If the sum to infinity of the series `1+2r+3r^2+4r^3+` is 9/4, then value of `r` is `1//2` b. `1//3` c. `1//4` d. none of these

Options

A. `1//2`
B. `1//3`
C. `1//4`
D. none of these

Explanation

To find the value of \( r \) for the series \( S = 1 + 2r + 3r^2 + 4r^3 + \ldots \) given that its sum to infinity is \( \frac{9}{4} \), we can follow these steps:

### Step 1: Express the series
The series can be expressed as:
\[ S = 1 + 2r + 3r^2 + 4r^3 + \ldots \]

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

### Step 3: Subtract the two equations
Subtract the second equation from the first:
\[ S - rS = (1 + 2r + 3r^2 + 4r^3 + \ldots) - (r + 2r^2 + 3r^3 + 4r^4 + \ldots) \]
This simplifies to:
\[ S(1 - r) = 1 + (2r - r) + (3r^2 - 2r^2) + (4r^3 - 3r^3) + \ldots \]
Which further simplifies to:
\[ S(1 - r) = 1 + r + r^2 + r^3 + \ldots \]

### Step 4: Recognize the geometric series
The right-hand side is a geometric series with first term \( 1 \) and common ratio \( r \):
\[ 1 + r + r^2 + r^3 + \ldots = \frac{1}{1 - r} \quad \text{(for } |r| < 1\text{)} \]

### Step 5: Substitute back into the equation
Now we have:
\[ S(1 - r) = \frac{1}{1 - r} \]

### Step 6: Solve for \( S \)
Rearranging gives:
\[ S = \frac{1}{(1 - r)(1 - r)} = \frac{1}{(1 - r)^2} \]

### Step 7: Set the equation equal to \( \frac{9}{4} \)
According to the problem, the sum \( S \) is given as \( \frac{9}{4} \):
\[ \frac{1}{(1 - r)^2} = \frac{9}{4} \]

### Step 8: Cross-multiply and solve for \( 1 - r \)
Cross-multiplying gives:
\[ 4 = 9(1 - r)^2 \]
Dividing both sides by 9:
\[ \frac{4}{9} = (1 - r)^2 \]

### Step 9: Take the square root
Taking the square root of both sides gives:
\[ 1 - r = \pm \frac{2}{3} \]

### Step 10: Solve for \( r \)
This results in two cases:
1. \( 1 - r = \frac{2}{3} \) which gives \( r = 1 - \frac{2}{3} = \frac{1}{3} \)
2. \( 1 - r = -\frac{2}{3} \) which gives \( r = 1 + \frac{2}{3} = \frac{5}{3} \)

### Step 11: Choose the valid value for \( r \)
Since \( r \) must be between 0 and 1 for the series to converge, we discard \( r = \frac{5}{3} \) and accept:
\[ r = \frac{1}{3} \]

### Final Answer
Thus, the value of \( r \) is:
\[ \boxed{\frac{1}{3}} \]