Hi: I'm am a total difference equation beginner but I noticed the following difference equation behavior and was wondering if it has a name or where I can read more about it. Suppose I have the following difference equation setup so I have
$ x_{0} = $ some initial value and the specific difference equation is:
$ x_{1} = (-2 \times (1-p)) \times x_{0} $
And, then, in general
$x_{k+1} = p \times ( -2 \times(1-p) )\times x_{k} $ $~~~for~~~ k = 1,\ldots (n-1) $
where $n$ is finite. In this manner. the values of $x_{k}$ go from $k = 0$ to $k = n$ and the recursion stops there.
What I noticed about the recurrence relation above, is the following: Suppose
1) I choose a value of $n$ and then set $p$ such that $p^n = \frac{1}{2}$.
2) Given this values $p$ and value of $n$ and the initial value $x_{0}$, I calculate the values of $x_{k}$ for $k = 1, \ldots, n$.
3) Finally, I sum all of the $x_{k}$ from $k = 1$ to k = n$.
I notice that, if I do the three steps above, then it is always the case that Step 3 indicates that $\sum_{k=1}^{n} x_{i} = -x_{0}$. It doesn't matter what value of $n$ is chosen in Step 1.
Obviously this means that the solution is periodic in some sense of the meaning of periodicity. But, like I said, I'm a total beginner in difference equations. So, my questions are: Is this a well known difference equation as far as it having this property ? Does it fall under some class of difference equations. Is it discussed somewhere because I don't know the difference equations literature much at all. Thanks for any references, links etc because I don't know where to look.