1. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
\begin{array}{l}
a_{1}=1, a_{2}=\frac{1}{3}, \\
a_{n} a_{n-1}+a_{n} a_{n+1}=2 a_{n-1} a_{n+1}\left(n \geqslant 2, n \in \mathbf{Z}_{+}\right) .
\end{array}
$$

Then the general term formula of the sequence is $\qquad$ .
$$
\text { I, 1. } a_{n}=\frac{1}{2 n-1} \text {. }
$$

Dividing both sides of the conditional equation by $a_{n-1} a_{n} a_{n+1}$, we get
$$
\frac{1}{a_{n+1}}+\frac{1}{a_{n-1}}=\frac{2}{a_{n}} \text {. }
$$

Thus, $\left\{\frac{1}{a_{n}}\right\}$ is an arithmetic sequence.
It is easy to see that its general term is $2 n-1$, hence $a_{n}=\frac{1}{2 n-1}$.