geometric progression

Examples of geometric progression in the following topics:

• Geometric Sequences

  • A geometric progression, also known as a geometric sequence, is an ordered list of numbers in which each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio $r$.
  • For example, the sequence $2, 6, 18, 54, \cdots$ is a geometric progression with common ratio $3$.
  • Geometric sequences (with common ratio not equal to $-1$, $1$ or $0$) show exponential growth or exponential decay, as opposed to the linear growth (or decline) of an arithmetic progression such as $4, 15, 26, 37, 48, \cdots$ (with common difference $11$).
  • Note that the two kinds of progression are related: exponentiating each term of an arithmetic progression yields a geometric progression, while taking the logarithm of each term in a geometric progression with a positive common ratio yields an arithmetic progression.
  • An interesting result of the definition of a geometric progression is that for any value of the common ratio, any three consecutive terms $a$, $b$, and $c$ will satisfy the following equation:

• Infinite Geometric Series

  • Geometric series are one of the simplest examples of infinite series with finite sums.
  • A geometric series is an infinite series whose terms are in a geometric progression, or whose successive terms have a common ratio.
  • If the terms of a geometric series approach zero, the sum of its terms will be finite.
  • A geometric series with a finite sum is said to converge.
  • Find the sum of the infinite geometric series $64+ 32 + 16 + 8 + \cdots$

• Summing the First n Terms in a Geometric Sequence

  • By utilizing the common ratio and the first term of a geometric sequence, we can sum its terms.
  • The terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant.
  • The following are several geometric series with different common ratios.
  • For $r\neq 1$, the sum of the first $n$ terms of a geometric series is:
  • Calculate the sum of the first $n$ terms in a geometric sequence

• Pottery in the Greek Geometric Period

  • The Geometric period in Greek art is distinguished by a reliance on geometric shapes to create human and animal figures as well as abstract décor.
  • In the eleventh century BCE, the citadel centers of the Mycenaeans were abandoned and Greece fell into a period with little cultural or social progression.
  • Every empty space in these scenes is filled with geometric shapes—M's, diamonds, starbursts—demonstrating the Geometric painter's horror vacui.
  • Geometric Amphora, from the Dipylon Cemetery, Athens, Greece, c. 740 BCE.
  • Geometric Krater.

• Geometric Symbolism

  • Throughout history, geometric designs have been ascribed with symbolic and sacred meaning.
  • Geometric designs have been used throughout history as religious and spiritual symbols.
  • Symbolic and sacred meanings are often ascribed to certain geometric shapes and geometric proportions.
  • According to the psychologist David Fontana, its symbolic nature can help one "to access progressively deeper levels of the unconscious, ultimately assisting the meditator to experience a mystical sense of oneness with the ultimate unity from which the cosmos in all its manifold forms arises. "
  • This sand mandala is an example of a sacred geometric design in Buddhist religious tradition.

• American Art Deco Architecture

  • The Art Deco style is often characterized by its use of rich colors, symmetry, bold geometric shapes, simple composition, rectilinear rather than curvilinear shapes, and lavish ornamentation.
  • During its heyday, Art Deco represented luxury, glamour, exuberance, and faith in social and technological progress.
  • As the Great Depression decade of the 1930s progressed, Americans saw a new decorative element of the Art Deco style emerge in the marketplace: streamlining.
  • The sunburst design executed in terracotta exemplifies Art Deco's characteristic combination of craft, ornament and geometrical motif.

• Verbal Aspect: Simple, Progressive, Perfect, and Perfect Progressive

• Introduction to geometric distribution (special topic)

  • These questions can be answered using the geometric distribution.
  • We first formalize each trial – such as a single coin flip or die toss – using the Bernoulli distribution, and then we combine these with our tools from probability (Chapter 2) to construct the geometric distribution.

• Painting

  • Postmodernism also rejects the notion of advancement or progress in art, and thus aims to overturn the "myth of the avant-garde" that modernism perpetuated.
  • During the 1950s and 1960s, forms of Geometric expression including Hard-edge painting and Frank Stella's work in Geometric abstraction emerged as reaction against the subjectivism of Abstract expressionism.
  • By the early 1960s Minimalism emerged as an abstract movement in art (with roots in geometric abstraction via Malevich, the Bauhaus and Mondrian) which rejected the idea of relational, and subjective painting, the complexity of Abstract expressionist surfaces, and the emotional zeitgeist and polemics present in the arena of Action painting.

• Greek Dark Ages

  • The Greek Dark Ages were ushered in by a period of violence and characterized by the disruption of Greek cultural progress.
  • Some technical innovations were introduced around 1050 BCE with the start of the Proto-geometric style, however, the overall trend was toward simpler, less intricate pieces with fewer resources being devoted to the creation of art.
  • The Linear B writing of the Greek language used by Mycenaean bureaucrats ceased, and decorations on Greek pottery after about 1100 BCE lacks the figurative decoration of the Mycenaeans and was restricted to simpler geometric styles.