# progression

**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,* 6th ed. Copyright © 2012, Columbia University Press. All rights reserved.

**See more Encyclopedia articles on: **Mathematics