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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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