geometric progression

(noun)

A series 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.

Related Terms

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

  • Applications of Geometric Series

    • Geometric series have applications in math and science and are one of the simplest examples of infinite series with finite sums.
    • Geometric series are used throughout mathematics.
    • The formula for the sum of a geometric series can be used to convert the decimal to a fraction:
    • In the case of the Koch snowflake, its area can be described with a geometric series.
    • Apply geometric sequences and series to different physical and mathematical topics

  • Sums and Series

    • If you add up all the terms of a geometric sequence (one in which each entry is the previous entry multiplied by a constant), you have a geometric series.
    • The common example of a trend that follows a geometric series is the number of people infected with a virus, as each person passes it to several more.
    • Once again, pause to convince yourself that this will work on all geometric series, but only on geometric series.
    • Finally—once again—we can apply this trick to the generic geometric series to find a formula.
    • So the total number of people infected follows a geometric series.

  • Recursive Definitions

    • An applied example of a geometric sequence involves the spread of the flu virus.
    • Suppose each infected person will infect two more people, such that the terms follow a geometric sequence.
    • Using this equation, the recursive equation for this geometric sequence is:
    • Each person infects two more people with the flu virus, making the number of recently-infected people the nth term in a geometric sequence.

  • Complex Numbers in Polar Coordinates

    • This leads to a way to visualize multiplying and dividing complex numbers geometrically.
    • Sometimes it is helpful to think of complex numbers in a different geometric way.
    • The previous geometric idea where the number $z=a+bi$ is associated with the point $(a,b)$ on the usual $xy$-coordinate system is called rectangular coordinates.
    • This way of thinking about multiplying and dividing complex numbers gives a geometric way of thinking about those operations.

  • Complex Conjugates

    • Two complex conjugates of each other multiply to be a real number with geometric significance.

  • Complex Conjugates and Division

    • Geometrically, z* is the "reflection" of z about the real axis (as shown in the figure below).
    • Geometric representation of z and its conjugate in the complex plane.

  • Addition, Subtraction, and Multiplication

    • Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram, three of whose vertices are O, A, and B (as shown in ).
    • Addition of two complex numbers can be done geometrically by constructing a parallelogram.