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An arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference from any succeeding term to its preceding term remains constant throughout the sequence. The constant difference is called common difference of that arithmetic progression.
Sequences dn + a with odd d are often ignored because half the numbers are even and the other half is the same numbers as a sequence with 2d, if we start with n = 0. For example, 6n + 1 produces the same primes as 3n + 1, while 6n + 5 produces the same as 3n + 2 except for the only even prime 2. The following table lists several arithmetic ...
Name First elements Short description OEIS Kolakoski sequence: 1, 2, 2, 1, 1, 2, 1, 2, 2, 1, ... The n th term describes the length of the n th run : A000002: Euler's ...
Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ...). The position of an element in a sequence is its rank or index ; it is the natural number for which the element is the image.
An example is the sequence of primes (3, 7, 11), which is given by = + for . According to the Green–Tao theorem, there exist arbitrarily long arithmetic progressions in the sequence of primes. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers.
The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a definable sequence relative to M if there exists some formula P ( x ) in the language of set theory, with one free variable and no parameters, which is true in M for that integer sequence and ...
The elements of an arithmetico-geometric sequence () are the products of the elements of an arithmetic progression (in blue) with initial value and common difference , = + (), with the corresponding elements of a geometric progression (in green) with initial value and common ratio , =, so that [4]
An excellent example of Harmonic Progression is the Leaning Tower of Lire. In it, uniform blocks are stacked on top of each other to achieve the maximum sideways or lateral distance covered. The blocks are stacked 1/2, 1/4, 1/6, 1/8, 1/10, … distance sideways below the original block.
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