In mathematics, an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. For instance, the sequence 3, 5, 7, 9, 11, ... is an arithmetic progression with common difference 2.

If the initial term of an arithmetic progression is $a\_1$ and the common difference of successive members is d, then the nth term of the sequence is given by:

$\backslash \; a\_n\; =\; a\_1\; +\; (n\; -\; 1)d.$

Sum (arithmetic series)

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The sum of the components of an arithmetic progression is called an arithmetic series. The formula for the sum of the first n terms of an arithmetic progression is:

$S\_n\; =\; a\_1+a\_2+\backslash dots+a\_n=\backslash frac\{n(\; a\_1\; +\; a\_n)\}\{2\}\; =\backslash frac\{n2a\_1\; +\; (n-1)d\}\{2\}.$

This formula follows from the fact that the sum of the first and the last term is the same as the sum of the second and the second last, and so forth. An often-told story is that Carl Friedrich Gauss discovered it when his third grade teacher asked the class to find the sum of the first 100 numbers, and he instantly computed the answer (5050).

Proof:

$S\_n=a\_1+a\_1+d+a\_1+2d+\backslash dots\backslash dots+a\_1+(n-2)d+a\_1+(n-1)d$ $S\_n=a\_n-(n-1)d+a\_n-(n-2)d+\backslash dots\backslash dots+a\_n-2d+a\_n-d+a\_n$

$\backslash \; 2S\_n=n(a\_1+a\_n)$

$S\_n=\backslash frac\{n(\; a\_1\; +\; a\_n)\}\{2\}$.

Product

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The product of the components of an arithmetic progression with an initial element $a\_1$, common distance $d$, and $n$ elements in total, is determined in a closed expression by

$a\_1a\_2\backslash cdots\; a\_n\; =\; n>d\; \{\backslash left(\backslash frac\{a\_1\}\{d\}\backslash right)\}^\{\backslash overline\{n\}\}\; =\; d^n\; \backslash frac\{\backslash Gamma\; \backslash left(a\_1/d\; +\; n\backslash right)\; \}\{\backslash Gamma\; \backslash left(\; a\_1\; /\; d\; \backslash right)\; \},$

where $x^\{\backslash overline\{n\}\}$ denotes the rising factorial and $\backslash Gamma$ denotes the Gamma function. (Note however that the formula is not valid when $a\_1/d$ is a negative integer or zero).

This is a generalization from the fact that the product of the progression $1\; \backslash times\; 2\; \backslash times\; \backslash ldots\; \backslash times\; n$ is given by the factorial $n!$ and that the product

$m\; \backslash times\; (m+1)\; \backslash times\; \backslash ldots\; \backslash times\; (n-1)\; \backslash times\; n\; \backslash ,\backslash !$

for positive integers $m$ and $n$ is given by

$\backslash frac\{n!\}\{(m-1)!\}$

See also

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• Addition
• Geometric progression
• Generalized arithmetic progression
• Thomas Robert Malthus
• Carl Friedrich Gauss

External links

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Integer sequences | Mathematical series

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