(a) Let $a_{0}, a_{1}, \ldots$ be positive integers, and $\beta>0$ a positive real number. Show that for every $n \geqslant 0$, if $\theta_{n}=\left[a_{0}, \ldots, a_{n}, \beta\right]$, then $\theta_{n}=\left(\beta p_{n}+p_{n-1}\right) /\left(\beta q_{n}+q_{n-1}\right)$, where $\left(p_{n}\right)$, $\left(q_{n}\right)(n \geqslant-1)$ are sequences of integers satisfying

$$\begin{aligned} &p_{0}=a_{0}, q_{0}=1, \quad p_{-1}=1, \quad q_{-1}=0 \quad \text { and } \\ &\left(\begin{array}{cc} p_{n} & p_{n-1} \\ q_{n} & q_{n-1} \end{array}\right)=\left(\begin{array}{cc} p_{n-1} & p_{n-2} \\ q_{n-1} & q_{n-2} \end{array}\right)\left(\begin{array}{cc} a_{n} & 1 \\ 1 & 0 \end{array}\right) \quad(n \geqslant 1) \end{aligned}$$

Show that $p_{n} q_{n-1}-p_{n-1} q_{n}=(-1)^{n-1}$, and that $\theta_{n}$ lies between $p_{n} / q_{n}$ and $p_{n-1} / q_{n-1}$.

(b) Show that if $\left[a_{0}, a_{1}, \ldots\right]$ is the continued fraction expansion of a positive irrational $\theta$, then $p_{n} / q_{n} \rightarrow \theta$ as $n \rightarrow \infty$.

(c) Let the convergents of the continued fraction $\left[a_{0}, a_{1}, \ldots, a_{n}\right]$ be $p_{j} / q_{j}(0 \leqslant$ $j \leqslant n)$. Using part (a) or otherwise, show that the $n$-th and $(n-1)$-th convergents of $\left[a_{n}, a_{n-1}, \ldots, a_{0}\right]$ are $p_{n} / p_{n-1}$ and $q_{n} / q_{n-1}$ respectively.

(d) Show that if $\theta=\left[\overline{a_{0}, a_{1}, \ldots, a_{n}}\right]$ is a purely periodic continued fraction with convergents $p_{j} / q_{j}$, then $f(\theta)=0$, where $f(X)=q_{n} X^{2}+\left(q_{n-1}-p_{n}\right) X-p_{n-1}$. Deduce that if $\theta^{\prime}$ is the other root of $f(X)$, then $-1 / \theta^{\prime}=\left[\overline{a_{n}, a_{n-1}, \ldots, a_{0}}\right]$.