Let $K$ be a \reftext{def:field-c54-2026b}{field}, let $n$ be a \reftext{def:natural-numbers-2026a}{natural number} with successor map $S$ as in that definition, and let there be assigned to each natural number $k$ with $1\le k\le n$ an element $a_{k}$ of $K$.

The \textbf{finite sum} $\sum_{k=1}^{n}a_{k}$ is defined recursively by

the addition being that of $K$.