64. The general term formula of the sequence $\left\{a_{n}\right\}$ is
$$
a_{n}=\frac{\cos 3^{n} x+3 \cos 3^{n-1} x}{3^{n-1} \sin 3^{n} x} \text {. Find } S_{n}
$$
$$
\text { Sol: } \begin{array}{l} 
\frac{2(\cos 3 \alpha+3 \cos \alpha)}{\sin 3 \alpha} \\
=\frac{3(2 \cos \alpha+\cos 3 \alpha)-\cos 3 \alpha}{\sin 3 \alpha} \\
=\frac{3\left(4 \cos ^{2} \alpha-1\right) \cos \alpha}{\sin \alpha\left(3 \sin ^{2} \alpha\right)}-\operatorname{ctg} 3 \alpha \\
=3 \operatorname{ctg} \alpha-\operatorname{ctg} 3 \alpha .
\end{array}
$$

Let $a=x, 3x, 3^{2} x, \cdots, 3^{n-1} x$, then

Therefore, $S_{n}=\frac{3 \operatorname{ctg} x}{2} \cdots \frac{\operatorname{ctg}^{n} x}{2 \cdot 3^{n-1}}$.
(From Baipu High School, Rugao City, Jiangsu Province, 226511)
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