progresiei

Progresiei is the Romanian term used to refer to progression in mathematics, describing a sequence of numbers that follows a rule relating consecutive terms. In Romanian usage, the most common types are arithmetic progression (progresie aritmetică), geometric progression (progresie geometrică) and harmonic progression (progresie armonică).

An arithmetic progression is defined by a constant difference d between consecutive terms. If the first term is a1, the nth term is a_n = a1 + (n−1)d, and the partial sum of the first n terms is S_n = n/2 [2a1 + (n−1)d].

A geometric progression has a constant ratio r between consecutive terms. With first term a1, the nth term is a_n = a1 r^{n−1}, and the finite sum is S_n = a1 (1 − r^n)/(1 − r) for r ≠ 1. If |r| < 1, the infinite sum is S = a1/(1 − r).

A harmonic progression consists of terms whose reciprocals form an arithmetic progression; equivalently, a_n = 1/b_n with b_n an arithmetic progression.

Convergence and applications: Sequences and series built from progresii are central in analysis, with applications in finance (annuity calculations), physics, computer science, and numerical methods. Progresii also appear in music theory as harmonic progressions.

See also: progresie aritmetică, progresie geometrică, progresie armonică.