Arithmetic progression

An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2.

If the initial term of an arithmetic progression is and the common difference of successive members is d, then the nth term of the sequence () is given by:

,

and in general

.

A finite portion of an arithmetic progression is called a finite arithmetic progression and sometimes just called an arithmetic progression. The sum of a finite arithmetic progression is called an arithmetic series.

SumEdit

2 + 5 + 8 + 11 + 14 = 40
14 + 11 + 8 + 5 + 2 = 40

16 + 16 + 16 + 16 + 16 = 80

Computation of the sum 2 + 5 + 8 + 11 + 14. When the sequence is reversed and added to itself term by term, the resulting sequence has a single repeated value in it, equal to the sum of the first and last numbers (2 + 14 = 16). Thus 16 × 5 = 80 is twice the sum.

The sum of the members of a finite arithmetic progression is called an arithmetic series. For example, consider the sum:

 

This sum can be found quickly by taking the number n of terms being added (here 5), multiplying by the sum of the first and last number in the progression (here 2 + 14 = 16), and dividing by 2:

 

In the case above, this gives the equation:

 

This formula works for any real numbers   and  . For example:

 

DerivationEdit

 
Animated proof for the formula giving the sum of the first integers 1+2+...+n.

To derive the above formula, begin by expressing the arithmetic series in two different ways:

 
 

Adding both sides of the two equations, all terms involving d cancel:

 

Dividing both sides by 2 produces a common form of the equation:

 

An alternate form results from re-inserting the substitution:  :

 

Furthermore, the mean value of the series can be calculated via:  :

 

The formula is very similar to the mean of a discrete uniform distribution.

ProductEdit

The product of the members of a finite arithmetic progression with an initial element a1, common differences d, and n elements in total is determined in a closed expression

 

where   denotes the Gamma function. The formula is not valid when   is negative or zero.

This is a generalization from the fact that the product of the progression   is given by the factorial   and that the product

 

for positive integers   and   is given by

 

DerivationEdit

 

where   denotes the rising factorial.

By the recurrence formula  , valid for a complex number  ,

 ,
 ,

so that

 

for   a positive integer and   a positive complex number.

Thus, if  ,

 ,

and, finally,

 

ExamplesEdit

Example 1

Taking the example  , the product of the terms of the arithmetic progression given by   up to the 50th term is

 
Example 2

The product of the first 10 odd numbers   is given by

  = 654,729,075

Standard deviationEdit

The standard deviation of any arithmetic progression can be calculated as

 

where   is the number of terms in the progression and   is the common difference between terms. The formula is very similar to the standard deviation of a discrete uniform distribution.

IntersectionsEdit

The intersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which can be found using the Chinese remainder theorem. If each pair of progressions in a family of doubly infinite arithmetic progressions have a non-empty intersection, then there exists a number common to all of them; that is, infinite arithmetic progressions form a Helly family.[1] However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression.

HistoryEdit

According to an anecdote of uncertain reliability,[2] young Carl Friedrich Gauss in primary school reinvented this method to compute the sum of the integers from 1 through 100, by multiplying n/2 pairs of numbers in the sum by the values of each pair n + 1.[clarification needed] However, regardless of the truth of this story, Gauss was not the first to discover this formula, and some find it likely that its origin goes back to the Pythagoreans in the 5th century BC.[3] Similar rules were known in antiquity to Archimedes, Hypsicles and Diophantus;[4] in China to Zhang Qiujian; in India to Aryabhata, Brahmagupta and Bhaskara II;[5] and in medieval Europe to Alcuin,[6] Dicuil,[7] Fibonacci,[8] Sacrobosco[9] and to anonymous commentators of Talmud known as Tosafists.[10]

See alsoEdit

ReferencesEdit

  1. ^ Duchet, Pierre (1995), "Hypergraphs", in Graham, R. L.; Grötschel, M.; Lovász, L. (eds.), Handbook of combinatorics, Vol. 1, 2, Amsterdam: Elsevier, pp. 381–432, MR 1373663. See in particular Section 2.5, "Helly Property", pp. 393–394.
  2. ^ Hayes, Brian (2006). "Gauss's Day of Reckoning". American Scientist. 94 (3): 200. doi:10.1511/2006.59.200. Archived from the original on 12 January 2012. Retrieved 16 October 2020.
  3. ^ Høyrup, J. The “Unknown Heritage”: trace of a forgotten locus of mathematical sophistication. Arch. Hist. Exact Sci. 62, 613–654 (2008). https://doi.org/10.1007/s00407-008-0025-y
  4. ^ Tropfke, Johannes (1924). Analysis, analytische Geometrie. Walter de Gruyter. pp. 3–15. ISBN 978-3-11-108062-8.
  5. ^ Tropfke, Johannes (1979). Arithmetik und Algebra. Walter de Gruyter. pp. 344–354. ISBN 978-3-11-004893-3.
  6. ^ Problems to Sharpen the Young, John Hadley and David Singmaster, The Mathematical Gazette, 76, #475 (March 1992), pp. 102–126.
  7. ^ Ross, H.E. & Knott,B.I (2019) Dicuil (9th century) on triangular and square numbers, British Journal for the History of Mathematics, 34:2, 79-94, https://doi.org/10.1080/26375451.2019.1598687
  8. ^ Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. pp. 259–260. ISBN 0-387-95419-8.
  9. ^ Katz, Victor J. (edit.) (2016). Sourcebook in the Mathematics of Medieval Europe and North Africa. Princeton University Press. pp. 91, 257. ISBN 9780691156859.
  10. ^ Stern, M. (1990). 74.23 A Mediaeval Derivation of the Sum of an Arithmetic Progression. The Mathematical Gazette, 74(468), 157-159. doi:10.2307/3619368

External linksEdit