1. Let $x, y, z$ be real numbers, $3 x, 4 y, 5 z$ form a geometric sequence, and $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ form an arithmetic sequence, then the value of $\frac{x}{z}+\frac{z}{x}$ is $\qquad$
$$
\frac{34}{15}
$$
1.【Analysis and Solution】From the problem, we have $\left\{\begin{array}{l}(4 y)^{2}=15 x z \\ \frac{2}{y}=\frac{1}{x}+\frac{1}{z}\end{array}\right.$

From (2), we get $y=\frac{2 x z}{x+z}$, substituting this into (1) gives
$$
\begin{array}{l}
16\left(\frac{2 x z}{x+z}\right)^{2}=15 x z, \\
\text { i.e., } \frac{(x+z)^{2}}{x z}=\frac{64}{15}, \text { thus } \frac{x}{z}+\frac{z}{x}=\frac{34}{15} .
\end{array}
$$