The average of first 50 odd natural number is:

Options

A. 50
B. 55
C. 51
D. 101

Answer

To find the average 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: Use the formula for the sum of an arithmetic series
The sum \( S_n \) of the first \( n \) odd numbers can be calculated using the formula:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]
where:
- \( n \) is the number of terms (in this case, 50),
- \( a \) is the first term (which is 1),
- \( d \) is the common difference (which is 2 for odd numbers).

### Step 3: Substitute the values into the formula
Substituting the values into the formula:
\[ S_{50} = \frac{50}{2} \times (2 \times 1 + (50 - 1) \times 2) \]
This simplifies to:
\[ S_{50} = 25 \times (2 + 49 \times 2) \]

### Step 4: Calculate the expression inside the parentheses
Calculating the expression:
\[ 49 \times 2 = 98 \]
So,
\[ 2 + 98 = 100 \]

### Step 5: Calculate the total sum
Now substituting back into the sum formula:
\[ S_{50} = 25 \times 100 = 2500 \]

### Step 6: Calculate the average
The average \( A \) is given by the formula:
\[ A = \frac{S_n}{n} \]
Substituting the values we found:
\[ A = \frac{2500}{50} = 50 \]

### Conclusion
Thus, the average of the first 50 odd natural numbers is **50**.

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