I'm having some trouble with trying to find the general term of this sequence. It has a non-linear recurrence. I would really appreciate it if anyone could help me!
$ a_{n}= \frac{c_{1}}{c_{2}-a_{n-1}} $
Although it may sound a bit strange, this comes from a really interesting problem of Electromagnetism that I'm trying to solve. The goal of the exercise is just to find this recursive expression, but I'm excited with going further because the teacher told us that a long time ago he saw the solution developed by an old professor, but he hasn't ever been able to reproduce it or find it again.
Introduce a new sequence $b_n$ such that $a_n=\frac{b_n}{b_{n+1}}$. Then, we have
$$\begin{align} \frac{b_n}{b_{n+1}}&=\frac{c_2}{c_1-\frac{b_{n-1}}{b_{n}}}\\\\ &=\frac{c_2b_n}{c_1b_n-b_{n-1}} \end{align}$$
which after rearranging becomes
$$c_2b_{n+1}-c_1b_n+b_{n-1}=0$$
which is a linear second order difference equation.