  Given two sets $\cA, \cB \subseteq \F_q$ of elements of the finite field
$\F_q$ of $q$ elements, we show that the productset $$ \cA\cB = \{ab | a \in
\cA, b \in\cB\} $$ contains an arithmetic progression of length $k \ge 3$
provided that $k<p$, where $p$ is the characteristic of $\F_q$, and $# \cA #
\cB \ge 3q^{2d-2/k}$. We also consider geometric progressions in a shifted
productset $\cA\cB +h$, for $f \in \F_q$, and obtain a similar result.
