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.

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