4.030. Find the denominator $q$ of an infinite geometric progression $(|q|<1)$, where each term is four times the sum of all its subsequent terms.
## Solution.

From the condition, we have $b_{1}=4\left(s-b_{1}\right)$.

By formula (4.12), we get

$$
b_{1}=4\left(\frac{b_{1}}{1-q}-b_{1}\right), b_{1}=\frac{4 b_{1}(1-1+q)}{1-q}, 1-q=4 q, q=\frac{1}{5}
$$

Answer: $\frac{1}{5}$.