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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progression

1. Maths a sequence of numbers in which each term differs from the succeeding term by a constant relation

2. Music movement, esp of a logical kind, from one note to the next (melodic progression) or from one chord to the next (harmonic progression)

3. Astrology one of several calculations, based on the movement of the planets, from which it is supposed that one can find the expected developments in a person's birth chart and the probable trends of circumstances for a year in his life

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progression [prə′gresh·ən]

(mathematics)

A sequence or series of mathematical objects or quantities, each entry determined from its predecessors by some algorithm.

(metallurgy)

The fixed dimension between adjacent stations in a progressive die and thus the precise distance the strip must advance between successive cycles of the press.