Recognize arithmetic and geometric sequences from lists, tables, graphs, recursive rules, or explicit forms, distinguishing constant additive differences from constant multiplicative ratios. Determine common differences and ratios, find terms, and write explicit formulas using $a_n=a_1+(n-1)d$ and $a_n=a_1r^{n-1}$, including positive, negative, fractional, zero, and alternating patterns where applicable. Calculate finite arithmetic sums with $S_n=\frac{n}{2}(a_1+a_n)=\frac{n}{2}[2a_1+(n-1)d]$ and finite geometric sums with $S_n=\frac{a(1-r^n)}{1-r}$ for $r\ne1$ or $S_n=na$ for $r=1$, excluding infinite-series convergence and complex or advanced generalized cases.