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For instance, the sequence 5, 7, 9, 11, 13, 15, . . . is an arithmetic progression with a common difference of 2. If the initial term of an arithmetic progression is a 1 {\displaystyle a_{1}} and the common difference of successive members is d {\displaystyle d} , then the n {\displaystyle n} -th term of the sequence ( a n {\displaystyle a_{n ...
These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects.
DASK is an acronym for Dansk Aritmetisk Sekvens Kalkulator or Danish Arithmetic Sequence Calculator. Regnecentralen almost did not allow the name, as the word dask means "slap" in Danish. In the end, however, it was named so as it fit the pattern of the name BESK, the Swedish computer which provided the initial architecture for DASK.
In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression, which is also known as an arithmetic sequence. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms.
An arithmetico-geometric series is a sum of terms that are the elements of an arithmetico-geometric sequence. Arithmetico-geometric sequences and series arise in various applications, such as the computation of expected values in probability theory , especially in Bernoulli processes .
The Goodstein sequence of a number m is a sequence of natural numbers. The first element in the sequence G m {\displaystyle G_{m}} is m itself. To get the second, G m ( 2 ) {\displaystyle G_{m}(2)} , write m in hereditary base-2 notation, change all the 2s to 3s, and then subtract 1 from the result.
In zeta function regularization, the series = is replaced by the series =. The latter series is an example of a Dirichlet series. When the real part of s is greater than 1, the Dirichlet series converges, and its sum is the Riemann zeta function ζ(s).
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .