Arithmetic progression - wikipedia Arithmetic progression In mathematics, an arithmetic progression ( AP ) 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 common difference of 2. If the initial term of an arithmetic progression is a 1 ( displaystyle a_ ( 1 ) ) and the common difference of successive members is d, then the nth term of the sequence ( a n ( displaystyle a_ ( n ) ) ) is given by : a n = a 1 + ( n − 1 ) d ( displaystyle a_ ( n ) = a_ ( 1 ) + ( n - 1 ) d ), and in general a n = a m + ( n − m ) d ( displaystyle a_ ( n ) = a_ ( m ) + ( n-m ) 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. The behavior of the arithmetic progression depends on the common difference d. If the common difference is : positive, then the members ( terms ) will grow towards positive infinity ; negative, then the members ( terms ) will grow towards negative infinity. Contents 1 Sum 1.1 Derivation 2 Product 3 Standard deviation 4 Intersections 5 Summary of formulas 6 See also 7 References 8 External links Sum This section is about Finite arithmetic series. For Infinite arithmetic series, see Infinite arithmetic series. + 5 + 8 + 11 + 14 = 40 14 + 11 + 8 + 5 + = 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 : 2 + 5 + 8 + 11 + 14 ( displaystyle 2 + 5 + 8 + 11 + 14 ) 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 : n ( a 1 + a n ) 2 ( displaystyle ( \frac ( n ( a_ ( 1 ) + a_ ( n ) ) ) ( 2 ) ) ) In the case above, this gives the equation : 2 + 5 + 8 + 11 + 14 = 5 ( 2 + 14 ) 2 = 5 × 16 2 = 40. ( displaystyle 2 + 5 + 8 + 11 + 14 = ( \frac ( 5 ( 2 + 14 ) ) ( 2 ) ) = ( \frac ( 5 \times 16 ) ( 2 ) ) = 40. ) This formula works for any real numbers a 1 ( displaystyle a_ ( 1 ) ) and a n ( displaystyle a_ ( n ) ). For example : ( − 3 2 ) + ( − 1 2 ) + 1 2 = 3 ( − 3 2 + 1 2 ) 2 = − 3 2. ( 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 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 1 + ( a 1 + d ) + ( a 1 + 2 d ) + ⋯ + ( a 1 + ( n − 2 ) d ) + ( a 1 + ( n − 1 ) d ) ( displaystyle S_ ( n ) = a_ ( 1 ) + ( a_ ( 1 ) + d ) + ( a_ ( 1 ) + 2d ) + cdots + ( a_ ( 1 ) + ( n - 2 ) d ) + ( a_ ( 1 ) + ( n - 1 ) d ) ) S n = ( a n − ( n − 1 ) d ) + ( a n − ( n − 2 ) d ) + ⋯ + ( a n − 2 d ) + ( a n − d ) + a n. ( displaystyle S_ ( n ) = ( a_ ( n ) - ( n - 1 ) d ) + ( a_ ( n ) - ( n - 2 ) d ) + cdots + ( a_ ( n ) - 2d ) + ( a_ ( n ) - d ) + a_ ( n ). ) Adding both sides of the two equations, all terms involving d cancel : 2 S n = n ( a 1 + a n ). ( displaystyle 2S_ ( n ) = n ( a_ ( 1 ) + a_ ( n ) ). ) Dividing both sides by 2 produces a common form of the equation : S n = n 2 ( a 1 + a n ). ( displaystyle S_ ( n ) = ( \frac ( n ) ( 2 ) ) ( a_ ( 1 ) + a_ ( n ) ). ) An alternate form results from re-inserting the substitution : a n = a 1 + ( n − 1 ) d ( displaystyle a_ ( n ) = a_ ( 1 ) + ( n - 1 ) d ) : S n = n 2 ( 2 a 1 + ( n − 1 ) d ). ( displaystyle S_ ( n ) = ( \frac ( n ) ( 2 ) ) ( 2a_ ( 1 ) + ( n - 1 ) d ). ) Furthermore, the mean value of the series can be calculated via : S n / n ( displaystyle S_ ( n ) / n ) : n  ̄ = a 1 + a n 2. ( displaystyle ( overline ( n ) ) = ( \frac ( a_ ( 1 ) + a_ ( n ) ) ( 2 ) ). ) In 499 AD Aryabhata, a prominent mathematician - astronomer from the classical age of Indian mathematics and Indian astronomy, gave this method in the Aryabhatiya ( section 2.18 ). Product