The sum of first 50 odd natural numbers is

Options

A. 1000
B. 1250
C. 5200
D. 2500

Answer

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

### Step 1: Identify the first 50 odd natural numbers
The first 50 odd natural numbers are:
1, 3, 5, 7, ..., 99.

### Step 2: Recognize the pattern
The nth odd natural number can be expressed as:
\[ a_n = 2n - 1 \]
For n = 1, we get 1; for n = 2, we get 3; and so on.

### Step 3: Use the formula for the sum of an arithmetic series
The sum \( S_n \) of the first n terms of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
where:
- \( n \) = number of terms,
- \( a \) = first term,
- \( d \) = common difference.

### Step 4: Substitute the values into the formula
For our case:
- \( n = 50 \) (the number of odd numbers),
- \( a = 1 \) (the first odd number),
- \( d = 2 \) (the difference between consecutive odd numbers).

Now substitute these values into the formula:
\[ S_{50} = \frac{50}{2} \times (2 \times 1 + (50 - 1) \times 2) \]

### Step 5: Simplify the equation
Calculating step-by-step:
1. Calculate \( \frac{50}{2} = 25 \).
2. Calculate \( 2 \times 1 = 2 \).
3. Calculate \( 50 - 1 = 49 \).
4. Calculate \( 49 \times 2 = 98 \).
5. Now add these results: \( 2 + 98 = 100 \).
6. Finally, multiply: \( 25 \times 100 = 2500 \).

### Conclusion
The sum of the first 50 odd natural numbers is:
\[ \boxed{2500} \]

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