0

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?

user10354138
  • 33,239
zzz
  • 1

1 Answers1

0

It is highly unlikely that this non-linear difference equation has a simple closed solution.

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 $c$.

vonbrand
  • 27,812