Example 2. The sequence $\left\{a_{1}\right\} a_{1}=-\frac{1}{3}$, $a_{n}=\frac{3 a_{n-1}-1}{3-a_{n-1}}$, find its general term formula.
Solve $a_{n}=\frac{-\frac{1}{3}+a_{n-1}}{1-\frac{1}{3} \cdot a_{n-1}}=\frac{a_{1}+a_{n-1}}{1+a_{1} a_{n-1}}$. By property 3, we have
$$
\begin{aligned}
F\left(a_{n}\right) & =F\left(\frac{a_{1}+a_{n-1}}{1+a_{1} a_{n-1}}\right) \\
& =F\left(a_{1}\right) F\left(a_{n-1}\right) .
\end{aligned}
$$
$F\left(a_{1}\right)=\frac{1+\frac{1}{3}}{1-\frac{1}{3}}=2$ for the sequence, so $F\left(a_{n}\right)=F^{n}\left(a_{1}\right)=2^{n}$.

Therefore, $a_{n}=F\left(F\left(a_{n}\right)\right)=F\left(2^{n}\right)$ $=\frac{1-2^{n}}{1+2^{n}}, n=1,2,3 \cdots$.