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 these

Answer

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 progression (GP).

### Step-by-Step Solution:

1. **Identify the first term (a) and the common ratio (r)**:
- The first term \(a = 1\).
- The common ratio \(r\) can be found by dividing the second term by the first term:
\[ r = \frac{-\frac{1}{2}}{1} = -\frac{1}{2} \]

2. **Use the formula for the sum of the first n terms of a GP**:
The formula for the sum \(S_n\) of the first \(n\) terms of a geometric series is given by:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
Here, we want to find the sum of the first 8 terms, so \(n = 8\).

3. **Substitute the values into the formula**:
\[ S_8 = 1 \cdot \frac{1 - \left(-\frac{1}{2}\right)^8}{1 - \left(-\frac{1}{2}\right)} \]

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

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

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

7. **Simplify the denominator**:
\[ 1 + \frac{1}{2} = \frac{2 + 1}{2} = \frac{3}{2} \]

8. **Now, substitute the simplified numerator and denominator back into the equation**:
\[ S_8 = \frac{\frac{255}{256}}{\frac{3}{2}} = \frac{255}{256} \cdot \frac{2}{3} \]

9. **Multiply the fractions**:
\[ S_8 = \frac{255 \cdot 2}{256 \cdot 3} = \frac{510}{768} \]

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

### Final Answer:
The sum of the first eight terms of the series is \(\frac{255}{384}\).