progressziók

Progressziók is a term used across disciplines to describe ordered sequences that follow a rule. In mathematics, a progresszió (progression) is a sequence of numbers obtained by a fixed rule that relates consecutive terms. The most common types are aritmetikai progressziók (arithmetic progressions) and geometriai progressziók (geometric progressions).

In an arithmetic progression, the difference between successive terms is constant. If the first term is a1 and the common difference is d, the n-th term is a_n = a1 + (n−1)d. For example, 3, 7, 11, 15 has d = 4. The partial sums S_n can be expressed as S_n = n/2 [2a1 + (n−1)d].

In a geometric progression, each term is obtained by multiplying the previous term by a constant r. If the first term is a1, then the n-th term is a_n = a1 r^(n−1). For example, 6, 3, 1.5,... has r = 1/2. If |r| < 1, the sequence converges to 0. The finite sum up to n terms is S_n = a1 (1 − r^n)/(1 − r), and the infinite sum (when |r| < 1) is S = a1/(1 − r).

In music theory, progressziók refer to sequences of chords that establish and develop a tonal center. Common harmonic progressions use functional harmony, such as tonic (I), subdominant (IV), and dominant (V) functions, with cadences like V–I providing resolution. The circle of fifths guides many typical progressions, though styles differ in harmonic language and voice leading.

Overall, progressziók denote structured patterns of change, whether in numerical sequences or musical harmony, and are fundamental to analysis in their respective fields.