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Arithmetic progression

From Wikipedia, the free encyclopedia

Sequence of equally spaced numbers

Proof without words of the arithmetic progression formulas using a rotated copy of the blocks.

Anarithmetic progression orarithmetic sequence is asequence ofnumbers 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 $a_{1}$ and the common difference of successive members is $d {\displaystyle d}$ , then the $n {\displaystyle n}$ -th term of the sequence ( $a_{n}$ ) is given by

a_{n}=a_{1}+(n-1)d.

A finite portion of an arithmetic progression is called afinite arithmetic progression and sometimes just called an arithmetic progression. Thesum of a finite arithmetic progression is called anarithmetic series.

History

According to an anecdote of uncertain reliability,^[1] in primary schoolCarl Friedrich Gauss reinvented the formula ${\tfrac {n(n+1)}{2}}$ for summing the integers from 1 through $n {\displaystyle n}$ , for the case $n=100$ , 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 toArchimedes,Hypsicles andDiophantus;^[2] in China toZhang Qiujian; in India toAryabhata,Brahmagupta andBhaskara II;^[3] and in medieval Europe toAlcuin,^[4]Dicuil,^[5]Fibonacci,^[6]Sacrobosco,^[7] and anonymous commentators ofTalmud known asTosafists.^[8] Some find it likely that its origin goes back to thePythagoreans in the 5th century BC.^[9]

Sum

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.

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

2+5+8+11+14=40

This sum can be found quickly by taking the numbern 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:

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

In the case above, this gives the equation:

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 $a_{1}$ and ending with $a_{n}$ . For example,

\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

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:

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

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

Rewriting the terms in reverse order:

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:

S_{n}={\frac {n}{2}}[2a+(n-1)d].

This formula can be simplified as:

{\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{aligned}}

Furthermore, the mean value of the series can be calculated via: $S_{n}/n$ :

{\overline {a}}={\frac {a_{1}+a_{n}}{2}}.

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

Product

Theproduct of the members of a finite arithmetic progression with an initial elementa₁, common differencesd, andn elements in total is determined in a closed expression

a_{1}a_{2}a_{3}\cdots a_{n}=a_{1}(a_{1}+d)(a_{1}+2d)...(a_{1}+(n-1)d)=\prod _{k=0}^{n-1}(a_{1}+kd)=d^{n}{\frac {\Gamma \left({\frac {a_{1}}{d}}+n\right)}{\Gamma \left({\frac {a_{1}}{d}}\right)}}

where $\Gamma$ denotes theGamma function. The formula is not valid when $a_{1}/d$ is negative or zero.

This is a generalization of the facts that the product of the progression $1\times 2\times \cdots \times n$ is given by thefactorial $n! {\displaystyle n!}$ and that the product

m\times (m+1)\times (m+2)\times \cdots \times (n-2)\times (n-1)\times n

forpositive integers $m {\displaystyle m}$ and $n {\displaystyle n}$ is given by

{\frac {n!}{(m-1)!}}.

Derivation

{\begin{aligned}a_{1}a_{2}a_{3}\cdots a_{n}&=\prod _{k=0}^{n-1}(a_{1}+kd)\\&=\prod _{k=0}^{n-1}d\left({\frac {a_{1}}{d}}+k\right)=d\left({\frac {a_{1}}{d}}\right)d\left({\frac {a_{1}}{d}}+1\right)d\left({\frac {a_{1}}{d}}+2\right)\cdots d\left({\frac {a_{1}}{d}}+(n-1)\right)\\&=d^{n}\prod _{k=0}^{n-1}\left({\frac {a_{1}}{d}}+k\right)=d^{n}{\left({\frac {a_{1}}{d}}\right)}^{\overline {n}}\end{aligned}}

where $x^{\overline {n}}$ denotes therising factorial.

By the recurrence formula $\Gamma (z+1)=z\Gamma (z)$ , valid for acomplex number $z>0$ ,

\Gamma (z+2)=(z+1)\Gamma (z+1)=(z+1)z\Gamma (z)

,

\Gamma (z+3)=(z+2)\Gamma (z+2)=(z+2)(z+1)z\Gamma (z)

,

so that

{\frac {\Gamma (z+m)}{\Gamma (z)}}=\prod _{k=0}^{m-1}(z+k)

for $m {\displaystyle m}$ a positive integer and $z {\displaystyle z}$ a positive complex number.

