# 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 1·210 - 1 = 29 = 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.

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