Find the sum of 25 terms of the AP ` Sqrt2,2sqrt2,3sqrt2,4sqrt2` ,….,

Answer

To find the sum of the first 25 terms of the arithmetic progression (AP) given by \( \sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, \ldots \), we will follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
- The first term \( a \) is \( \sqrt{2} \).
- To find the common difference \( d \), we subtract the first term from the second term:
\[ d = 2\sqrt{2} - \sqrt{2} = \sqrt{2} \]

### Step 2: Use the formula for the sum of the first n terms of an AP
The formula for the sum of the first \( n \) terms \( S_n \) of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
Here, \( n = 25 \).

### Step 3: Substitute the values into the formula
Substituting \( n = 25 \), \( a = \sqrt{2} \), and \( d = \sqrt{2} \):
\[ S_{25} = \frac{25}{2} \times (2\sqrt{2} + (25 - 1)\sqrt{2}) \]
\[ = \frac{25}{2} \times (2\sqrt{2} + 24\sqrt{2}) \]
\[ = \frac{25}{2} \times (26\sqrt{2}) \]

### Step 4: Simplify the expression
Now we can simplify the expression:
\[ S_{25} = \frac{25 \times 26\sqrt{2}}{2} \]
\[ = \frac{650\sqrt{2}}{2} \]
\[ = 325\sqrt{2} \]

### Final Answer
The sum of the first 25 terms of the given AP is:
\[ S_{25} = 325\sqrt{2} \]

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