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Arithmetic progression

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Proof without words of the arithmetic progression formulas using a rotated copy of the blocks.
Proof without words of the arithmetic progression formulas using a rotated copy of the blocks. Image: source and license

An arithmetic progression, arithmetic sequence or linear sequence[1] is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression. 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 {\displaystyle a_{1 and the common difference of successive members is {\displaystyle d}, then the {\displaystyle n}-th term of the sequence ({\displaystyle a_{n) is given by {\displaystyle a_{n}=a_{1}+(n-1)d.}

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.

History#edit

According to an anecdote of uncertain reliability,[2] in primary school Carl Friedrich Gauss reinvented the formula {\displaystyle {\tfrac {n(n+1)}{2 for summing the integers from 1 through {\displaystyle n}, for the case {\displaystyle n=100}, by grouping the numbers from both ends of the sequence into pairs summing to 101 and multiplying by the number of pairs. Regardless of the truth of this story, Gauss was not the first to discover this formula. Similar rules were known in antiquity to Archimedes, Hypsicles and Diophantus;[3] in China to Zhang Qiujian; in India to Aryabhata, Brahmagupta and Bhaskara II;[4] and in medieval Europe to Alcuin,[5] Dicuil,[6] Fibonacci,[7] Sacrobosco,[8] and anonymous commentators of Talmud known as Tosafists.[9] Some find it likely that its origin goes back to the Pythagoreans in the 5th century BC.[10]

Sum#edit

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

{\displaystyle 2+5+8+11+14=40}

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:

{\displaystyle {\frac {n(a_{1}+a_{n})}{2

In the case above, this gives the equation:

{\displaystyle 2+5+8+11+14={\frac {5(2+14)}{2={\frac {5\times 16}{2=40.}

This formula works for any arithmetic progression of real numbers beginning with {\displaystyle a_{1 and ending with {\displaystyle a_{n. For example,

{\displaystyle \left(-{\frac {3}{2\right)+\left(-{\frac {1}{2\right)+{\frac {1}{2={\frac {3\left(-{\frac {3}{2+{\frac {1}{2\right)}{2=-{\frac {3}{2

Derivation#edit

Animated proof for the formula giving the sum of the first integers 1+2+...+n.
Animated proof for the formula giving the sum of the first integers 1+2+...+n. Image: source and license

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

{\displaystyle S_{n}=a+a_{2}+a_{3}+\dots +a_{(n-1)}+a_{n
{\displaystyle S_{n}=a+(a+d)+(a+2d)+\dots +(a+(n-2)d)+(a+(n-1)d).}

Rewriting the terms in reverse order: {\displaystyle S_{n}=(a+(n-1)d)+(a+(n-2)d)+\dots +(a+2d)+(a+d)+a.}

Adding the corresponding terms of both sides of the two equations and halving both sides: {\displaystyle S_{n}={\frac {n}{2[2a+(n-1)d].} This formula can be simplified as: {\displaystyle {\begin{aligned}S_{n}&={\frac {n}{2[a+a+(n-1)d].\\&={\frac {n}{2(a+a_{n}).\\&={\frac {n}{2({\text{initial term+{\text{last term).\end{aligne Furthermore, the mean value of the series can be calculated via: {\displaystyle S_{n}/n}: {\displaystyle {\overline {a={\frac {a_{1}+a_{n{2.}

The formula is essentially the same as the formula for the mean of a discrete uniform distribution, interpreting the arithmetic progression as a set of equally probable outcomes.

Product#edit

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 {\displaystyle {\begin{aligned}a_{1}a_{2}a_{3}\cdots a_{n}&=a_{1}(a_{1}+d)(a_{1}+2d)\cdots (a_{1}+(n-1)d)\\[1ex]&=\prod _{k=0}^{n-1}(a_{1}+kd)=d^& where {\displaystyle \Gamma } denotes the Gamma function. The formula is not valid when {\displaystyle a_{1}/d} is negative or zero.

This is a generalization of the facts that the product of the progression {\displaystyle 1\times 2\times \cdots \times n} is given by the factorial {\displaystyle n!} and that the product {\displaystyle m\times (m+1)\times (m+2)\times \cdots \times (n-2)\times (n-1)\times n} for positive integers {\displaystyle m} and {\displaystyle n} is given by {\displaystyle {\frac {n!}{(m-1)!.}

Derivation#edit

{{mathimg|1f92acd108ad06278aa02f12729ff46eb0cbf5a9|width=68.967ex|height=28.843ex|va=-13.61ex|alt={\displaystyle {\begin{aligned}a_{1}a_{2}a_{3}\cdots a_{n}&=\prod _{k=0}^{n-1}(a_{1}+kd)\\[2pt]&=\prod _{k=0}^{n-1}d\left({\frac {a_{1{d+k\right)\}} where {\displaystyle x^{\overline {n denotes the rising factorial.

