Sequences not so far from being arithmetic/geometric.

Let $(a_n)$ be a sequence determined by its first term, say $a_0$ and a recurrence formula of the form $a_{n+1}= f(a_n)$ , where $f$ is a function in whose domain belong all the ternms of the sequence. In some special cases, we can defined another sequence $(b_n)$ , ``close to'' $(a_n)$ and which is either geometric or arithmetic. This will help to decide whether $(a_n)$ is convergent or not.