Let S denotes the infinite sum
` 2 + 5x + 9x^(2) + 14x^(3) + 2x^(4) + …`
where ` |x| lt 1 ` . Then S equals

Options

A. `(2-x)/((1 - x)^(3))`
B. `(2-x)/(( 1 + x)^(3))`
C. `(2 + x)/((1 - x)^(3))`
D. `(2 + x)/((1 + x)^(3))`

Answer

To solve the infinite sum \( S = 2 + 5x + 9x^2 + 14x^3 + 2x^4 + \ldots \) where \( |x| < 1 \), we will follow these steps:

### Step 1: Write the series
We start by expressing the sum \( S \):
\[ S = 2 + 5x + 9x^2 + 14x^3 + 2x^4 + \ldots \]

### Step 2: Multiply by \( x \)
Next, we multiply the entire series by \( x \):
\[ xS = 2x + 5x^2 + 9x^3 + 14x^4 + 2x^5 + \ldots \]

### Step 3: Subtract the two equations
Now, we subtract the equation for \( xS \) from \( S \):
\[ S - xS = (2 + 5x + 9x^2 + 14x^3 + 2x^4 + \ldots) - (2x + 5x^2 + 9x^3 + 14x^4 + 2x^5 + \ldots) \]
This simplifies to:
\[ S - xS = 2 + (5x - 2x) + (9x^2 - 5x^2) + (14x^3 - 9x^3) + (2x^4 - 14x^4) + \ldots \]
\[ S - xS = 2 + 3x + 4x^2 + 5x^3 + 6x^4 + \ldots \]

### Step 4: Recognize the new series
The series \( 2 + 3x + 4x^2 + 5x^3 + 6x^4 + \ldots \) can be expressed in a different form. We denote this new series as \( T \):
\[ T = 2 + 3x + 4x^2 + 5x^3 + 6x^4 + \ldots \]

### Step 5: Multiply \( T \) by \( x \)
Now, we multiply \( T \) by \( x \):
\[ xT = 2x + 3x^2 + 4x^3 + 5x^4 + 6x^5 + \ldots \]

### Step 6: Subtract \( T \) and \( xT \)
Subtract \( xT \) from \( T \):
\[ T - xT = 2 + (3x - 2x) + (4x^2 - 3x^2) + (5x^3 - 4x^3) + (6x^4 - 5x^4) + \ldots \]
This simplifies to:
\[ T - xT = 2 + x + x^2 + x^3 + x^4 + \ldots \]
The right side is a geometric series with first term \( 2 \) and common ratio \( x \):
\[ T - xT = 2 + \frac{x}{1 - x} \]

### Step 7: Solve for \( T \)
We can factor out \( T \):
\[ T(1 - x) = 2 + \frac{x}{1 - x} \]
Thus,
\[ T = \frac{2 + \frac{x}{1 - x}}{1 - x} \]
This simplifies to:
\[ T = \frac{2(1 - x) + x}{(1 - x)(1 - x)} = \frac{2 - 2x + x}{(1 - x)^2} = \frac{2 - x}{(1 - x)^2} \]

### Step 8: Substitute back to find \( S \)
Now we substitute \( T \) back into our equation for \( S \):
\[ S(1 - x) = T \]
Thus,
\[ S = \frac{T}{1 - x} = \frac{\frac{2 - x}{(1 - x)^2}}{1 - x} = \frac{2 - x}{(1 - x)^3} \]

### Final Answer
Therefore, the sum \( S \) is:
\[ S = \frac{2 - x}{(1 - x)^3} \]