6.25 Three numbers form a geometric progression. If the second number is increased by 2, the progression becomes arithmetic, and if the last number is then increased by 9, the progression becomes geometric again. Find these numbers.
6.25 According to the condition, the numbers $a, a q, a q^{2}$ form a geometric progression, the numbers $a, a q+2, a q^{2}$ form an arithmetic progression, and the numbers $a, a q+2, a q^{2}+9$ form a geometric progression again. Using formulas (6.4) and (6.8), we obtain the system of equations

$$
\left\{\begin{array}{l}
a q+2=\frac{a+a q^{2}}{2} \\
(a q+2)^{2}=a\left(a q^{2}+9\right)
\end{array}\right.
$$

from which we find two solutions: 1) $a=4, q=2$;
2) $a=\frac{4}{25}, q=-4$.

Answer: $4 ; 8 ; 16$ or $\frac{4}{25} ;-\frac{16}{25} ; \frac{64}{25}$.