Consecutive

Consecutive is an adjective meaning following continuously; in mathematics it describes numbers that follow one another in order without gaps. For example, 5, 6, 7 are consecutive integers, as are -3, -2, -1.

Consecutive integers differ by exactly 1, such as n, n+1, n+2. Consecutive can also apply to even or odd numbers: consecutive even integers are 4, 6, 8 (each step is 2), and consecutive odd integers are 3, 5, 7 (also stepping by 2).

Important properties arise from the idea of adjacency in a sequence. Among any m consecutive integers, exactly one is divisible by m. The product of k consecutive integers is divisible by k! (the factorial of k), a fact used in combinatorics and number theory.

Consecutive is also used in broader contexts to describe items arranged in a sequence or timeline with no gaps, such as consecutive days, pages, or sentences. In geometry, consecutive vertices of a polygon are adjacent along the edge sequence.

Etymology: the word derives from Latin consecutus meaning “following closely,” from con- meaning “together” and sequi meaning “to follow.” The term thus conveys the idea of items that come one right after another in a specified order.