how to work out a closed form of a sequence 
Consider the following linear recurrence sequence.

$x_1 = 11$,  $x_{n+1} = -0.8x_n + 9,\quad n = 1,2,3, \ldots.$
Find a closed form for this sequence.

Use generating functions. Define $X(z) = \sum_{n \ge 0} x_{n + 1} z^n$, multiply the recurrence by $z^n$ and sum over $n \ge 0$ to get:
$$
\frac{X(z) - x_1}{z} = - \frac{4 X(z)}{5} + \frac{9}{1 - z}
$$
Thus:
$$
X(z) = \frac{55 - 10 z}{5 - z - 4 z^2}
     = 5 \cdot \frac{1}{1 - z} + 6 \cdot \frac{1}{1 + 4 z / 5}
$$
Two geometric series:
$$
x_{n + 1} = 5 + 6 \cdot (-0.8)^n
$$
