4d = \sqrt{2} + 4\sqrt{2} = 5\sqrt{2} \]
- The sixth term \( a_6 \):
\[ a_6 = a_1 + 5d = \sqrt{2} + 5\sqrt{2} = 6\sqrt{2} \]
- The seventh term \( a_7 \):
\[ a_7 = a_1 + 6d = \sqrt{2} + 6\sqrt{2} = 7\sqrt{2} \]

### Final Answer
The sequence \( \sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, \ldots \) is an arithmetic progression with a common difference \( d = \sqrt{2} \). The next three terms are:
\[ 5\sqrt{2}, 6\sqrt{2}, 7\sqrt{2} \]