What is the the sum of the infinite geometric series `1/4 - 3/(16) + 9/(64) - (27)/(256) + `…

Options

A. `1/3`
B. `1/4`
C. `1/7`
D. none of these

Answer

To find the sum of the infinite geometric series given by \( \frac{1}{4} - \frac{3}{16} + \frac{9}{64} - \frac{27}{256} + \ldots \), we will follow these steps:

### Step 1: Identify the first term (a) and the common ratio (r)
The first term \( a \) of the series is:
\[ a = \frac{1}{4} \]

To find the common ratio \( r \), we can divide the second term by the first term:
\[ r = \frac{\text{Second term}}{\text{First term}} = \frac{-\frac{3}{16}}{\frac{1}{4}} = -\frac{3}{16} \times \frac{4}{1} = -\frac{3}{4} \]

### Step 2: Use the formula for the sum of an infinite geometric series
The formula for the sum \( S \) of an infinite geometric series is:
\[ S = \frac{a}{1 - r} \]
where \( |r| < 1 \).

### Step 3: Substitute the values of \( a \) and \( r \) into the formula
Now substituting \( a = \frac{1}{4} \) and \( r = -\frac{3}{4} \):
\[ S = \frac{\frac{1}{4}}{1 - \left(-\frac{3}{4}\right)} = \frac{\frac{1}{4}}{1 + \frac{3}{4}} = \frac{\frac{1}{4}}{\frac{4}{4} + \frac{3}{4}} = \frac{\frac{1}{4}}{\frac{7}{4}} \]

### Step 4: Simplify the expression
To simplify the expression:
\[ S = \frac{1}{4} \times \frac{4}{7} = \frac{1 \cdot 4}{4 \cdot 7} = \frac{1}{7} \]

### Final Answer:
Thus, the sum of the infinite geometric series is:
\[ \boxed{\frac{1}{7}} \]