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A geometric progression, also known as a geometric sequence, is a mathematical sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3.
In analogy to the square root of x, one can calculate the heptagonal root of x, meaning the number of terms in the sequence up to and including x. The heptagonal root of x is given by the formula n = 40 x + 9 + 3 10 , {\displaystyle n={\frac {{\sqrt {40x+9}}+3}{10}},}
The geometric series is an infinite series derived from a special type of sequence called a geometric progression.This means that it is the sum of infinitely many terms of geometric progression: starting from the initial term , and the next one being the initial term multiplied by a constant number known as the common ratio .
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 .
In 1876 the sequence and its equation were initially mentioned by Édouard Lucas, who noted that the index n divides term P(n) if n is prime. [5] In 1899 Raoul Perrin asked if there were any counterexamples to this property. [6] The first P(n) divisible by composite index n was found only in 1982 by William Adams and Daniel Shanks. [7]
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.
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In general: + = + (())where both terms on the right are generalized pentagonal numbers and the first term is a pentagonal number proper (n ≥ 1).This division of centered hexagonal arrays gives generalized pentagonal numbers as trapezoidal arrays, which may be interpreted as Ferrers diagrams for their partition.