progresi

Progresi is a term found in several South Slavic languages, such as Croatian, Bosnian, and Serbian, meaning progressions or progress. It can function as the plural form of progres in those languages and may appear as a loanword in other contexts. In general use, progresi can refer to sequences, steps, or developments in mathematics, music, or other domains, depending on the context.

In mathematics, progressions are specific types of sequences with a constant relationship between consecutive terms. An arithmetic progression (AP) has a constant difference d between terms, with nth term a_n = a_1 + (n−1)d and sum S_n = n/2 [2a_1 + (n−1)d]. A geometric progression (GP) has a constant ratio r, with a_n = a_1 r^{n−1} and sum S_n = a_1(1 − r^n)/(1 − r) for r ≠ 1. These structures appear in problems of counting, averaging, and modeling growth or decline.

In music theory, progressions (progresi in some languages) refer to chord progressions—sequences of chords that establish harmonic movement. Common diatonic progressions include I–IV–V–I and ii–V–I, which give a sense of resolution. The plural progresi may be used when discussing multiple progressions.

Other uses of progresi can include general references to stages or steps in a process or to the progression of a condition or event. The precise meaning depends on language and domain.