  For relatively prime positive integers $u_0$ and $r$, we consider the least
common multiple $L_n:=\mathrm{lcm}(u_0,u_1,\ldots, u_n)$ of the finite
arithmetic progression $\{u_k:=u_0+kr\}_{k=0}^n$. We derive new lower bounds on
$L_n$ which improve upon those obtained previously when either $u_0$ or $n$ is
large. When $r$ is prime, our best bound is sharp up to a factor of $n+1$ for
$u_0$ properly chosen, and is also nearly sharp as $n\to\infty$.
