I have a recurrence of the form $w(n) \geq p_n\cdot(w(n-1) - 1)$, where $p_n$ is the largest prime less than or equal to $n$. If $w(1) = a$ for some $a > 10$, how good of a lower bound on $w(n)$ can one obtain?
Bertrand's postulate says that $p > n/2$, and I know there are generalizations of his postulate that give better bounds...