What is the sum of the infinite geometric series whose first two terms are 3 and 1?

Options

A. 1.5
B. 4.5
C. 9
D. 12

Explanation

To find the sum of the infinite geometric series whose first two terms are 3 and 1, we can follow these steps:

### Step 1: Identify the first term and the common ratio
The first term \( a \) of the series is given as 3. The second term is 1.

To find the common ratio \( r \), we can use the formula:
\[ r = \frac{\text{second term}}{\text{first term}} = \frac{1}{3} \]

### 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 \( a \) is the first term and \( r \) is the common ratio.

### Step 3: Substitute the values into the formula
Now we substitute \( a = 3 \) and \( r = \frac{1}{3} \) into the formula:
\[ S = \frac{3}{1 - \frac{1}{3}} \]

### Step 4: Simplify the expression
First, simplify the denominator:
\[ 1 - \frac{1}{3} = \frac{3}{3} - \frac{1}{3} = \frac{2}{3} \]
Now, substitute this back into the sum formula:
\[ S = \frac{3}{\frac{2}{3}} \]

### Step 5: Perform the division
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ S = 3 \times \frac{3}{2} = \frac{9}{2} \]

### Step 6: Convert to decimal (if necessary)
\[ \frac{9}{2} = 4.5 \]

Thus, the sum of the infinite geometric series is \( 4.5 \).