  We show that properties $F_n$ and $FP_n$ hold for a relatively hyperbolic
group if and only if they hold for all the peripheral subgroups. As an
application we show that there are at least countably many distinct
quasi-isometry classes of one-ended groups that are type $F_n$ but not
$F_{n+1}$ and similarly of type $FP_n$ and not $FP_{n+1}$ for all positive
integers $n$.
