1 + 2 + 3 + .......... + 49 + 50 + 49 + 48 + ........ + 3 + 2 + 1 is equal to.

Options

A. 1250
B. 2500
C. 2525
D. 5000

Answer

To solve the expression \( 1 + 2 + 3 + \ldots + 49 + 50 + 49 + 48 + \ldots + 3 + 2 + 1 \), we can break it down into two parts: the increasing sequence from 1 to 50 and the decreasing sequence from 49 back to 1.

### Step 1: Calculate the sum of the first part (1 to 50)
The sum of the first \( n \) natural numbers can be calculated using the formula:
\[ S_n = \frac{n(n + 1)}{2} \]
For our case, \( n = 50 \):
\[ S_{50} = \frac{50(50 + 1)}{2} = \frac{50 \times 51}{2} = \frac{2550}{2} = 1275 \]

### Step 2: Calculate the sum of the second part (49 to 1)
The second part is the sum of the numbers from 1 to 49, which can also be calculated using the same formula:
\[ S_{49} = \frac{49(49 + 1)}{2} = \frac{49 \times 50}{2} = \frac{2450}{2} = 1225 \]

### Step 3: Combine both sums
Now, we need to add the two sums together:
\[ \text{Total Sum} = S_{50} + S_{49} = 1275 + 1225 = 2500 \]

### Final Answer
Thus, the total sum of the expression \( 1 + 2 + 3 + \ldots + 49 + 50 + 49 + 48 + \ldots + 3 + 2 + 1 \) is \( 2500 \).