`1 + (1)/(2) + (1)/(4) + (1)/(16) + (1)/(32) + (1)/(64)` is approximately equal to ______.

Options

A. `1.5`
B. `2`
C. `2.5`
D. 3

Answer

To solve the expression \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}\), we can recognize that this is a geometric series.

### Step-by-step Solution:

1. **Identify the series**: The series can be written as:
\[ S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64} \]

2. **Recognize the first term (A) and common ratio (r)**:
- The first term \(A = 1\)
- The common ratio \(r = \frac{1}{2}\)

3. **Count the number of terms (n)**:
- The number of terms \(n = 6\) (from \(1\) to \(\frac{1}{64}\))

4. **Use the formula for the sum of a geometric series**:
The formula for the sum of the first \(n\) terms of a geometric series is:
\[ S_n = \frac{A(1 - r^n)}{1 - r} \]
Substituting the values:
\[ S_6 = \frac{1(1 - (\frac{1}{2})^6)}{1 - \frac{1}{2}} \]

5. **Calculate \(r^n\)**:
\[ (\frac{1}{2})^6 = \frac{1}{64} \]

6. **Substitute \(r^n\) back into the formula**:
\[ S_6 = \frac{1(1 - \frac{1}{64})}{\frac{1}{2}} = \frac{1(\frac{63}{64})}{\frac{1}{2}} \]

7. **Simplify the expression**:
\[ S_6 = \frac{63}{64} \times 2 = \frac{126}{64} = \frac{63}{32} \]

8. **Convert to decimal**:
\[ \frac{63}{32} \approx 1.96875 \]

9. **Approximate the final answer**:
Since the question asks for an approximate value, we can round \(1.96875\) to \(2\).

### Final Answer:
The approximate value of \(1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}\) is approximately equal to **2**.