3.1. The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $577 a, \frac{2020 b}{7}, \frac{c}{7}$ is an arithmetic progression. Find the common ratio of the geometric progression.
Answer: 4039.

Solution. Let $b=a q, c=a q^{2}$. The properties of an arithmetic progression and the conditions of the problem lead to the equation $2 \cdot \frac{2020 a q}{7}=577 a+\frac{a q^{2}}{7} \Leftrightarrow q^{2}-4040 q+4039=0$, from which $q=1$ or $q=4039$. A decreasing geometric progression can only occur when $q=4039$ (for example, if $a=-1$).