We call a sequence of real numbers, $\{a_n\}_{n\geq1}$, an asymptotically arithmetic sequence, if its increment $a_{n+1}-a_{n}$ approaches a real number $d$, as $n\to\infty$. For each $p\in[-\infty,\infty]$, we compute the limit of the increment $H_p(a_1,\dots,a_n,a_{n+1})-H_p(a_1,\dots,a_n)$, of the $p$-Hölder mean sequence, $\{H_p(a_1,\dots,a_n)\}_{n\geq1}$, of an asymptotically arithmetic sequence $\{a_n\}_{n\geq1}$, with positive terms. Moreover, for $p\leq-1$, we not only show that this limit is $0$, but we also compute the rate with which the increment approaches zero.