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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+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

pro·gres·sion / prəˈgreshən/ • n. a movement or development toward a destination or a more advanced state, esp. gradually or in stages: the normal progression from junior to senior status| their mode of progression through the forest. ∎  a succession; a series: counting the twenty-four hours in a single progression from midnight. ∎  Mus. a passage or movement from one note or chord to another: a blues progression. ∎  Math. short for arithmetic progression, geometric progression, or harmonic progression. ∎  Astrol. a predictive technique in which the daily movement of the planets, starting from the day of birth, represents a year in the subject's life. DERIVATIVES: pro·gres·sion·al / -shənl/ adj.

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## progression

progression. The motion of one note to another note or one chord to another chord, in logical progression.

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