What is the sum of first eight terms of the series
`1-(1)/(2)+(1)/(4)-(1)/(8)+.....?`

Options

A. `(89)/(128)`
B. `(57)/(384)`
C. `(85)/(128)`
D. None of the above

Explanation

To find the sum of the first eight terms of the series \( 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \ldots \), we can recognize that this series is a geometric series.

### Step-by-Step Solution:

1. **Identify the first term (A) and the common ratio (R)**:
- The first term \( A = 1 \).
- The common ratio \( R = -\frac{1}{2} \) (since each term is obtained by multiplying the previous term by \(-\frac{1}{2}\)).

2. **Use the formula for the sum of the first n terms of a geometric series**:
The formula for the sum of the first \( n \) terms of a geometric series is given by:
\[ S_n = \frac{A(1 - R^n)}{1 - R} \]
where \( A \) is the first term, \( R \) is the common ratio, and \( n \) is the number of terms.

3. **Substitute the values into the formula**:
Here, we want to find \( S_8 \):
\[ S_8 = \frac{1(1 - (-\frac{1}{2})^8)}{1 - (-\frac{1}{2})} \]

4. **Calculate \( (-\frac{1}{2})^8 \)**:
\[ (-\frac{1}{2})^8 = \frac{1}{256} \]

5. **Substitute this value back into the sum formula**:
\[ S_8 = \frac{1(1 - \frac{1}{256})}{1 + \frac{1}{2}} = \frac{1 - \frac{1}{256}}{\frac{3}{2}} \]

6. **Simplify the numerator**:
\[ 1 - \frac{1}{256} = \frac{256 - 1}{256} = \frac{255}{256} \]

7. **Now substitute back into the sum formula**:
\[ S_8 = \frac{\frac{255}{256}}{\frac{3}{2}} = \frac{255}{256} \times \frac{2}{3} = \frac{255 \times 2}{256 \times 3} = \frac{510}{768} \]

8. **Simplify the fraction**:
- Divide both the numerator and the denominator by 2:
\[ S_8 = \frac{255}{384} \]

Thus, the sum of the first eight terms of the series is \( \frac{255}{384} \).