By the recurrence formula {\displaystyle \Gamma (z+1)=z\Gamma (z)}, valid for a complex number {\displaystyle z>0},

{\displaystyle \Gamma (z+2)=(z+1)\Gamma (z+1)=(z+1)z\Gamma (z)},
{\displaystyle \Gamma (z+3)=(z+2)\Gamma (z+2)=(z+2)(z+1)z\Gamma (z)},

so that {\displaystyle {\frac {\Gamma (z+m)}{\Gamma (z)=\prod _{k=0}^{m-1}(z+k)} for {\displaystyle m} a positive integer and {\displaystyle z} a positive complex number.

Thus, if {\displaystyle a_{1}/d>0}, {\displaystyle \prod _{k=0}^{n-1}\left({\frac {a_{1{d+k\right)={\frac {\Gamma {\left({\frac {a_{1{d+n\right){\Gamma {\left({\frac {a_{1{d\right),} and, finally, {\displaystyle a_{1}a_{2}a_{3}\cdots a_{n}=d^{n}\prod _{k=0}^{n-1}\left({\frac {a_{1{d+k\right)=d^{n}{\frac {\Gamma {\left({\frac {a_{1{d+n\right)&#

Examples#edit

Example 1#edit

In the example {\displaystyle 3,8,13,18,23,28,\ldots }, the product of the terms of the arithmetic progression given by {\displaystyle a_{n}=3+5(n-1)} up to the 50th term is {\displaystyle P_{50}=5^{50}\cdot {\frac {\Gamma \left(3/5+50\right)}{\Gamma \left(3/5\right)\approx 3.78438\times 10^{98}.}

Example 2#edit

The product of the first 10 odd numbers {\displaystyle (1,3,5,7,9,11,13,15,17,19)} is given by {\displaystyle 1\cdot 3\cdot 5\cdots 19=\prod _{k=0}^{9}(1+2k)=2^{10}\cdot {\frac {\Gamma \left({\frac {1}{2+10\right)}{\Gamma \left({\frac {1}{2\right)=654,729,075}

Standard deviation#edit

The standard deviation of any arithmetic progression is {\displaystyle \sigma =|d|{\sqrt {\frac {(n-1)(n+1)}{12 where {\displaystyle n} is the number of terms in the progression and {\displaystyle d} is the common difference between terms. The formula is essentially the same as the formula for the standard deviation of a discrete uniform distribution, interpreting the arithmetic progression as a set of equally probable outcomes.

Intersections#edit

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.[11] However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression.

Amount of arithmetic subsets of length k of the set {1,...,n}#edit

Let {\displaystyle a(n,k)} denote the number of arithmetic subsets of length {\displaystyle k} one can make from the set {\displaystyle \{1,\cdots ,n\ and let {\displaystyle \phi (\eta ,\kappa )} be defined as:

{{mathimg|2b205747ab4840a5349dcd0852dfd3d8b5b630f6|width=55.682ex|height=6.176ex|va=-2.505ex|alt={\displaystyle \phi (\eta ,\kappa )={\begin{cases}0&{\text{if \kappa \mid \eta \\\left(\left[\eta \;({\text{mod \kappa )\right]-2\right)\left(\kappa -\left[\eta \;({\text{mod \kappa )\right]\right)&{\text{if \kappa \not }}

Then:

{{mathimg|39c408472bd20e2830d86efc2b5c8e923f51d215|width=63.32ex|height=12.176ex|va=-5.505ex|alt={\displaystyle {\begin{aligned}a(n,k)&={\frac {1}{2(k-1)\left(n^{2}-(k-1)n+(k-2)+\phi (n+1,k-1)\right)\\&={\frac {1}{2(k-1)\left((n-1)(n-(k-2))+\phi (n+1,k-1)\right)\end{aligned}}

As an example, if {\textstyle (n,k)=(7,3)}, one expects {\textstyle a(7,3)=9} arithmetic subsets and, counting directly, one sees that there are 9; these are {\textstyle \{1,2,3\},\{2,3,4\},\{3,4,5\},\{4,5,6\},\{5,6,7\},\{1,3,5\},\{3,5,7\},\{2,4,6\},\{1,4,7\}.}

See also#edit

References#edit

  1. ^ "Linear sequences". BBC Bitesize. Retrieved 2025-12-16.
  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. ^ Tropfke, Johannes (1924). Analysis, analytische Geometrie. Walter de Gruyter. pp. 3–15. ISBN 978-3-11-108062-8. {{cite book}}: ISBN / Date incompatibility (help)
  4. ^ Tropfke, Johannes (1979). Arithmetik und Algebra. Walter de Gruyter. pp. 344–354. ISBN 978-3-11-004893-3.
  5. ^ Problems to Sharpen the Young, John Hadley and David Singmaster, The Mathematical Gazette, 76, #475 (March 1992), pp. 102–126.
  6. ^ 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
  7. ^ Sigler, Laurence E. (trans.) (2002). Fibonacci's Liber Abaci. Springer-Verlag. pp. 259–260. ISBN 0-387-95419-8.
  8. ^ Katz, Victor J., ed. (2016). Sourcebook in the Mathematics of Medieval Europe and North Africa. Princeton University Press. pp. 91, 257. ISBN 9780691156859.
  9. ^ 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
  10. ^ 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
  11. ^ 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.
Categories Arithmetic seriesSequences and series

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