# Arithmetic Progression

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Related to Arithmetic progressions: geometric progressions

## Arithmetic Progression

A sequence of numbers in which the difference between any two succeeding numbers in the sequence is always the same. For example, given the sequence 0, 10, 20, 30, 40, and 50, the progression is arithmetic because the difference between each succeeding number is always 10.
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A sequence a in a group G is called an arithmetic progression of order h = 0, 1, 2 ..., if [D.sup.h+1] a = 0.
It is well known that the sequence a in G is an arithmetic progression of order h if and only if there exists a polynomial p(n) in n, with coefficients in G and of degree less than or equal to h, such that p(n) = [a.sub.n], for every n = 0, 1, 2 ...; that is, there are [[gamma].sub.h], [[gamma].sub.h-1], ..., [[gamma].sub.1], [[gamma].sub.0] [member of] G, which depend only on a, such that, for every n = 0, 1, 2, ...,
which is simply an arithmetic progression with common difference 4 when b progresses through the C(n).
Thus we conclude that whenever both the first column and the first row are arithmetic progressions (2), the "fixed" sum
If in a six by six array in which all rows are translations of the first row, and the numbers in both the first row and column form arithmetic progressions, then the sum of any six numbers of the array, no two of which are in the same row or column, is six times the arithmetic mean of the numbers in the upper left and lower right corners of the array.
To concentrate on the arithmetic progressions in the a[b.sub.0]ve formulas Euler removed repeating l's by elementary transformations:
Progressive primes A computer search revealed an arithmetic progression consisting of 23 primes, the longest such sequence yet found (sciencenews.org/ 20040828/mathtrek.asp).
The mathematicians then deduced that the prime numbers are arranged within the spread of almost-primes with enough regularity to ensure that the overall sequence of primes does indeed contain arithmetic progressions of every length.
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