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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?

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    – dantopa
    Commented Jun 10, 2019 at 16:16
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    $\begingroup$ Why would you expect a nice formula to exist? $\endgroup$ Commented Jun 10, 2019 at 16:16
  • $\begingroup$ I would be very surprised by a closed form. Did you try to comute the first terms ? $\endgroup$ Commented Jun 11, 2019 at 3:15

1 Answer 1

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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 $c$.

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