Find the sum of
1 + 2 + 3 + …………… + 50 + 49 + ……… + 3 + 2 + 1

Options

A. 2500
B. 2550
C. 2600
D. 2450

Answer

To find the sum of the series \(1 + 2 + 3 + \ldots + 50 + 49 + \ldots + 3 + 2 + 1\), we can break it down into two parts:

1. The first part is the sum of the first 50 natural numbers: \(1 + 2 + 3 + \ldots + 50\).
2. The second part is the sum of the natural numbers from 1 to 49: \(49 + 48 + \ldots + 3 + 2 + 1\).

### Step 1: Calculate the sum of the first 50 natural numbers

The formula for the sum of the first \(n\) natural numbers is given by:

\[ S_n = \frac{n(n + 1)}{2} \]

For \(n = 50\):

\[ S_{50} = \frac{50(50 + 1)}{2} = \frac{50 \times 51}{2} = \frac{2550}{2} = 1275 \]

### Step 2: Calculate the sum of the first 49 natural numbers

Now, we need to calculate the sum of the first 49 natural numbers using the same formula:

For \(n = 49\):

\[ S_{49} = \frac{49(49 + 1)}{2} = \frac{49 \times 50}{2} = \frac{2450}{2} = 1225 \]

### Step 3: Add the two sums together

Now, we add the two sums we calculated:

\[ \text{Total Sum} = S_{50} + S_{49} = 1275 + 1225 = 2500 \]

### Final Answer

The sum of the series \(1 + 2 + 3 + \ldots + 50 + 49 + \ldots + 3 + 2 + 1\) is:

\[ \boxed{2500} \]

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