now, we can always get check a sequence if it's geometric or arithmetic, by doing a quick division of two adjacent terms, or a quick difference.
the division of two adjacent terms will give use the common ratio if it's geometric.
the difference of two adjacent terms will give use the common difference if it's geometric.
[tex]\bf 4\frac{1}{2}~~,~~3\frac{7}{8}~~,~~3\frac{1}{4}~\hfill \implies ~\hfill \cfrac{9}{2}~~,~~\cfrac{31}{8}~~,~~\cfrac{13}{4} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \cfrac{31}{8}-\cfrac{9}{2}\implies \cfrac{31-36}{8}\implies \boxed{-\cfrac{5}{8}} ~\hfill \cfrac{13}{4}-\cfrac{31}{8}\implies \cfrac{26-31}{8}\implies \boxed{-\cfrac{5}{8}}[/tex]
so, low and behold, is an arithmetic sequence, whose common difference is -5/8, meaning to get the next term, we simply subtract that much from the current one.
[tex]\bf \cfrac{13}{4}-\cfrac{5}{8}\implies \cfrac{26-5}{8}\implies \cfrac{21}{8}\implies 2\frac{5}{8} \\\\[-0.35em] ~\dotfill\\\\ 4\frac{1}{2}~~,~~3\frac{7}{8}~~,~~3\frac{1}{4}~~,~~\stackrel{\downarrow }{2\frac{5}{8}}~~,~~2[/tex]
