Everincreasing

Everincreasing, often written as ever-increasing, is an adjective used to describe something that continually grows or rises over time. It is common in both technical writing and journalism to emphasize trends that do not plateau, implying persistent growth or escalation.

In mathematics, an ever-increasing (or strictly increasing) sequence satisfies a_{n+1} > a_n for every index n. This is distinct from a non-decreasing sequence, where a_{n+1} ≥ a_n but equality is allowed. An ever-increasing sequence can be unbounded, such as a_n = 2^n, or bounded above, such as a_n = 1 − 1/n, which is strictly increasing and converges to 1. The term can also be applied to functions on an interval, where a function is ever-increasing if f(x1) < f(x2) whenever x1 < x2.

In broader usage, the phrase describes trends and quantities that keep rising, including prices, demand, costs, or complexity. Depending on context, it can carry evaluative overtones—either neutral emphasis on growth or a negative connotation when rapid increases are viewed unfavorably.

Etymologically, ever comes from a sense of continuity or perpetuity, paired with increasing to convey uninterrupted growth. In formal writing, the compound is typically hyphenated as ever-increasing; in some contexts it may function as a compound adjective or as an adverbial emphasis.

See also: monotone increasing, strictly increasing, non-decreasing, unbounded growth.