2. Convert the following rational fractions into continued fractions:
(i) $\frac{50}{13}$,
(ii) $-\frac{53}{25}$.
2. (i) Solution:
$$\begin{aligned}
\frac{50}{13} & =3+\frac{11}{13}=3+\frac{1}{\frac{13}{11}}=3+\frac{1}{1+\frac{2}{11}} \\
& =3+\frac{1}{1+\frac{1}{\frac{11}{2}}}=3+\frac{1}{1+\frac{1}{5+\frac{1}{2}}} \\
& =[3,1,5,2] .
\end{aligned}$$
(ii) Solution:
$$\begin{aligned}
-\frac{53}{25} & =-3+\frac{22}{25}=-3+\frac{1}{\frac{25}{22}} \\
& =-3+\frac{1}{1+\frac{3}{22}}=-3+\frac{1}{1+\frac{1}{\frac{22}{3}}} \\
& =-3+\frac{1}{1+\frac{1}{7+\frac{1}{3}}}=[-3,1,7,3]
\end{aligned}$$