Example 6 Express $\frac{225}{43}$ as a continued fraction.
$$\begin{aligned}
\frac{225}{43} & =5+\frac{10}{43}=5+\frac{1}{\frac{43}{10}}=5+\frac{1}{4+\frac{3}{10}}=5+\frac{1}{4+\frac{1}{\frac{10}{3}}} \\
& =5+\frac{1}{4+\frac{1}{3+\frac{1}{3}}}=[5,4,3,3] .
\end{aligned}$$

We have