TITLE: How is it possible to find a closed formula for the $n$-th element of the series defined by $s_{n+1} = \frac{s_n+ax}{1-bx}$?
QUESTION [1 upvotes]: This is the definition of the series ($a,b,x \in \mathbf{R}$):
$$s_{n+1} = \frac{s_n+ax}{1-bx} $$
I would like to find a closed formula, to calculate $s_n$ from $s_0$
How is it possible? The denominator makes it impossible to factor out $s_n$, and I have no idea how to progress further.

REPLY [0 votes]: By induction easily we can show that
$$s_n = \frac{s_0 + ax + ax\sum_{i=0}^{n-2}(1-bx)^{2^i}}{(1-bx)^{2^{n-1}}}$$
If we simplify the above equation we can get the above equation too.
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