1. Given the sequence $\left\{a_{n}\right\}$ is an arithmetic sequence, $\left\{b_{n}\right\}$ is a geometric sequence, and $a_{1}+2 b_{1} 、 a_{3}+4 b_{3} 、 a_{5}+8 b_{5}$ form an arithmetic sequence. Then $\frac{b_{3} b_{7}}{b_{4}^{2}}=$ . $\qquad$
- 1. $\frac{1}{2}$.

From $a_{1} 、 a_{3} 、 a_{5}$ forming an arithmetic sequence, and $a_{1}+2 b_{1} 、 a_{3}+4 b_{3} 、 a_{5}+8 b_{5}$ forming an arithmetic sequence, we know that $2 b_{1} 、 4 b_{3} 、 8 b_{5}$ also form an arithmetic sequence.
Since $2 b_{1} 、 4 b_{3} 、 8 b_{5}$ form a geometric sequence, we have
$$
\begin{array}{l}
2 b_{1}=4 b_{3}=8 b_{5} \Rightarrow q^{2}=\frac{b_{3}}{b_{1}}=\frac{1}{2} \\
\Rightarrow \frac{b_{3} b_{7}}{b_{4}^{2}}=\frac{b_{5}^{2}}{b_{4}^{2}}=q^{2}=\frac{1}{2} .
\end{array}
$$