The sum to infinite terms of the arithmetic - gemoetric progression `3, 4, 4, (32)/(9), ……` is equal to

Options

A. 27
B. 30
C. 24
D. 25

Answer

To find the sum to infinite terms of the arithmetic-geometric progression (AGP) given by the sequence \(3, 4, 4, \frac{32}{9}, \ldots\), we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \(a\) of the AGP is given as:
\[ a = 3 \]
To find the common difference \(d\), we can observe the terms:
- The second term is \(4\).
- The third term is also \(4\).

We can set up the equation for the second term:
\[ a + d \cdot r = 4 \]
Substituting \(a = 3\):
\[ 3 + d \cdot r = 4 \implies d \cdot r = 1 \implies d = \frac{1}{r} \]

### Step 2: Find the third term
For the third term:
\[ a + 2d \cdot r^2 = 4 \]
Substituting \(a = 3\):
\[ 3 + 2d \cdot r^2 = 4 \implies 2d \cdot r^2 = 1 \implies d \cdot r^2 = \frac{1}{2} \]

### Step 3: Set up equations
Now we have two equations:
1. \(d \cdot r = 1\)
2. \(d \cdot r^2 = \frac{1}{2}\)

From the first equation, we can express \(d\) in terms of \(r\):
\[ d = \frac{1}{r} \]
Substituting this into the second equation:
\[ \frac{1}{r} \cdot r^2 = \frac{1}{2} \implies r = \frac{1}{2} \]

### Step 4: Find \(d\)
Substituting \(r = \frac{1}{2}\) back into the equation for \(d\):
\[ d = \frac{1}{\frac{1}{2}} = 2 \]

### Step 5: Calculate the sum of infinite terms
The formula for the sum of an infinite AGP is given by:
\[ S_{\infty} = \frac{a}{1 - r} + \frac{d \cdot r}{(1 - r)^2} \]
Substituting \(a = 3\), \(d = 2\), and \(r = \frac{1}{2}\):
\[ S_{\infty} = \frac{3}{1 - \frac{1}{2}} + \frac{2 \cdot \frac{1}{2}}{(1 - \frac{1}{2})^2} \]
Calculating each term:
\[ S_{\infty} = \frac{3}{\frac{1}{2}} + \frac{1}{\left(\frac{1}{2}\right)^2} \]
\[ = 6 + 4 = 10 \]

### Final Answer
The sum to infinite terms of the arithmetic-geometric progression is:
\[ \boxed{10} \]