What would be the sum of 1+3+5+7+9+11+13+15+…… up to 15th term ?

Options

A. 250
B. 240
C. 225
D. 265

Answer

To find the sum of the series 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 up to the 15th term, we can use the formula for the sum of an arithmetic series.

### Step-by-step Solution:

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

2. **Determine the number of terms (n)**:
- We need to find the sum up to the 15th term, so \( n = 15 \).

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

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

5. **Calculate the expression inside the parentheses**:
- Calculate \( 2 \times 1 = 2 \)
- Calculate \( (15 - 1) \times 2 = 14 \times 2 = 28 \)
- Now, add these values: \( 2 + 28 = 30 \)

6. **Complete the calculation for \( S_{15} \)**:
\[ S_{15} = \frac{15}{2} \times 30 \]
- Simplify \( \frac{15 \times 30}{2} = \frac{450}{2} = 225 \)

7. **Final Result**:
- Therefore, the sum of the series up to the 15th term is \( 225 \).