1. Positive real numbers $a, b, c$ satisfy $\ln (a b), \ln (a c), \ln (b c)$ form an arithmetic sequence, and $4(a+c)=17 b$, then all possible values of $\frac{c}{a}$ are $\qquad$ .
The answer (provided by Wang Zheng) is 16 or $\frac{1}{16}$.
From $\ln (a b), \ln (a c), \ln (b c)$ forming an arithmetic sequence, it follows that $a b, b c, c a$ form a geometric sequence, thus $a, b, c$ form a geometric sequence. Let $a=k b$, then $c=\frac{b}{k}$. Substituting this in gives $k+\frac{1}{k}=\frac{17}{4}$, solving which yields $k=4$ or $\frac{1}{4}$. Therefore, $\frac{c}{a}=k^{2}=16$ or $\frac{1}{16}$.