Are there $p_0, q_0 \in\mathbb{R}$, and computable functions $P(p_n, \ldots) = p_{n+1}, Q(q_n, \ldots) = q_{n+1}$ such that for all $n\in\mathbb{N}$, $p_n\ge1$, $q_n\ge1$, both $p_n, q_n$ are bounded, and $p_{n+1} + q_{n+1} = 2q_n - 2.5$? (EDIT: the limit condition is wrong, sorry).
The ellipsis in the functions means that these extra parameters maybe included: $q_n$ for $P$, $p_n$ for $Q$, $n$ itself, $w$, etc. I need guidance on how to proceed on problems like this. Thanks!
Context: I attempted to create a self-similar impossible rod. Unfortunately, it is not truly self-similar—see the image below.
As also shown in the image, $p$ and $q$ are proportions to the "width" $w$ of the "outer" and "triangular" part of rod, respectively. For each iteration $n$, this width decreases by half. The relationship between the previous and next iterate gives the $p_n + q_n \ldots$ equation above.
Therefore, answering the above question answers the title question. Here's the HTML code to generate the above image: https://github.com/poypoyan/canvas-svg-experiments/blob/main/impossible-rod/impossible-rod.html.
It is possible, and the solution is so trivial I should have solved this earlier. The simplest solution is by setting $p_n=3.5$ and $q_n=6$ for all $n$. Both are greater than 1, both are of course bounded, and $6+3.5 = 2(6)-2.5 \implies 9.5 = 12-2.5 \implies 9.5=9.5$.
Here's the rendering for this (10 iterations). Enjoy!