Find the sum of first 25 even natural numbers.

Answer

To find the sum of the first 25 even natural numbers, we can follow these steps:

### Step 1: Identify the series of the first 25 even natural numbers.
The first 25 even natural numbers are:
\[ 2, 4, 6, 8, 10, \ldots, 50 \]

### Step 2: Recognize that this series is an Arithmetic Progression (AP).
In this series, the common difference \(d\) between consecutive terms is:
\[ d = 4 - 2 = 2 \]
This confirms that the series is an AP.

### Step 3: Identify the first term and the number of terms.
The first term \(a\) of the series is:
\[ a = 2 \]
The number of terms \(n\) is:
\[ n = 25 \]

### Step 4: Use the formula for the sum of the first \(n\) terms of an AP.
The formula for the sum \(S_n\) of the first \(n\) terms of an AP is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

### Step 5: Substitute the values into the formula.
Substituting \(n = 25\), \(a = 2\), and \(d = 2\) into the formula:
\[ S_{25} = \frac{25}{2} \times (2 \times 2 + (25 - 1) \times 2) \]

### Step 6: Simplify the expression.
Calculating inside the parentheses:
\[ S_{25} = \frac{25}{2} \times (4 + 24 \times 2) \]
\[ = \frac{25}{2} \times (4 + 48) \]
\[ = \frac{25}{2} \times 52 \]

### Step 7: Calculate the final sum.
Now, simplifying further:
\[ S_{25} = \frac{25 \times 52}{2} = \frac{1300}{2} = 650 \]

Thus, the sum of the first 25 even natural numbers is:
\[ \boxed{650} \]