25 terms of the arithmetic sequence if a1= 1/2 d= -3/8
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..................G. O. A. T
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An arithmetic sequence (or arithmetic progression) is a sequence (finite or infinite list) of real numbers for which each term is the previous term plus a constant (called the common difference). For example, starting with 1 and using a common difference of 4 we get the finite arithmetic sequence: 1, 5, 9, 13, 17, 21; and also the infinite sequence
1, 5, 9, 13, 17, 21, 25, 29, . . ., 4n+1, . . .
In general, the terms of an arithmetic sequence with the first term a0 and common difference d, have the form an = dn+a0 (n=0,1,2,...). If a0 and d are relatively prime positive integers, then the corresponding infinite sequence contains infinitely many primes (see Dirichlet's theorem on primes in arithmetic progressions).
An important example of this is the following two arithmetic sequences:
1, 7, 13, 19, 25, 31, 37, ...
5, 11, 17, 23, 29, 35, 41, ...
Together these two sequences contain all of the primes except 2 and 3.
A related question is how long of a arithmetic sequence can we find all of whose members are prime. Dickson's conjecture says the answer should be arbitrarily long--but finding long sequences of primes is quite difficult. It is fairly easy to heuristically estimate how many such primes sequences there should be for any given length--Hardy and Littlewood first did this in 1922 [HL23]. In 1939, van der Corput showed that there are infinitely many triples of primes in arithmetic progression [Corput1939]. Finally, in 2004, Green and Tao [GT2004a] showed that there are indeed arbitrarily long sequences of primes and that a k-term one occurs before [GT2004b]: