If sum to infinity of the series ` 3- 5r +7r^(2) -9r^(3) + ….." is " (14)/(9) .` Find r.

Answer

To find the value of \( r \) for the given series \( 3 - 5r + 7r^2 - 9r^3 + \ldots \) with a sum to infinity of \( \frac{14}{9} \), we can follow these steps:

### Step 1: Define the series and its sum
Let the sum of the series be denoted as \( S \):
\[ S = 3 - 5r + 7r^2 - 9r^3 + \ldots \]
According to the problem, we know that:
\[ S = \frac{14}{9} \]

### Step 2: Multiply the series by \( r \)
Now, multiply the entire series by \( r \):
\[ rS = 3r - 5r^2 + 7r^3 - 9r^4 + \ldots \]

### Step 3: Set up the equations
Now we have two equations:
1. \( S = 3 - 5r + 7r^2 - 9r^3 + \ldots \) (Equation 1)
2. \( rS = 3r - 5r^2 + 7r^3 - 9r^4 + \ldots \) (Equation 2)

### Step 4: Add the two equations
Adding Equation 1 and Equation 2:
\[ S + rS = (3 - 5r + 7r^2 - 9r^3) + (3r - 5r^2 + 7r^3 - 9r^4) \]
This simplifies to:
\[ S(1 + r) = 3 - 2r + 2r^2 - 2r^3 + \ldots \]

### Step 5: Recognize the remaining series
The remaining series \( 2r^2 - 2r^3 + \ldots \) can be factored:
\[ 2r^2(1 - r + r^2 - \ldots) = 2r^2 \frac{1}{1 + r} = \frac{2r^2}{1 + r} \]
Thus, we can rewrite the equation as:
\[ S(1 + r) = 3 - 2r + \frac{2r^2}{1 + r} \]

### Step 6: Substitute \( S \) and solve for \( r \)
Substituting \( S = \frac{14}{9} \):
\[ \frac{14}{9}(1 + r) = 3 - 2r + \frac{2r^2}{1 + r} \]

### Step 7: Cross-multiply and simplify
Cross-multiplying gives:
\[ 14(1 + r) = 27 - 18r + \frac{18r^2}{1 + r} \]
Multiplying through by \( 1 + r \) to eliminate the fraction:
\[ 14(1 + r)^2 = 27(1 + r) - 18r(1 + r) \]

### Step 8: Expand and rearrange
Expanding both sides:
\[ 14(1 + 2r + r^2) = 27 + 27r - 18r - 18r^2 \]
This simplifies to:
\[ 14 + 28r + 14r^2 = 27 + 9r - 18r^2 \]
Rearranging gives:
\[ 32r^2 + 19r - 13 = 0 \]

### Step 9: Use the quadratic formula
Using the quadratic formula \( r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 32, b = 19, c = -13 \):
\[ r = \frac{-19 \pm \sqrt{19^2 - 4 \cdot 32 \cdot (-13)}}{2 \cdot 32} \]
Calculating the discriminant:
\[ 19^2 + 4 \cdot 32 \cdot 13 = 361 + 1664 = 2025 \]
Thus:
\[ r = \frac{-19 \pm 45}{64} \]

### Step 10: Calculate the values of \( r \)
Calculating the two possible values:
1. \( r = \frac{26}{64} = \frac{13}{32} \)
2. \( r = \frac{-64}{64} = -1 \)

### Final Answer
Thus, the values of \( r \) are:
\[ r = \frac{13}{32}, -1 \]