# General solution to a recursive rational equation

I have this problem which states that:

$$a_n = \frac{a_{n-1}}{4}\left(1-\frac{63}{a_{n-1}^3+7}\right),\text{ and }a_1=c$$

I have tried numerous ways to solve it by hand, as well as Wolfram Mathematica's RSolve function, with no success. The following is the command in Wolfram Mathematica:

RSolve[{a[n] == a[n-1]/4*(1-63/((a[n-1])^3+7)), a[1] == 1}, a[n], n]


I cannot get the closed form for $$a_n$$. Could anybody help please?

• Welcome to Mathematics Stack Exchange! A quick tour will enhance your experience. Here are helpful tips to write a good question and write a good answer. For equations, please use MathJax. Commented Jun 10, 2019 at 16:16
• Why would you expect a nice formula to exist? Commented Jun 10, 2019 at 16:16
• I would be very surprised by a closed form. Did you try to comute the first terms ? Commented Jun 11, 2019 at 3:15

As long as $$a_n$$ is small(ish), it will grow exponentially as $$(9/4)^n$$, once it gets large, it will decay as $$4^{-n}$$, until it gets small, and...
Play around with some values of $$a_0 = c$$ of interest, see if something interesting develops. Refine your question with specific $$c$$.