Let the first term of a geometric progression be $a_{1}$ and its common ratio be $q$. Calculate the sum of the first $n$ terms of this progression, taking the first term as $a_{1}$, then $a_{2}$, $a_{3}$, etc. If these sums are denoted by $S_{1}, S_{2}, S_{3}, \ldots$, determine the sum

$$
S_{1}+S_{2}+S_{3}+\ldots \text { to infinity }
$$
$$
\begin{gathered}
S_{1}+S_{2}+S_{3}+\ldots \text { in infin. }=a_{1} \frac{q^{n}-1}{q-1}+a_{1} \frac{q^{n}-1}{q-1}+a_{1} q^{2} \frac{q^{n}-1}{q-1}+\ldots \text { in infin. }= \\
=a_{1} \frac{q^{n}-1}{q-1}\left(1+q+q^{2}+\ldots \text { in infin. }\right) .
\end{gathered}
$$

For $n=\infty$ then

$$
S_{1}+S_{2}+S_{3}+\ldots \text { in infin. }=a_{1} \cdot \frac{1}{1-q} \cdot \frac{1}{1-q}=\frac{a_{1}}{(1-q)^{2}}
$$

(Pilczer Pál, Kaposvár.)

Number of solutions: 35.