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, ar² , … , 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- 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.

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