The product of the members of a finite arithmetic progression with an initial element a, common differences d, and n elements in total is determined in a closed expression a 1 a 2 ⋯ a n = d a 1 d d ( a 1 d + 1 ) d ( a 1 d + 2 ) ⋯ d ( a 1 d + n − 1 ) = d n ( a 1 d ) n  ̄ = d n Γ ( a 1 / d + n ) Γ ( a 1 / d ), ( displaystyle a_ ( 1 ) a_ ( 2 ) cdots a_ ( n ) = d ( \frac ( a_ ( 1 ) ) ( d ) ) 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 ) ( left ( ( \frac ( a_ ( 1 ) ) ( d ) ) \right ) ) ^ ( overline ( n ) ) = d ^ ( n ) ( \frac ( Gamma left ( a_ ( 1 ) / d + n \right ) ) ( Gamma left ( a_ ( 1 ) / d \right ) ) ), ) where x n  ̄ ( displaystyle x ^ ( overline ( n ) ) ) denotes the rising factorial and Γ ( displaystyle Gamma ) denotes the Gamma function. ( Note however that the formula is not valid when a 1 / d ( displaystyle a_ ( 1 ) / d ) is a negative integer or zero. ) This is a generalization from the fact that the product of the progression 1 × 2 × ⋯ × n ( displaystyle 1 \times 2 \times cdots \times n ) is given by the factorial n ! ( displaystyle n ! ) and that the product m × ( m + 1 ) × ( m + 2 ) × ⋯ × ( n − 2 ) × ( n − 1 ) × n ( displaystyle m \times ( m + 1 ) \times ( m + 2 ) \times cdots \times ( n - 2 ) \times ( n - 1 ) \times n ) for positive integers m ( displaystyle m ) and n ( displaystyle n ) is given by n ! ( m − 1 ) !. ( displaystyle ( \frac ( n ! ) ( ( m - 1 ) ! ) ). ) Taking the example from above, the product of the terms of the arithmetic progression given by a = 3 + ( n - 1 ) ( 5 ) up to the 50th term is P 50 = 5 50 ⋅ Γ ( 3 / 5 + 50 ) Γ ( 3 / 5 ) ≈ 3.78438 × 10 98. ( displaystyle P_ ( 50 ) = 5 ^ ( 50 ) cdot ( \frac ( Gamma left ( 3 / 5 + 50 \right ) ) ( Gamma left ( 3 / 5 \right ) ) ) approx 3.78438 \times 10 ^ ( 98 ). ) Standard deviation The standard deviation of any arithmetic progression can be calculated as σ = d ( n − 1 ) ( n + 1 ) 12 ( displaystyle 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. Intersections 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. However, the intersection of infinitely many infinite arithmetic progressions might be a single number rather than itself being an infinite progression. Summary of formulas If a 1 ( displaystyle a_ ( 1 ) ) is the first term of an arithmetic progression. a n ( displaystyle a_ ( n ) ) is the nth term of an arithmetic progression. d ( displaystyle d ) is the difference between terms of the arithmetic progression. n ( displaystyle n ) is the number of terms in the arithmetic progression. S n ( displaystyle S_ ( n ) ) is the sum of n terms in the arithmetic progression. n  ̄ ( displaystyle ( overline ( n ) ) ) is the mean value of arithmetic series. then 1. a n = a 1 + ( n − 1 ) d, ( displaystyle a_ ( n ) = a_ ( 1 ) + ( n - 1 ) d, ) 2. a n = a m + ( n − m ) d. ( displaystyle a_ ( n ) = a_ ( m ) + ( n-m ) d. ) 3. S n = n 2 ( 2 a 1 + ( n − 1 ) d ). ( displaystyle S_ ( n ) = ( \frac ( n ) ( 2 ) ) ( 2a_ ( 1 ) + ( n - 1 ) d ). ) 4. S n = n 2 ( a 1 + a n ). ( displaystyle S_ ( n ) = ( \frac ( n ) ( 2 ) ) ( a_ ( 1 ) + a_ ( n ) ). ) 5. n  ̄ ( displaystyle ( overline ( n ) ) ) = S n / n ( displaystyle S_ ( n ) / n ) 6. n  ̄ = a 1 + a n 2. ( displaystyle ( overline ( n ) ) = ( \frac ( a_ ( 1 ) + a_ ( n ) ) ( 2 ) ). ) See also Linear difference equation Arithmetico - geometric sequence Generalized arithmetic