9. Let $\left\{a_{n}\right\}$ be a sequence of positive terms in a geometric progression, and let $S_{n}=\lg a_{1}+\lg a_{2}+\cdots+\lg a_{n}, \forall n \in \mathbf{N}^{*}$. If there exist distinct positive integers $m 、 n$, such that $S_{n}=S_{m}$, find $S_{m+n}$.
9. Solution: Let $b_{n}=\lg a_{n}\left(n \in \mathbf{N}^{*}\right)$, then $\left\{b_{n}\right\}$ is an arithmetic sequence, with common difference $d$. Since $S_{n}=S_{m}$,
$\therefore n a_{1}+\frac{n(n-1) d}{2}=m a_{1}+\frac{m(m-1) d}{2}$, which means $(n-m)\left(a_{1}+\frac{n+m-1}{2} d\right)=0$,
or $a_{1}+\frac{m+n-1}{2} d=0$.
Therefore, $S_{m+n}=(m+n) a_{1}+\frac{(m+n)(m+n-1)}{2} d=(m+n)\left(a_{1}+\frac{m+n-1}{2} d\right)=0$.