The sum of n odd natural numbers is

Options

A. `n^(2)`
B. `2n`
C. `(n+1)/(2)`
D. `n^(2)+1`

Answer

To find the sum of the first \( n \) odd natural numbers, we can follow these steps:

### Step 1: Identify the first \( n \) odd natural numbers
The first \( n \) odd natural numbers are:
\[ 1, 3, 5, 7, \ldots, (2n - 1) \]

### Step 2: Recognize the sequence as an arithmetic progression (AP)
The sequence of odd natural numbers is an arithmetic progression where:
- The first term \( A = 1 \)
- The common difference \( D = 2 \)

### Step 3: Use the formula for the sum of an arithmetic progression
The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]
where \( n \) is the number of terms, \( A \) is the first term, and \( D \) is the common difference.

### Step 4: Substitute the values into the formula
Substituting \( A = 1 \), \( D = 2 \), and \( n \) into the formula:
\[ S_n = \frac{n}{2} \times (2 \times 1 + (n - 1) \times 2) \]

### Step 5: Simplify the expression
Now simplify the expression:
\[ S_n = \frac{n}{2} \times (2 + 2(n - 1)) \]
\[ = \frac{n}{2} \times (2 + 2n - 2) \]
\[ = \frac{n}{2} \times 2n \]

### Step 6: Further simplify
\[ = n \times n = n^2 \]

### Conclusion
Thus, the sum of the first \( n \) odd natural numbers is:
\[ S_n = n^2 \]