  This paper presents the log-concavity of the $m$-gonal figurate number
sequences. The author gives and proves the recurrence formula for $m$-gonal
figurate number sequences and its corresponding quotient sequences which are
found to be bounded. Finally, the author also show that for $m\geq 3$, the
sequence $\big \{S_n(m)\big\}_{n\geq 1}$ of $m$-gonal figurate numbers is a
log-concave.
