Find nth term of mathematical series with both addition and multiplication

Question

If I have some series such that:

$$a_n= (a_{n-1} + b)c $$

where $n$ is the index of the term of the series, and $b$ and $c$ are variable integers, how can I find a numeric value of the $n$th term without a recursive function.

For example with a series starting at 10 with $b=-1$ and $c=0.5$ would be the following:

$$10, 4.5, 1.75, 0.375 \ldots$$

Is there a formula for $a_n$ which includes an initial value $a_0$ such that there is not recursion?

Additionally, I would appreciate if the steps and methodology used to achieve this formula are shown, not just the answer.

Answer 1

hint

Assume $ c\ne 1$.

Let us look for $x$ such that

$$a_n+x=c(a_{n-1}+x)$$

then $$cb=cx-x \; and \; x=\frac{cb}{c-1}$$

then $$a_n+x=c^n(a_0+x)$$