Find the sum of the first 50 even natural numbers.

Answer

To find the sum of the first 50 even natural numbers, we can follow these steps:

### Step 1: Identify the first 50 even natural numbers
The first even natural number is 2, and the sequence of even natural numbers can be expressed as:
2, 4, 6, 8, ..., up to the 50th term.

### Step 2: Determine the number of terms (n)
We are given that we need to find the sum of the first 50 even natural numbers. Thus, \( n = 50 \).

### Step 3: Identify the first term (a) and the common difference (d)
- The first term \( a = 2 \).
- The common difference \( d = 2 \) (since each even number is 2 more than the previous one).

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

### Step 5: Substitute the values into the formula
Substituting \( n = 50 \), \( a = 2 \), and \( d = 2 \) into the formula:
\[ S_{50} = \frac{50}{2} \times (2 \cdot 2 + (50 - 1) \cdot 2) \]

### Step 6: Simplify the expression
Calculating the terms inside the parentheses:
- \( 2 \cdot 2 = 4 \)
- \( (50 - 1) \cdot 2 = 49 \cdot 2 = 98 \)

Now substituting these values back:
\[ S_{50} = 25 \times (4 + 98) \]
\[ S_{50} = 25 \times 102 \]

### Step 7: Calculate the final sum
Now, calculate \( 25 \times 102 \):
\[ S_{50} = 2550 \]

### Conclusion
The sum of the first 50 even natural numbers is **2550**.

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