Thus, if $a_{1}/d>0$ ,

\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,

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)}{\Gamma \left({\frac {a_{1}}{d}}\right)}}

Examples

Example 1

Taking the example $3,8,13,18,23,28,\ldots$ , the product of the terms of the arithmetic progression given by $a_{n}=3+5(n-1)$ up to the 50th term is

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

The product of the first 10 odd numbers $(1,3,5,7,9,11,13,15,17,19)$ is given by

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

The standard deviation of any arithmetic progression is

\sigma =|d|{\sqrt {\frac {(n-1)(n+1)}{12}}}

where $n {\displaystyle n}$ is the number of terms in the progression and $d {\displaystyle d}$ is the common difference between terms. The formula is essentially the same as the formula for the standard deviation of adiscrete uniform distribution, interpreting the arithmetic progression as a set of equally probable outcomes.

Intersections

Theintersection of any two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which can be found using theChinese 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 aHelly family.^[10] 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 lengthk of the set {1,...,n}

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

$\phi (\eta ,\kappa )={\begin{cases}0&{\text{if }}\kappa \mid \eta \\\left(\left[\eta \;({\text{mod }}\kappa )\right]-2\right)\left(\kappa -\left[\eta \;({\text{mod }}\kappa )\right]\right)&{\text{if }}\kappa \not \mid \eta \\\end{cases}}$

Then:

${\begin{aligned}a(n,k)&={\frac {1}{2(k-1)}}\left(n^{2}-(k-1)n+(k-2)+\phi (n+1,k-1)\right)\\&={\frac {1}{2(k-1)}}\left((n-1)(n-(k-2))+\phi (n+1,k-1)\right)\end{aligned}}$

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

Geometric progression
Harmonic progression
Triangular number
Arithmetico-geometric sequence
Inequality of arithmetic and geometric means
Primes in arithmetic progression
Linear difference equation
Generalized arithmetic progression, a set of integers constructed as an arithmetic progression is, but allowing several possible differences
Heronian triangles with sides in arithmetic progression
Problems involving arithmetic progressions
Utonality
Polynomials calculating sums of powers of arithmetic progressions

References

^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. Retrieved16 October 2020.
^Tropfke, Johannes (1924).Analysis, analytische Geometrie. Walter de Gruyter. pp. 3–15.ISBN 978-3-11-108062-8.
^Tropfke, Johannes (1979).Arithmetik und Algebra. Walter de Gruyter. pp. 344–354.ISBN 978-3-11-004893-3.
^Problems to Sharpen the Young, John Hadley and David Singmaster,The Mathematical Gazette,76, #475 (March 1992), pp. 102–126.
^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
^Sigler, Laurence E. (trans.) (2002).Fibonacci's Liber Abaci. Springer-Verlag. pp. 259–260.ISBN 0-387-95419-8.
^Katz, Victor J. (edit.) (2016).Sourcebook in the Mathematics of Medieval Europe and North Africa. Princeton University Press. pp. 91, 257.ISBN 9780691156859.
^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
^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
^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.

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Sequences andseries

Integer sequences

Basic	Arithmetic progression Geometric progression Harmonic progression Square number Cubic number Factorial Powers of two Powers of three Powers of 10
Advanced(list)	Complete sequence Fibonacci sequence Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number array

Fibonacci spiral with square sizes up to 34.

Properties of sequences

Properties of series

Series	Alternating Convergent Divergent Telescoping
Convergence	Absolute Conditional Uniform

Explicit series

Convergent	1/2 − 1/4 + 1/8 − 1/16 + ⋯ 1/2 + 1/4 + 1/8 + 1/16 + ⋯ 1/4 + 1/16 + 1/64 + 1/256 + ⋯ 1 + 1/2^s + 1/3^s + ... (Riemann zeta function)
Divergent	1 + 1 + 1 + 1 + ⋯ 1 − 1 + 1 − 1 + ⋯ (Grandi's series) 1 + 2 + 3 + 4 + ⋯ 1 − 2 + 3 − 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 2 + 4 − 8 + ⋯ Infinite arithmetic series 1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials) 1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series) 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)

Kinds of series

Hypergeometric series

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