Determine the sequence whose terms simultaneously form an arithmetic and geometric progression.
Let the sequence be

$$
a-d, \quad a, \quad a+d
$$

If these three terms also form a geometric sequence, then simultaneously

$$
\frac{a-d}{a}=\frac{a}{a+d}
$$

or

$$
a^{2}-d^{2}=a^{2}
$$

from which

$$
d=0
$$

Thus, the only sequence that satisfies the given conditions is one where all terms are equal. In such a sequence, the ratio of any two consecutive terms is 1, and their difference is 0. (Aladár Visnya, fóreálisk. VIII. o. t. Pécs).

The problem was also solved by: Bernát Friedmann, S.-A.-Ujhely; Béla Grünhut, Pécs; János Imre, Jr., Nyíregyháza; Elemér Meitner, Budapest; Lipót Weisz, Pécs.