The sum of an infinite geometric progression is 6. if the sum of the first two terms is 9/2, then what is the first term ?

Options

A. 1
B. `5//2`
C. or 3 or 3/5
D. 9 or 3

Answer

To solve the problem step by step, we need to find the first term \( A \) of an infinite geometric progression (GP) given the following conditions:

1. The sum of the infinite GP is \( S = 6 \).
2. The sum of the first two terms is \( \frac{9}{2} \).

### Step 1: Use the formula for the sum of an infinite GP
The formula for the sum of an infinite geometric progression is given by:
\[ S = \frac{A}{1 - R} \]
where \( A \) is the first term and \( R \) is the common ratio.

Given that \( S = 6 \), we can write:
\[ 6 = \frac{A}{1 - R} \]

### Step 2: Rearranging the equation
From the equation above, we can express \( A \) in terms of \( R \):
\[ A = 6(1 - R) = 6 - 6R \]

### Step 3: Sum of the first two terms
The first two terms of the GP are \( A \) and \( AR \). Therefore, the sum of the first two terms is:
\[ A + AR = A(1 + R) \]
We know this sum is \( \frac{9}{2} \), so we can write:
\[ A(1 + R) = \frac{9}{2} \]

### Step 4: Substitute \( A \) in the equation
Now we substitute \( A = 6 - 6R \) into the equation:
\[ (6 - 6R)(1 + R) = \frac{9}{2} \]

### Step 5: Expand and simplify
Now we expand the left side:
\[ 6(1 + R) - 6R(1 + R) = \frac{9}{2} \]
This simplifies to:
\[ 6 + 6R - 6R - 6R^2 = \frac{9}{2} \]
Thus, we have:
\[ 6 - 6R^2 = \frac{9}{2} \]

### Step 6: Clear the fraction
To eliminate the fraction, multiply the entire equation by 2:
\[ 12 - 12R^2 = 9 \]

### Step 7: Rearranging the equation
Now rearranging gives:
\[ 12R^2 = 12 - 9 \]
\[ 12R^2 = 3 \]
\[ R^2 = \frac{3}{12} = \frac{1}{4} \]

### Step 8: Solve for \( R \)
Taking the square root gives:
\[ R = \frac{1}{2} \quad \text{or} \quad R = -\frac{1}{2} \]

### Step 9: Find \( A \) for both values of \( R \)
1. For \( R = \frac{1}{2} \):
\[ A = 6 - 6 \left(\frac{1}{2}\right) = 6 - 3 = 3 \]

2. For \( R = -\frac{1}{2} \):
\[ A = 6 - 6 \left(-\frac{1}{2}\right) = 6 + 3 = 9 \]

### Conclusion
The possible values for the first term \( A \) are \( 3 \) and \( 9 \).

### Final Answer
The first term can be either \( 3 \) or \( 9 \).