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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 +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 , … , 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 1·2 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. |
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"progression." The Columbia Encyclopedia, 6th ed.. 2011. Encyclopedia.com. 27 May. 2012 <http://www.encyclopedia.com>. "progression." The Columbia Encyclopedia, 6th ed.. 2011. Encyclopedia.com. (May 27, 2012). http://www.encyclopedia.com/doc/1E1-progrsn.html "progression." The Columbia Encyclopedia, 6th ed.. 2011. Retrieved May 27, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1E1-progrsn.html |
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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." The Oxford Pocket Dictionary of Current English. 2009. Encyclopedia.com. 27 May. 2012 <http://www.encyclopedia.com>. "progression." The Oxford Pocket Dictionary of Current English. 2009. Encyclopedia.com. (May 27, 2012). http://www.encyclopedia.com/doc/1O999-progression.html "progression." The Oxford Pocket Dictionary of Current English. 2009. Retrieved May 27, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O999-progression.html |
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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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MICHAEL KENNEDY and JOYCE BOURNE. "progression." The Concise Oxford Dictionary of Music. 1996. Encyclopedia.com. 27 May. 2012 <http://www.encyclopedia.com>. MICHAEL KENNEDY and JOYCE BOURNE. "progression." The Concise Oxford Dictionary of Music. 1996. Encyclopedia.com. (May 27, 2012). http://www.encyclopedia.com/doc/1O76-progression.html MICHAEL KENNEDY and JOYCE BOURNE. "progression." The Concise Oxford Dictionary of Music. 1996. Retrieved May 27, 2012 from Encyclopedia.com: http://www.encyclopedia.com/doc/1O76-progression.html |
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