In the sum to infinity of the series `3+(3+x) (1)/(4) + (3+2x)(1)/(4^(2))+ ..... "is" (44)/(9)` find x.

Answer

To solve the problem, we need to find the value of \( x \) in the infinite series given by:

\[ S = 3 + \frac{(3+x)}{4} + \frac{(3+2x)}{4^2} + \ldots = \frac{44}{9} \]

### Step 1: Write the equation for the sum to infinity
We can express the sum \( S \) as:

\[ S = 3 + \frac{(3+x)}{4} + \frac{(3+2x)}{4^2} + \ldots \]

### Step 2: Factor the series
Notice that we can factor out the terms involving \( x \) and rewrite the series:

\[ S = 3 + \sum_{n=0}^{\infty} \frac{(3 + nx)}{4^n} \]

### Step 3: Separate the series
We can separate the series into two parts:

\[ S = 3 + \sum_{n=0}^{\infty} \frac{3}{4^n} + \sum_{n=0}^{\infty} \frac{nx}{4^n} \]

### Step 4: Calculate the first series
The first series \( \sum_{n=0}^{\infty} \frac{3}{4^n} \) is a geometric series with first term \( a = 3 \) and common ratio \( r = \frac{1}{4} \):

\[ \sum_{n=0}^{\infty} \frac{3}{4^n} = \frac{3}{1 - \frac{1}{4}} = \frac{3}{\frac{3}{4}} = 4 \]

### Step 5: Calculate the second series
The second series \( \sum_{n=0}^{\infty} \frac{nx}{4^n} \) can be computed using the formula for the sum of \( n \cdot r^n \):

\[ \sum_{n=0}^{\infty} n r^n = \frac{r}{(1 - r)^2} \]

Here, \( r = \frac{1}{4} \):

\[ \sum_{n=0}^{\infty} n \left(\frac{1}{4}\right)^n = \frac{\frac{1}{4}}{\left(1 - \frac{1}{4}\right)^2} = \frac{\frac{1}{4}}{\left(\frac{3}{4}\right)^2} = \frac{\frac{1}{4}}{\frac{9}{16}} = \frac{4}{9} \]

Thus,

\[ \sum_{n=0}^{\infty} \frac{nx}{4^n} = x \cdot \frac{4}{9} \]

### Step 6: Combine the results
Now we can combine the results:

\[ S = 3 + 4 + x \cdot \frac{4}{9} \]

### Step 7: Set the equation equal to \( \frac{44}{9} \)
Setting \( S \) equal to \( \frac{44}{9} \):

\[ 3 + 4 + x \cdot \frac{4}{9} = \frac{44}{9} \]

### Step 8: Simplify the left side
Convert \( 3 + 4 \) into a fraction:

\[ 7 = \frac{63}{9} \]

Thus, we have:

\[ \frac{63}{9} + \frac{4x}{9} = \frac{44}{9} \]

### Step 9: Clear the denominators
Multiply through by 9 to eliminate the denominators:

\[ 63 + 4x = 44 \]

### Step 10: Solve for \( x \)
Now, isolate \( x \):

\[ 4x = 44 - 63 \]
\[ 4x = -19 \]
\[ x = -\frac{19}{4} \]

Thus, the value of \( x \) is:

\[ \boxed{-\frac{19}{4}} \]