Find the coefficient of `x^(50)` in `(2-3x)/((1-x)^(3))`

Answer

To find the coefficient of \( x^{50} \) in the expression \( \frac{2 - 3x}{(1 - x)^3} \), we can break the problem down into manageable steps.

### Step 1: Rewrite the Expression
We start by separating the expression:
\[ \frac{2 - 3x}{(1 - x)^3} = \frac{2}{(1 - x)^3} - \frac{3x}{(1 - x)^3} \]

### Step 2: Find the Coefficient of \( x^{50} \) in \( \frac{2}{(1 - x)^3} \)
Using the binomial series expansion, we know that:
\[ \frac{1}{(1 - x)^r} = \sum_{n=0}^{\infty} \binom{n + r - 1}{r - 1} x^n \]
For \( r = 3 \), we have:
\[ \frac{1}{(1 - x)^3} = \sum_{n=0}^{\infty} \binom{n + 2}{2} x^n \]
Thus, the coefficient of \( x^{50} \) in \( \frac{2}{(1 - x)^3} \) is:
\[ 2 \cdot \binom{50 + 2}{2} = 2 \cdot \binom{52}{2} \]
Calculating \( \binom{52}{2} \):
\[ \binom{52}{2} = \frac{52 \times 51}{2} = 1326 \]
So, the coefficient from this part is:
\[ 2 \cdot 1326 = 2652 \]

### Step 3: Find the Coefficient of \( x^{49} \) in \( \frac{-3x}{(1 - x)^3} \)
For this part, we need the coefficient of \( x^{49} \):
\[ -3 \cdot \binom{49 + 2}{2} = -3 \cdot \binom{51}{2} \]
Calculating \( \binom{51}{2} \):
\[ \binom{51}{2} = \frac{51 \times 50}{2} = 1275 \]
So, the coefficient from this part is:
\[ -3 \cdot 1275 = -3825 \]

### Step 4: Combine the Coefficients
Now we combine the coefficients from both parts:
\[ \text{Total Coefficient} = 2652 - 3825 = -1173 \]

### Final Answer
Thus, the coefficient of \( x^{50} \) in \( \frac{2 - 3x}{(1 - x)^3} \) is:
\[ \boxed{-1173} \]