progressions

Progressions are ordered lists of elements that follow a rule connecting successive terms. In mathematics, the most familiar progressions are arithmetic, geometric, and harmonic progressions. More generally, a progression is any sequence of numbers or objects where each term is determined by a fixed relationship to earlier terms. Progressions are studied for their structure, limits, and sums.

Arithmetic progression, or AP, has a constant difference d between successive terms. If the first term is a1, the nth term is a1 + (n-1)d, and the sum of the first n terms is S_n = n/2 [2a1 + (n-1)d].

Geometric progression, or GP, has a constant ratio r between successive terms. The nth term is a1 r^(n-1), and the sum of the first n terms is S_n = a1 (1 − r^n)/(1 − r) for r ≠ 1. If |r| < 1, the infinite sum converges to a1/(1 − r).

Harmonic progression, or HP, is defined by the reciprocals forming an arithmetic progression.

When a progression extends indefinitely, it yields a sequence. If a sequence is associated with a sum, it is called a series; convergence or divergence concerns whether the partial sums approach a finite limit. For a GP, the infinite sum converges when |r| < 1.

In music, a chord progression is a sequence of chords that establishes a key or mood. In other disciplines, progressions may refer to steps in a process or stages of a phenomenon, preserving the general sense of orderly advancement.