progression - is a set of integers constructed as an arithmetic progression is, but allowing several possible differences. Harmonic progression Heronian triangles with sides in arithmetic progression Problems involving arithmetic progressions Utonality References Jump up ^ Duchet, Pierre ( 1995 ), `` Hypergraphs '', in Graham, R.L. ; Grötschel, M. ; Lovász, L., Handbook of combinatorics, Vol. 1, 2, Amsterdam : Elsevier, pp. 381 -- 432, MR 1373663. See in particular Section 2.5, `` Helly Property '', pp. 393 -- 394. Sigler, Laurence E. ( trans. ) ( 2002 ). Fibonacci 's Liber Abaci. Springer - Verlag. pp. 259 -- 260. ISBN 0 - 387 - 95419 - 8. External links Hazewinkel, Michiel, ed. ( 2001 ) ( 1994 ), `` Arithmetic series '', Encyclopedia of Mathematics, Springer Science + Business Media B.V. / Kluwer Academic Publishers, ISBN 978 - 1 - 55608 - 010 - 4 Weisstein, Eric W. `` Arithmetic progression ''. MathWorld. Weisstein, Eric W. `` Arithmetic series ''. MathWorld. Sequences and series Integer sequences Basic Arithmetic progression Geometric progression Square number Cubic number Factorial Powers of two Powers of 10 Advanced Complete sequence Fibonacci numbers Figurate number Heptagonal number Hexagonal number Lucas number Pell number Pentagonal number Polygonal number Triangular number Full list Properties of sequences Cauchy sequence Monotone sequence Periodic sequence Properties of series Convergent series Divergent series Conditional convergence Absolute convergence Uniform convergence Alternating series Telescoping series 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 / 1 + 1 / 2 + 1 / 3 +... ( Riemann zeta function ) Divergent 1 + 1 + 1 + 1 + ⋯ 1 + 2 + 3 + 4 + ⋯ 1 + 2 + 4 + 8 + ⋯ 1 − 1 + 1 − 1 + ⋯ ( Grandi 's series ) Infinite arithmetic series 1 − 2 + 3 − 4 + ⋯ 1 − 2 + 4 − 8 + ⋯ 1 + 1 / 2 + 1 / 3 + 1 / 4 + ⋯ ( harmonic series ) 1 − 1 + 2 − 6 + 24 − 120 + ⋯ ( alternating factorials ) 1 / 2 + 1 / 3 + 1 / 5 + 1 / 7 + 1 / 11 + ⋯ ( inverses of primes ) Kinds of series Taylor series Power series Formal power series Laurent series Puiseux series Dirichlet series Trigonometric series Fourier series Generating series Hypergeometric series Generalized hypergeometric series Hypergeometric function of a matrix argument Lauricella hypergeometric series Modular hypergeometric series Riemann 's differential equation Theta hypergeometric series Retrieved from `` https://en.wikipedia.org/w/index.php?title=Arithmetic_progression&oldid=801071991 '' Categories : Arithmetic series Sequences and series Hidden categories : Articles containing proofs Talk Contents About Wikipedia Azərbaycanca Беларуская Български Bosanski Català Čeština Dansk Deutsch Eesti Ελληνικά Español Esperanto Euskara فارسی Français Gaeilge Galego 한국어 Հայերեն हिन्दी Hrvatski Ido Bahasa Indonesia Italiano עברית ქართული Қазақша Кыргызча Latviešu Lietuvių Magyar Македонски മലയാളം Bahasa Melayu Nederlands नेपाली 日本 語 Norsk nynorsk Oʻzbekcha / ўзбекча ਪੰਜਾਬੀ Piemontèis Polski Português Română Русский Slovenčina Slovenščina Српски / srpski Srpskohrvatski / српскохрватски Suomi Svenska தமிழ் Татарча / tatarça ไทย Українська Tiếng Việt 中文 Edit links This page was last edited on 17 September 2017, at 13 : 49. 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Question: what does d stand for in arithmetic sequence
Answer: common difference