# Arithmetic Progression

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Related to Arithmetic Progression: geometric progression, harmonic progression

## 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.
References in periodicals archive ?
--The arithmetic progression calculated according to the physical skills of students from the fifth grade was 17.08 repetitions, exceeding the minimum scale by 2.08 repetitions established by SNE; the arithmetic progression (19.31) calculated according to the performances of the sixth grade exceeds the minimum scale by 3.31 repetitions, and that of the seventh grade (19.89) by 2.89 repetitions.
that is, the sequence [([T.sup.*n][T.sup.n]).sub.n [greater than or equal to] 0] is an arithmetic progression of strict order m-1 in B(H).
To concentrate on the arithmetic progressions in the a[b.sub.0]ve formulas Euler removed repeating l's by elementary transformations:
for k = 1, 2, 3, are all arithmetic progressions with common difference 2k.
Therefore, the new sequence {[M.sub.(k,t]} generated form {[a.sub.n]} is an arithmetic progression with [M.sub.(0,t)] = t [A.sub.1] = t(t - 1) So.
is a finite arithmetic progression with common diference [d.sub.1],
We assume without proof the following property of all arithmetic progressions: the sequence [a.sub.1], [a.sub.2], [a.sub.3], ...
Note that ([2.sup.2m+1], [2.sup.m] + 1) = 1, so from Dirichlet's Theorem we can easily deduce that there are infinitely many primes in the arithmetic progression:
In fact it is also conjectured that for every n, there are n consecutive primes in arithmetic progression; at the time of writing, the longest such string consists of 10 primes (see Caldwell, 2004b).
They announced that they had found seven consecutive primes in arithmetic progression. The previous record had been six.
(i) [for all]k = 0, 1, 2,..., [a.sub.kt+1], [a.sub.kt+2], [a.sub.kt+3],..., [a.sub.kt+t] is a finite arithmetic progression with [d.sub.1] as the common difference, where t is a constant natural numbers;
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