Problem of the Month
Hint for Problem 0: September 2023

1. $f(r)$ is undefined only when $r=-2$. For what value of $r$ is $f(r)=-2$?

2. The sequence in part (i) is periodic. Can you show that the sequence is periodic for other values of $r_1$?

3. 1. No hint given.

   2. After substituting the expression for $f(r)$, multiply the numerator and denominator by $r+1$. Try to find a common factor in the numerator and denominator.

   3. Use (ii) and the fact that when $r$ is positive, $\left|{\displaystyle \frac{1-\sqrt{2}}{r+1}}\right|<\frac{1}{2}$. Try to establish the given inequality for a few small values of $n$ and observe how knowing the inequality for $n$ can help you to deduce it for $n+1$.

4. Three of these sequences are periodic, one of them is constant (after the first term), and one of them always approaches the fixed value ${\displaystyle \frac{3-\sqrt{13}}{2}}$ as long as there are no undefined values in the sequence.