# progression

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Related to progression: geometric progression

## progression,

in mathematics, sequence**sequence,**

in mathematics, ordered set of mathematical quantities called terms. A sequence is said to be known if a formula can be given for any particular term using the preceding terms or using its position in the sequence.

**.....**Click the link for more information. 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

**series,**

in mathematics, indicated sum of a sequence of terms. A series may be finite or infinite. A finite series contains a definite number of terms whose sum can be found by various methods. An infinite series is a sum of infinitely many terms, e.g.

**.....**Click the link for more information. 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 © 2013, Columbia University Press. Licensed from Columbia University Press. All rights reserved. www.cc.columbia.edu/cu/cup/

## Progression

Illustrated Dictionary of Architecture Copyright © 2012, 2002, 1998 by The McGraw-Hill Companies, Inc. All rights reserved

## progression

[prə′gresh·ən] (mathematics)

A sequence or series of mathematical objects or quantities, each entry determined from its predecessors by some algorithm.

(metallurgy)

The fixed dimension between adjacent stations in a progressive die and thus the precise distance the strip must advance between successive cycles of the press.

McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.

## progression

**1.**

*Maths*a sequence of numbers in which each term differs from the succeeding term by a constant relation

**2.**

*Music*movement, esp of a logical kind, from one note to the next (

**melodic progression**) or from one chord to the next (

**harmonic progression**)

**3.**

*Astrology*one of several calculations, based on the movement of the planets, from which it is supposed that one can find the expected developments in a person's birth chart and the probable trends of circumstances for a year in his life

Collins Discovery Encyclopedia, 1st edition © HarperCollins Publishers 2005

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