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Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is , , , , , … where r is the common ratio and a is the initial value. The sum of a geometric progression's terms is called a geometric series.
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 rotating calipers technique for designing geometric algorithms may also be interpreted as a form of the plane sweep, in the projective dual of the input plane: a form of projective duality transforms the slope of a line in one plane into the x-coordinate of a point in the dual plane, so the progression through lines in sorted order by their ...
[1] [6] The name "Kepler triangle" for this shape was used by Roger Herz-Fischler, based on Kepler's 1597 letter, as early as 1979. [7] Another name for the same triangle, used by Matila Ghyka in his 1946 book on the golden ratio, The Geometry of Art and Life, is the "triangle of Price", after pyramidologist W. A. Price. [12]
One of them includes the geometric progression problem. The story is first known to have been recorded in 1256 by Ibn Khallikan. [3] Another version has the inventor of chess (in some tellings Sessa, an ancient Indian Minister) request his ruler give him wheat according to the wheat and chessboard problem. The ruler laughs it off as a meager ...
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 .
Algebra (and later, calculus) can thus be used to solve geometrical problems. Geometry was split into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically. [23] Analytic geometry allows the study of curves unrelated to circles and lines.
The Kepler triangle is a right triangle whose sides are in geometric progression. If the sides are formed from the geometric progression a, ar, ar 2 then its common ratio r is given by r = √ φ where φ is the golden ratio. Its sides are therefore in the ratio 1 : √ φ : φ. Thus, the shape of the Kepler triangle is uniquely determined (up ...
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