Defining and generalizing a recursive function applied $n$ times

Question

First of all, I am not a mathematician so forgive me for the errors.

I am dealing with a processing pipeline that applies a reduction function, $r(x)$, $n$ times at its core, where:

$$r(x) = \lceil x/2 \rceil$$

If I am applying this function $n = 3$ times then, the processing pipeline function $R_n(x)$ would be:

$$R_3(x) = r(r(r(x)))$$

How could I generalize this expression for an arbitrary $n$ number of times? I am looking for:

$$R_n(x) = ¿?$$

I know this must be very simple, but I just can not generalize it. I have search over recursive functions like the Fibonacci which gave me some clues but still can not generalize it. Thanks in advance community!

Answer 1

The answer is simple: $$R_n(x) = \lceil \frac{x}{2^n} \rceil$$ Applying $r$ will halve the value, so applying it $n$ times will divide by $2^n$. The repeated roundings upwards turn out not to matter, and can just be done once at the end.