\(f(r)\) is undefined only when \(r=-2\). For what value of \(r\) is \(f(r)=-2\)?
The sequence in part (i) is periodic. Can you show that the sequence is periodic for other values of \(r_1\)?
No hint given.
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.
Use (ii) and the fact that when \(r\) is positive, \(\left|\dfrac{1-\sqrt{2}}{r+1}\right| < \frac{1}{2}\). 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 \(n+1\).
Three of these sequences are periodic, one of them is constant (after the first term), and one of them always approaches the fixed value \(\dfrac{3-\sqrt{13}}{2}\) as long as there are no undefined values in the sequence.