What is the sum of the infinite geometric series
`2+(-(1)/(2))+((1)/(8))+(-(1)/(32))+…` ?

Options

A. `1(3)/(8)`
B. `1(2)/(5)`
C. `1(1)/(2)`
D. `1(3)/(5)`

Explanation

To find the sum of the infinite geometric series \(2 + \left(-\frac{1}{2}\right) + \frac{1}{8} + \left(-\frac{1}{32}\right) + \ldots\), we follow these steps:

### Step 1: Identify the first term (A)
The first term \(A\) of the series is:
\[ A = 2 \]

### Step 2: Determine the common ratio (R)
To find the common ratio \(R\), we divide the second term by the first term:
\[ R = \frac{\text{second term}}{\text{first term}} = \frac{-\frac{1}{2}}{2} = -\frac{1}{4} \]

### Step 3: Check the condition for the sum of the infinite series
The formula for the sum \(S\) of an infinite geometric series is given by:
\[ S = \frac{A}{1 - R} \]
This formula is valid when the absolute value of \(R\) is less than 1, which is true in this case since \(|R| = \frac{1}{4} < 1\).

### Step 4: Substitute the values into the sum formula
Now we can substitute \(A\) and \(R\) into the formula:
\[ S = \frac{2}{1 - \left(-\frac{1}{4}\right)} = \frac{2}{1 + \frac{1}{4}} = \frac{2}{\frac{5}{4}} = 2 \times \frac{4}{5} = \frac{8}{5} \]

### Step 5: Convert the sum to a mixed number
The fraction \(\frac{8}{5}\) can be converted to a mixed number:
\[ \frac{8}{5} = 1 \frac{3}{5} \]

### Final Answer
Thus, the sum of the infinite geometric series is:
\[ 1 \frac{3}{5} \]

---