The coefficient of `x^(n)` in `(x+1)/((x-1)^(2)(x-2))` is

Options

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

Answer

To find the coefficient of \( x^n \) in the expression

\[ \frac{x+1}{(x-1)^2(x-2)}, \]

we will use the method of partial fractions. Here are the steps to solve the problem:

### Step 1: Set up the partial fraction decomposition

We can express the given function as:

\[ \frac{x+1}{(x-1)^2(x-2)} = \frac{A}{x-1} + \frac{B}{(x-1)^2} + \frac{C}{x-2} \]

where \( A \), \( B \), and \( C \) are constants that we need to determine.

### Step 2: Clear the denominators

Multiplying both sides by the denominator \((x-1)^2(x-2)\) gives:

\[ x + 1 = A(x-1)(x-2) + B(x-2) + C(x-1)^2 \]

### Step 3: Expand the right-hand side

Expanding the right-hand side, we get:

\[ A(x^2 - 3x + 2) + B(x - 2) + C(x^2 - 2x + 1) \]

Combining like terms, we have:

\[ (A + C)x^2 + (-3A - 2C + B)x + (2A - 2B + C) \]

### Step 4: Set up equations for coefficients

Now, we can equate the coefficients from both sides of the equation:

1. Coefficient of \( x^2 \): \( A + C = 0 \)
2. Coefficient of \( x \): \( -3A - 2C + B = 1 \)
3. Constant term: \( 2A - 2B + C = 1 \)

### Step 5: Solve the system of equations

From the first equation, we can express \( C \) in terms of \( A \):

\[ C = -A \]

Substituting \( C \) into the other two equations:

1. \( -3A - 2(-A) + B = 1 \) simplifies to \( -3A + 2A + B = 1 \) or \( -A + B = 1 \) (Equation 1).
2. \( 2A - 2B - A = 1 \) simplifies to \( A - 2B = 1 \) (Equation 2).

Now, we can solve these two equations:

From Equation 1:

\[ B = A + 1 \]

Substituting \( B \) into Equation 2:

\[ A - 2(A + 1) = 1 \implies A - 2A - 2 = 1 \implies -A - 2 = 1 \implies -A = 3 \implies A = -3 \]

Now substituting \( A \) back to find \( B \) and \( C \):

\[ B = -3 + 1 = -2 \]
\[ C = -(-3) = 3 \]

### Step 6: Write the partial fraction decomposition

Now we have:

\[ \frac{x+1}{(x-1)^2(x-2)} = \frac{-3}{x-1} + \frac{-2}{(x-1)^2} + \frac{3}{x-2} \]

### Step 7: Rewrite using series expansion

Next, we can rewrite each term using the geometric series expansion:

1. \(\frac{-3}{x-1} = -3 \cdot \frac{1}{1 - \frac{x}{1}} = -3 \sum_{k=0}^{\infty} \left(\frac{x}{1}\right)^k = -3 \sum_{k=0}^{\infty} x^k\)
2. \(\frac{-2}{(x-1)^2} = -2 \cdot \frac{1}{(1 - \frac{x}{1})^2} = -2 \sum_{k=0}^{\infty} (k+1)x^k\)
3. \(\frac{3}{x-2} = 3 \cdot \frac{1}{1 - \frac{x}{2}} = 3 \sum_{k=0}^{\infty} \left(\frac{x}{2}\right)^k = 3 \sum_{k=0}^{\infty} \frac{x^k}{2^k}\)

### Step 8: Combine the series and find the coefficient of \( x^n \)

Now we need to combine these series and find the coefficient of \( x^n \):

1. From the first term, the coefficient of \( x^n \) is \( -3 \).
2. From the second term, the coefficient of \( x^n \) is \( -2(n + 1) \).
3. From the third term, the coefficient of \( x^n \) is \( \frac{3}{2^n} \).

Thus, the total coefficient of \( x^n \) is:

\[ -3 - 2(n + 1) + \frac{3}{2^n} \]

### Final Answer

The coefficient of \( x^n \) in the expression is:

\[ -2n - 5 + \frac{3}{2^n} \]