Skip to main content
Select Source:

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.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"progression." The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. 15 Aug. 2018 <http://www.encyclopedia.com>.

"progression." The Columbia Encyclopedia, 6th ed.. . Encyclopedia.com. (August 15, 2018). http://www.encyclopedia.com/reference/encyclopedias-almanacs-transcripts-and-maps/progression

"progression." The Columbia Encyclopedia, 6th ed.. . Retrieved August 15, 2018 from Encyclopedia.com: http://www.encyclopedia.com/reference/encyclopedias-almanacs-transcripts-and-maps/progression

Learn more about citation styles

Citation styles

Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

http://www.mla.org/style

The Chicago Manual of Style

http://www.chicagomanualofstyle.org/tools_citationguide.html

American Psychological Association

http://apastyle.apa.org/

Notes:
  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.

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.

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"progression." The Oxford Pocket Dictionary of Current English. . Encyclopedia.com. 15 Aug. 2018 <http://www.encyclopedia.com>.

"progression." The Oxford Pocket Dictionary of Current English. . Encyclopedia.com. (August 15, 2018). http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/progression

"progression." The Oxford Pocket Dictionary of Current English. . Retrieved August 15, 2018 from Encyclopedia.com: http://www.encyclopedia.com/humanities/dictionaries-thesauruses-pictures-and-press-releases/progression

Learn more about citation styles

Citation styles

Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

http://www.mla.org/style

The Chicago Manual of Style

http://www.chicagomanualofstyle.org/tools_citationguide.html

American Psychological Association

http://apastyle.apa.org/

Notes:
  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.

progression

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

Cite this article
Pick a style below, and copy the text for your bibliography.

  • MLA
  • Chicago
  • APA

"progression." The Concise Oxford Dictionary of Music. . Encyclopedia.com. 15 Aug. 2018 <http://www.encyclopedia.com>.

"progression." The Concise Oxford Dictionary of Music. . Encyclopedia.com. (August 15, 2018). http://www.encyclopedia.com/arts/dictionaries-thesauruses-pictures-and-press-releases/progression

"progression." The Concise Oxford Dictionary of Music. . Retrieved August 15, 2018 from Encyclopedia.com: http://www.encyclopedia.com/arts/dictionaries-thesauruses-pictures-and-press-releases/progression

Learn more about citation styles

Citation styles

Encyclopedia.com gives you the ability to cite reference entries and articles according to common styles from the Modern Language Association (MLA), The Chicago Manual of Style, and the American Psychological Association (APA).

Within the “Cite this article” tool, pick a style to see how all available information looks when formatted according to that style. Then, copy and paste the text into your bibliography or works cited list.

Because each style has its own formatting nuances that evolve over time and not all information is available for every reference entry or article, Encyclopedia.com cannot guarantee each citation it generates. Therefore, it’s best to use Encyclopedia.com citations as a starting point before checking the style against your school or publication’s requirements and the most-recent information available at these sites:

Modern Language Association

http://www.mla.org/style

The Chicago Manual of Style

http://www.chicagomanualofstyle.org/tools_citationguide.html

American Psychological Association

http://apastyle.apa.org/

Notes:
  • Most online reference entries and articles do not have page numbers. Therefore, that information is unavailable for most Encyclopedia.com content. However, the date of retrieval is often important. Refer to each style’s convention regarding the best way to format page numbers and retrieval dates.
  • In addition to the MLA, Chicago, and APA styles, your school, university, publication, or institution may have its own requirements for citations. Therefore, be sure to refer to those guidelines when editing your bibliography or works cited list.