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Let $a, Q\in\Q$ be given and consider the set $\cal{G}(a,Q)=\{aQ^{i}:\;i\in\N\}$ of terms of geometric progression with 0th term equalto $a$ and the quotient $Q$. Let $f\in\Q(x, y)$ and $\cal{V}_{f}$ be the set offinite values of $f$. We consider the problem of existence of $a, Q\in\Q$ suchthat $\cal{G}(a, Q)\subset\cal{V}_{f}$. In the first part of the paper wedescribe several classes of rational function for which our problem has apositive solution. In particular, if $f(x,y)=\frac{f_{1}(x,y)}{f_{2}(x,y)}$,where $f_{1}, f_{2}\in\Z[x,y]$ are homogenous forms of degrees $d_{1}, d_{2}$and $|d_{1}-d_{2}|=1$, we prove that $\cal{G}(a, Q)\subset \cal{V}_{f}$ if andonly if there are $u, v\in\Q$ such that $a=f(u, v)$. In the second,experimental, part of the paper we study the stated problem for the rationalfunction $f(x, y)=(y^2-x^3)/x$. We relate the problem to the existence ofrational points on certain elliptic curves and present interesting numericalobservations which allow us to state several questions and conjectures.

Geometric progressions in the sets of values of rational functions