Arithmetic progression

Encyclopedia : A : AR : ARI : Arithmetic progression

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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:

[\ a_n = a_1 + (n - 1)d.]

Sum (arithmetic series)

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+\dots+a_n=\frac =\frac.]

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+\dots\dots+a_1+(n-2)d+a_1+(n-1)d] [ S_n=a_n-(n-1)d+a_n-(n-2)d+\dots\dots+a_n-2d+a_n-d+a_n]

[\ 2S_n=n(a_1+a_n)]

[ S_n=\frac].

Product

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\cdots a_n = n>d \right)}^} = d^n \frac,]

where [x^}] denotes the rising factorial and [\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 \times 2 \times \ldots \times n] is given by the factorial [n!] and that the product

[m \times (m+1) \times \ldots \times (n-1) \times n \,\!]

for positive integers [m] and [n] is given by

[\frac]

See also

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

External links

• , [Arithmetic progression] at MathWorld.
  • , [Arithmetic series] at MathWorld.

     

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