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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.
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 series that has terms which are each the product of an element of an arithmetic progression with the corresponding element of a geometric progression. Example: 3 + 5 2 + 7 4 + 9 8 + 11 16 + ⋯ = ∑ n = 0 ∞ ( 3 + 2 n ) 2 n . {\displaystyle 3+{5 \over 2}+{7 \over 4}+{9 \over 8}+{11 \over 16}+\cdots =\sum ...
Progression may refer to: In mathematics: Arithmetic progression, a sequence of numbers such that the difference between any two successive members of the sequence is a constant; Geometric progression, a sequence of numbers such that the quotient of any two successive members of the sequence is a constant
Grégoire de Saint-Vincent gave the first definition of limit (terminus) of a geometric series in his work Opus Geometricum (1647): "The terminus of a progression is the end of the series, which none progression can reach, even not if she is continued in infinity, but which she can approach nearer than a given segment." [4]
Conversely, in any right triangle whose squared edge lengths are in geometric progression with any ratio , the Pythagorean theorem implies that this ratio obeys the identity = +. Therefore, the ratio must be the unique positive solution to this equation, the golden ratio, and the triangle must be a Kepler triangle.
In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the ...
In linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix with the terms of a geometric progression in each row: an (+) (+) matrix