The simplification of `(1)/(8) + (1)/(8^(2)) + (1)/(8^(3)) + (1)/(8^(4)) + (1)/(8^(5))` upto three-places of decimals yields

Options

A. 0.143
B. 0.163
C. 0.215
D. 0.715

Answer

To simplify the expression \( \frac{1}{8} + \frac{1}{8^2} + \frac{1}{8^3} + \frac{1}{8^4} + \frac{1}{8^5} \), we can recognize that this is a geometric series.

### Step-by-Step Solution:

1. **Identify the first term and the common ratio:**
- The first term \( a = \frac{1}{8} \).
- The common ratio \( r = \frac{1}{8} \).

2. **Determine the number of terms:**
- The series has 5 terms: \( n = 5 \).

3. **Use the formula for the sum of a geometric series:**
The sum \( S_n \) of the first \( n \) terms of a geometric series can be calculated using the formula:
\[ S_n = a \frac{1 - r^n}{1 - r} \]
where \( a \) is the first term, \( r \) is the common ratio, and \( n \) is the number of terms.

4. **Substitute the values into the formula:**
\[ S_5 = \frac{1}{8} \cdot \frac{1 - \left(\frac{1}{8}\right)^5}{1 - \frac{1}{8}} \]

5. **Calculate \( r^n \):**
\[ \left(\frac{1}{8}\right)^5 = \frac{1}{8^5} = \frac{1}{32768} \]

6. **Calculate \( 1 - r^n \):**
\[ 1 - \frac{1}{32768} = \frac{32768 - 1}{32768} = \frac{32767}{32768} \]

7. **Calculate \( 1 - r \):**
\[ 1 - \frac{1}{8} = \frac{7}{8} \]

8. **Substitute back into the sum formula:**
\[ S_5 = \frac{1}{8} \cdot \frac{\frac{32767}{32768}}{\frac{7}{8}} = \frac{1}{8} \cdot \frac{32767}{32768} \cdot \frac{8}{7} \]

9. **Simplify the expression:**
\[ S_5 = \frac{32767}{32768 \cdot 7} = \frac{32767}{229376} \]

10. **Calculate the decimal value:**
\[ S_5 \approx 0.142857 \]

11. **Round to three decimal places:**
\[ S_5 \approx 0.143 \]

### Final Answer:
The simplification of \( \frac{1}{8} + \frac{1}{8^2} + \frac{1}{8^3} + \frac{1}{8^4} + \frac{1}{8^5} \) up to three decimal places yields \( 0.143 \).