The sum of the infinite Arithmetico -Geometric progression3,4,4,… is _________.

Explanation

To find the sum of the infinite Arithmetico-Geometric progression (AGP) given as 3, 4, 4, ..., 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:
\[ a = 3 \]

### Step 2: Identify the second term and set up equations
The second term is given as:
\[ a + d \cdot r = 4 \]
where \( d \) is the common difference and \( r \) is the common ratio.

### Step 3: Identify the third term and set up another equation
The third term is given as:
\[ a + 2d \cdot r^2 = 4 \]

### Step 4: Substitute the known values into the equations
From the second term:
\[ 3 + d \cdot r = 4 \implies d \cdot r = 1 \implies d = \frac{1}{r} \]

From the third term:
\[ 3 + 2d \cdot r^2 = 4 \implies 2d \cdot r^2 = 1 \implies d = \frac{1}{2r^2} \]

### Step 5: Set the two expressions for \( d \) equal to each other
\[ \frac{1}{r} = \frac{1}{2r^2} \]
Cross-multiplying gives:
\[ 2r = 1 \implies r = \frac{1}{2} \]

### Step 6: Substitute \( r \) back to find \( d \)
Using \( d = \frac{1}{r} \):
\[ d = \frac{1}{\frac{1}{2}} = 2 \]

### Step 7: Calculate the sum of the infinite AGP
The formula for the sum of an infinite AGP is:
\[ S_{\infty} = \frac{a}{1 - r} + \frac{d \cdot r}{(1 - r)^2} \]
Substituting the values \( a = 3 \), \( d = 2 \), and \( r = \frac{1}{2} \):
\[ S_{\infty} = \frac{3}{1 - \frac{1}{2}} + \frac{2 \cdot \frac{1}{2}}{(1 - \frac{1}{2})^2} \]
\[ = \frac{3}{\frac{1}{2}} + \frac{1}{\left(\frac{1}{2}\right)^2} \]
\[ = 6 + 4 = 10 \]

### Final Answer
The sum of the infinite Arithmetico-Geometric progression is:
\[ \boxed{10} \]