Find the sum of first 15 odd natural numbers.

Answer

To find the sum of the first 15 odd natural numbers, we can use the formula for the sum of the first n terms of an arithmetic progression (AP). The odd natural numbers form an arithmetic progression where the first term (a) is 1 and the common difference (d) is 2.

### Step-by-step solution:

1. **Identify the first term and common difference**:
- The first odd natural number (first term, a) = 1
- The common difference (d) = 3 - 1 = 2

2. **Determine the number of terms (n)**:
- We need to find the sum of the first 15 odd natural numbers, so n = 15.

3. **Use the formula for the sum of the first n terms of an AP**:
- The formula is given by:
\[ S_n = \frac{n}{2} \times [2a + (n - 1)d] \]
- Here, \( S_n \) is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference.

4. **Substitute the values into the formula**:
- Substitute n = 15, a = 1, and d = 2 into the formula:
\[ S_{15} = \frac{15}{2} \times [2 \times 1 + (15 - 1) \times 2] \]

5. **Calculate the expression inside the brackets**:
- Calculate \( 2 \times 1 = 2 \)
- Calculate \( (15 - 1) \times 2 = 14 \times 2 = 28 \)
- So, \( 2 + 28 = 30 \)

6. **Complete the calculation**:
- Now substitute back into the formula:
\[ S_{15} = \frac{15}{2} \times 30 \]
- Calculate \( \frac{15 \times 30}{2} = \frac{450}{2} = 225 \)

7. **Conclusion**:
- Therefore, the sum of the first 15 odd natural numbers is \( 225 \).