Problem of the Month
Problem 0: September 2023

In this problem, $f$ will always be a function defined by $f(r)={\displaystyle \frac{ar+b}{cr+d}}$ where $a$, $b$, $c$, and $d$ are integers. These integers will vary throughout the parts of the problem.

Given such a function $f$ and a rational number $r_1$, we can generate a sequence $r_1,r_2,r_3,\dots$ by taking $r_n=f(r_{n-1})$ for each $n\ge 2$. That is, $r_2=f(r_1)$, $r_3=f(r_2)$, $r_4=f(r_3)$, and so on. Unless there is some point in the sequence where $f(r_{n-1})$ is undefined, a sequence of this form can be made arbitrarily long.

These sequences behave in different ways depending on the function $f$ and the starting value $r_1$. This problem explores some those behaviours.

1. Suppose $f(r)={\displaystyle \frac{2r-1}{r+2}}$.

   1. With $r_1=\frac{3}{2}$, compute $r_2$, $r_3$, and $r_4$.

   2. Find a rational number $r_1$ with the property that $r_2$ is defined, but $r_3$ is not defined.

2. Suppose $f(r)={\displaystyle \frac{r+3}{2r-1}}$.

   1. With $r_1=\frac{3}{7}$, compute $r_2$, $r_3$, $r_4$, and $r_5$.

   2. Determine all rational values of $r_1$ with the property that there is some integer $n\ge 1$ for which $f(r_n)$ is undefined. For all other values of $r_1$, find simplified formulas for $r_{2023}$ and $r_{2024}$ in terms of $r_1$.

3. Suppose $f(r)={\displaystyle \frac{r+2}{r+1}}$.

   1. With $r_1=1$, compute $r_2$ through $r_9$. Write down decimal approximations of $r_2$ through $r_9$ (after computing them exactly).

   2. Suppose $r$ is a positive rational number. Prove that $$\left|\frac{f(r)-\sqrt{2}}{r-\sqrt{2}}\right|=\left|{\displaystyle \frac{1-\sqrt{2}}{r+1}}\right|$$

   3. Suppose $r_1$ is a positive rational number. Prove that $|r_n-\sqrt{2}|<{\displaystyle \frac{1}{2^{n-1}}}|r_1-\sqrt{2}|$ for each $n\ge 2$. Use this result to convince yourself that as $n$ gets large, $r_n$ gets close to $\sqrt{2}$, regardless of the choice of the positive value $r_1$. Can you modify $f$ slightly so that the sequence always approaches $\sqrt{3}$?

4. Explore the behaviour of the sequences generated by various values of $r_1$ for each of the functions below. Detailed solutions will not be provided, but a brief discussion will. $$f(r)={\displaystyle \frac{r-3}{r-2}},f(r)={\displaystyle \frac{r-1}{5r+3}},f(r)={\displaystyle \frac{r-1}{r+2}},f(r)={\displaystyle \frac{2r+2}{3r+3}},f(r)=\frac{r+1}{r-2}$$