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+2d, … , a+(n−1)d, … , where a is some number and a+(n−1)d is the nth, 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 [2a+(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, ar², … , arⁿ⁻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−rⁿ)/(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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