The coefficient of xn in11-x1-2x1-3x is

1(1-x) (1-2x) (1-3x)=A1-x+B1-2x+C1-3x 1=A(1-2x) (1-3x)+B(1-x) (1-3x)+C(1-x) (1-3x) x=1 ⇒ 1=A(-1) (-2) ⇒ A=12 x=12 ⇒ 1=B12 -12 ⇒ B=-4 x=13 ⇒ 1=C23 13 ⇒ C=92 =12 11-x-41-2x+94 11-3x =12 (1-x)-1-4(1-2x)-1+94(1-3x)-1 =12 (1+x+x2+......+xn+......) -941+3x+32x2+......+3nxn+....... xn Coeff =12-4.2n+923n =3n+2-2n+3+12

The coefficient of xn in11-x1-2x1-3x is

1(1-x) (1-2x) (1-3x)=A1-x+B1-2x+C1-3x 1=A(1-2x) (1-3x)+B(1-x) (1-3x)+C(1-x) (1-3x) x=1 ⇒ 1=A(-1) (-2) ⇒ A=12 x=12 ⇒ 1=B12 -12 ⇒ B=-4 x=13 ⇒ 1=C23 13 ⇒ C=92 =12 11-x-41-2x+94 11-3x =12 (1-x)-1-4(1-2x)-1+94(1-3x)-1 =12 (1+x+x2+......+xn+......) -941+3x+32x2+......+3nxn+....... xn Coeff =12-4.2n+923n =3n+2-2n+3+12

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The coefficient of $x^{n}$ in
$\frac{1}{(1 - x) (1 - 2 x) (1 - 3 x)} is$

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$\frac{3^{n + 2} - 2^{n + 3} + 1}{2}$

$\frac{3^{n + 2} + 2^{n + 3} - 1}{2}$

$\frac{3^{n + 2} - 2^{n + 3} - 1}{2}$

$\frac{3^{n + 2} + 2^{n + 3} + 1}{2}$

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Detailed Solution

$\begin{array}{lcl} \begin{array}{lcl} \frac{1}{(1 - x) (1 - 2 x) (1 - 3 x)} & = & \frac{A}{1 - x} + \frac{B}{1 - 2 x} + \frac{C}{1 - 3 x} \end{array} \\ 1 \begin{array}{cl} = & A (1 - 2 x) (1 - 3 x) + B (1 - x) (1 - 3 x) + C (1 - x) (1 - 3 x) \end{array} \\ x \begin{array}{cl} = & 1 \Rightarrow 1 = A (- 1) (- 2) \Rightarrow A = \frac{1}{2} \end{array} \\ x \begin{array}{cl} = & \frac{1}{2} \Rightarrow 1 = B (\frac{1}{2}) (\frac{- 1}{2}) \Rightarrow B = - 4 \end{array} \\ x \begin{array}{cl} = & \frac{1}{3} \Rightarrow 1 = C (\frac{2}{3}) (\frac{1}{3}) \Rightarrow C = \frac{9}{2} \end{array} \\ = \frac{1}{2} \frac{1}{1 - x} - \frac{4}{1 - 2 x} + \frac{9}{4} \frac{1}{1 - 3 x} \\ \begin{array}{cl} = & \frac{1}{2} {(1 - x)}^{- 1} - 4 {(1 - 2 x)}^{- 1} + \frac{9}{4} {(1 - 3 x)}^{- 1} = \frac{1}{2} (1 + x + x^{2} + . . . . . . + x^{n} + . . . . . .) \end{array} \\ - \frac{9}{4} (1 + 3 x + 3^{2} x^{2} + . . . . . . + 3^{n} x^{n} + . . . . . . .) \\ x^{n} Coeff \begin{array}{cl} = & \frac{1}{2} - 4 . 2^{n} + \frac{9}{2} 3^{n} \end{array} \\ \begin{array}{cl} = & \frac{3^{n + 2} - 2^{n + 3} + 1}{2} \end{array} \end{array}$