Example 13 In the sequence $\left\{a_{n}\right\}$, $a_{1}=a>2$, and $a_{n+1}$ $=\frac{a_{n}^{2}}{2\left(a_{n}-1\right)}, n=1,2, \cdots$. Find $a_{n}$.
Let $a_{n}=b_{n}+1$, then we have
$$
b_{n+1}=\frac{b_{n}^{2}+1}{2 b_{n}}=\frac{\left(b_{n}+1\right)^{2}+\left(b_{n}-1\right)^{2}}{\left(b_{n}+1\right)^{2}-\left(b_{n}-1\right)^{2}} \text {. }
$$

Using the conclusion of Theorem 3, we get
$$
b_{n}=\frac{\left(b_{1}+1\right)^{2^{n-1}}+\left(b_{1}-1\right)^{2^{n-1}}}{\left(b_{1}+1\right)^{2^{n-1}}-\left(b_{1}-1\right)^{2^{n-1}}} \text {. }
$$

Substituting $a_{n}=b_{n}+1$ into the equation, we get
$$
\begin{array}{l}
a_{n}=\frac{\left(b_{1}+1\right)^{2^{n-1}}+\left(b_{1}-1\right)^{2^{n-1}}}{\left(b_{1}+1\right)^{2^{n-1}}-\left(b_{1}-1\right)^{2^{n-1}}}+1 \\
=\frac{2\left(b_{1}+1\right)^{2^{n-1}}}{\left(b_{1}+1\right)^{2^{n-1}}-\left(b_{1}-1\right)^{2^{n-1}}} \\
=\frac{2 a^{2^{n-1}}}{a^{2^{n-1}}-(a-2)^{2^{n-1}}} \\
=\frac{2}{1-\left(1-\frac{2}{a}\right)^{2^{n-1}}} \quad(n \in \mathbf{N}) .
\end{array}
$$

Theorem 4: Let the function $f(x)=\frac{x^{2}+A}{2 x+B}$ have two distinct fixed points $x_{1}$ and $x_{2}$, and let the sequence $\left\{a_{n}\right\}$ be defined by $a_{n+1}=f\left(a_{n}\right)$. Then the sequence $\left\{\ln \frac{a_{n}-x_{1}}{a_{n}-x_{2}}\right\}$ is a geometric sequence with a common ratio of 2.
Note: This conclusion is actually a generalization of Theorem 3.
Proof: Since $x_{k} (k=1,2)$ are the two fixed points of $f(x)$, i.e., $x_{k}=\frac{x_{k}^{2}+A}{2 x_{k}+B}$, we have $A-B x_{k}=x_{k}^{2}$. Therefore,
$$
\begin{array}{l}
\frac{a_{n+1}-x_{1}}{a_{n+1}-x_{2}}=\frac{\frac{a_{n}^{2}+A}{2 a_{n}+B}-x_{1}}{\frac{a_{n}^{2}+A}{2 a_{n}+B}-x_{2}} \\
=\frac{a_{n}^{2}+A-x_{1}\left(2 a_{n}+B\right)}{a_{n}^{2}+A-x_{2}\left(2 a_{n}+B\right)} \\
=\frac{a_{n}^{2}-2 a_{n} x_{1}+A-x_{1} B}{a_{n}^{2}-2 a_{n} x_{2}+A-x_{2} B} \\
=\frac{a_{n}^{2}-2 a_{n} x_{1}+x_{1}^{2}}{a_{n}^{2}-2 a_{n} x_{2}+x_{2}^{2}}=\left(\frac{a_{n}-x_{1}}{a_{n}-x_{2}}\right)^{2} .
\end{array}
$$

Thus, the sequence $\left\{\ln \frac{a_{n}-x_{1}}{a_{n}-x_{2}}\right\}$ is a geometric sequence with a common ratio of 2.