Problem 0: September 2023

In this problem, \(f\) will always be a function defined by \(f(r) = \dfrac{ar+b}{cr+d}\) 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 \(n\geq 2\). 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.

Suppose \(f(r) = \dfrac{2r-1}{r+2}\).

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

Find a rational number \(r_1\) with the property that \(r_2\) is defined, but \(r_3\) is not defined.

Suppose \(f(r) = \dfrac{r+3}{2r-1}\).

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

Determine all rational values of \(r_1\) with the property that there is some integer \(n\geq 1\) 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 \(r_1\).

Suppose \(f(r) = \dfrac{r+2}{r+1}\).

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

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

Suppose \(r_1\) is a positive rational number. Prove that \(\left|r_n-\sqrt{2}\right| < \dfrac{1}{2^{n-1}}\left|r_1-\sqrt{2}\right|\) for each \(n\geq 2\). 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 \(\sqrt{3}\)?

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. \[f(r)=\dfrac{r-3}{r-2},\,\,\,\, f(r)=\dfrac{r-1}{5r+3},\,\,\,\, f(r) = \dfrac{r-1}{r+2},\,\,\,\, f(r) = \dfrac{2r+2}{3r+3},\,\,\,\, f(r)=\frac{r+1}{r-2}\]