How to approximate a complex linear homogenous recurrence with constant coefficients with a simple one?

Question

Is there some standard way to approximate a complex linear homogenous recurrence with constant coefficients with a simple one?

For example, I might want to approximate

$$ a_{n+k}=a_{n+k-1}+a_{n+k-2}+...+a_n $$

with a geometric series

$$ b_{n+1}=qb_n $$

using some standard method.

I'd like to estimate the series when the root of the characteristic equation is difficult to find or doesn't have an analytic solution.

Answer 1

In your example, take $q$ to be the number $q\gt1$ satisfying $q^k=q^{k-1}+q^{k-2}+\cdots+1$.