The sum of the first n odd natural number is

Options

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

Answer

To find the sum of the first \( n \) odd natural numbers, we can derive the formula step by step.

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

### Step 2: Write the sum of the first \( n \) odd natural numbers
We denote the sum of the first \( n \) odd natural numbers as \( S_n \):
\[ S_n = 1 + 3 + 5 + ... + (2n - 1) \]

### Step 3: Use the formula for the sum of an arithmetic series
The sequence of odd numbers is an arithmetic series where:
- The first term \( a = 1 \)
- The common difference \( d = 2 \)
- The number of terms \( n \)

The formula for the sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by:
\[ S_n = \frac{n}{2} \times (2a + (n - 1)d) \]

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

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