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progression, in mathematics, sequence of quantities, called terms, in which the relationship between consecutive terms is the same. An arithmetic progression is a sequence in which each term is derived from the preceding one by adding a given number, *d,* called the common difference. It has the general form *a,* *a*+*d,* *a*+2*d,* … , *a*+(*n*-1)*d, … ,* where *a* is some number and *a*+(*n*-1)*d* is the *n*th, or general, term; e.g., the progression 3, 7, 11, 15, … is arithmetic with *a*=3 and *d*=4. The value of the 20th term, i.e., when *n*=20, is found by using the general term: for *a*=3, *d*=4, and *n*=20, its value is 3+(20-1)4=79. An arithmetic series is the indicated sum of an arithmetic progression, and its sum of the first *n* terms is given by the formula [2*a*+(*n*-1)*d*]*n*/2; in the above example the arithmetic series is 3+7+11+15+… , and the sum of the first 5 terms, i.e., when *n*=5, is [2·3+(5-1)4] 5/2=55. A geometric progression is one in which each term is derived by multiplying the preceding term by a given number *r,* called the common ratio; it has the general form *a,* *ar,* *ar*^{2}, … , *ar*^{n-1}, … , where *a* and *n* have the same meanings as above; e.g., the progression 1, 2, 4, 8, … is geometric with *a*=1 and *r*=2. The value of the 10th term, i.e., when *n*=10, is given as 1·2^{10-1}=2^{9}=512. The sum of the geometric progression is given by the formula *a*(1-*r*^{n})/(1-*r*) for the first *n* terms. A harmonic progression is one in which the terms are the reciprocals of the terms of an arithmetic progression; it therefore has the general form 1/*a*, 1/(*a* + *d*), … , 1/[*a*+(*n*-1)*d*]. This type of progression has no general formula to express its sum.

The Columbia Electronic Encyclopedia Copyright © 2004.

Licensed from Columbia University Press

Licensed from Columbia University Press

Progression may refer to:## See also

In mathematics:

- Arithmetic progression, sequence of numbers such that the difference of any two successive members of the sequence is a constant
- Geometric progression, sequence of numbers such that the quotient of any two successive members of the sequence is a constant

In music:

- Chord progression, series of chords played in order
- Backdoor progression, the cadential chord progression from iv7 to I, or flat-VII7 to I in jazz music theory
- Omnibus progression, sequence of chords which effectively divides the octave into 4 equal parts
- Ragtime progression, chord progression typical of ragtime music and parlour music genres

In other fields:

- Age progression, the process of modifying a photograph of a person to represent the effect of aging on their appearance
- Cisternal progression, theory of protein transport through the Golgi apparatus inside a cell
- Color progression, ranges of color whose values transition smoothly through a hue, saturation, luminance, or any combination of the three
- Horizontal progression, the gradual movement from left to right during writing a line of text in Western handwriting
- Semantic progression, evolution of word usage
- Educational progression, an individual's movement through stages of education and/or training

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Last updated on Monday June 16, 2008 at 04:41:52 PDT (GMT -0700)

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Last updated on Monday June 16, 2008 at 04:41:52 PDT (GMT -0700)

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