In three geometric sequences, the first terms form a geometric sequence with a common ratio of $q_{3}=2$; the common ratios of the three geometric sequences form an arithmetic sequence with a difference of $d=1$; the sum of the second terms of the three geometric sequences is 24; and the sum of the first three terms of the third geometric sequence is 84. Write down these three geometric sequences.
The first terms of the sequences: $a, 2a, 4a;$ the ratios: $q, q+1, q+2$. According to the problem,

\[
\begin{gathered}
a q + 2a(q+1) + 4a(q+2) = 24 \\
4a + 4a(q+2) + 4a(q+2)^2 = 84
\end{gathered}
\]

From these equations, \(a=1, a'=\frac{192}{31}; q=2, q'=-\frac{7}{8}\). The sought sequences are thus:

$1, 2, 4, \ldots ; 2, 6, 18, \ldots ; 4, 16, 64, \ldots$

$\frac{192}{31}, -\frac{168}{31}, \frac{147}{31}, \ldots ; \frac{384}{31}, \frac{48}{31}, \frac{6}{31}, \ldots ; \frac{768}{31}, \frac{684}{31}, \frac{972}{31}, \ldots$

(Mátyás Lindtner, Löcse.)

The problem was solved by: Bayer B., Benedek Zs., Bertrám I., Burján K., Czank K., Faith F., Filkorn J., Freibauer E., Hendel J., Krausz B., Krisztián Gy., Kürth A., Lukhaub Gy., Lupsa Gy., Messik G., Messik V., Oltay K., Perl Gy., Petrogalli G., Sasvári G., Sasvári J., Scharff J., Stromfeld F., Szabó J.