Find the sum `(2+4+6+8+…+100).`

Explanation

To find the sum of the series \(2 + 4 + 6 + 8 + \ldots + 100\), we can follow these steps:

### Step 1: Identify the series
The series consists of even numbers starting from 2 up to 100. We can express this series in a more manageable form.

### Step 2: Factor out the common term
Notice that each term in the series is an even number, which can be factored out:
\[ 2 + 4 + 6 + 8 + \ldots + 100 = 2(1 + 2 + 3 + 4 + \ldots + 50) \]
Here, we have factored out 2 from each term, and the remaining series \(1 + 2 + 3 + \ldots + 50\) consists of the first 50 natural numbers.

### Step 3: Use the formula for the sum of the first \(n\) natural numbers
The formula for the sum of the first \(n\) natural numbers is:
\[ S_n = \frac{n(n + 1)}{2} \]
In our case, \(n = 50\).

### Step 4: Calculate the sum of the first 50 natural numbers
Substituting \(n = 50\) into the formula:
\[ S_{50} = \frac{50(50 + 1)}{2} = \frac{50 \times 51}{2} = \frac{2550}{2} = 1275 \]

### Step 5: Multiply by the factored out term
Now, we multiply the sum of the first 50 natural numbers by 2:
\[ 2(1 + 2 + 3 + \ldots + 50) = 2 \times 1275 = 2550 \]

### Conclusion
Thus, the sum \(2 + 4 + 6 + 8 + \ldots + 100\) is:
\[ \boxed{2550} \]