Progression

Progression is the act or process of moving forward, developing, or advancing along a path toward a goal. In mathematics the term describes a sequence of numbers that follows a regular rule. In music a progression refers to a sequence of chords or harmonies that creates musical direction and anticipation.

In mathematics, the two most familiar progressions are arithmetic and geometric. An arithmetic progression has a constant difference d between consecutive terms: a, a+d, a+2d, and so on. The nth term is a_n = a + (n−1)d, and the sum of the first n terms is S_n = n/2 [2a + (n−1)d]. A geometric progression has a constant ratio r: a, ar, ar^2, and so forth. The nth term is a_n = a r^{n−1}, and the partial sum is S_n = a(1 − r^n)/(1 − r) for r ≠ 1. Geometric progressions model exponential growth or decay and appear in finance and natural phenomena.

In music, a chord progression is used to establish tonality and movement. Common progressions resolve tensions through cadences, such as a I–IV–V–I sequence in diatonic harmony.

Beyond numbers and music, progression denotes stages in processes such as development, manufacturing, research, or disease progression, where events unfold toward a defined outcome. The term emphasizes forward movement, order, and eventual conclusion.