  This paper is devoted to establish nontrivial effective lower bounds for the
least common multiple of consecutive terms of a sequence ${(u_n)}_{n \in
\mathbb{N}}$ whose general term has the form $u_n = r {[n]}_q + u_0$, where $q
, r$ are positive integers and $u_0$ is a non-negative integer such that
$\mathrm{gcd}(u_0 , r) = \mathrm{gcd}(u_1 , q) = 1$. For such a sequence, we
show that for all positive integer $n$, we have $\mathrm{lcm}\{u_1 , u_2 ,
\dots , u_n\} \geq c_1 \cdot c_2^n \cdot q^{\frac{n^2}{4}}$, where $c_1$ and
$c_2$ are positive constants depending only on $q , r$ and $u_0$. This can be
considered as a $q$-analog of the lower bounds already obtained by the author
(in 2005) and by Hong and Feng (in 2006) for the arithmetic progressions.
