Solve the Recurrence $Q_0 = \alpha$, $Q_1 = \beta$ and $Q_n = (1 + Q_{n-1})/Q_{n-2}$ for $n \ge 1$

Question

Solve the Recurrence $Q_0 = \alpha$, $Q_1 = \beta$ and $Q_n = \dfrac{1 + Q_{n-1}}{Q_{n-2}}$ for $n \ge 1$

This problem is from Concrete Mathematics by Donald Knuth. One of the homework exercises on Chapter one. However the author gave a hint that $Q_4 = \frac{1 + \alpha}{\beta}$ but I don't get why, plugging in small values of $n$ into our recurrence above, I was able to obtain that.

$$Q_2 = \frac{1 + \beta}{\alpha}$$

$$Q_3 = \frac{1 + \alpha + \beta}{\alpha^2}$$ And finally

$$Q_4 = \frac{\alpha^2 + \alpha + \beta + 1}{1 + \beta}$$ Which is quite different from the hint provided by the author. Where is my mistake? Useful hints and corrections are welcome.

Answer 1

The first few terms are:

$$Q_0=\alpha$$ $$Q_1=\beta$$ $$Q_2={{1+\beta}\over \alpha}$$ $$Q_3={{1+{{1+\beta}\over{\alpha}}}\over{\beta}}={{1+\alpha+\beta}\over{\alpha\beta}}$$ $$Q_4=...={{1+\alpha}\over{\beta}}$$ $$Q_5=...=\alpha$$ $$Q_6=...=\beta$$

therefore the sequence is periodic with period $5$.

$\alpha,\beta,{{1+\beta}\over \alpha},{{1+\alpha+\beta}\over{\alpha\beta}},{{1+\alpha}\over{\beta}},\alpha,\beta,...$