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 $\qquad$
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) yields
$$
16\left(\frac{2 x z}{x+z}\right)^{2}=15 x z
$$

Thus, $\frac{(x+z)^{2}}{x z}=\frac{64}{15}$. Therefore, $\frac{x}{z}+\frac{z}{x}=\frac{34}{15}$.