geometric

(adjective)

increasing or decreasing in a geometric progression, i.e. multiplication by a constant.

Related Terms

  • arithmetic

Examples of geometric in the following topics:

  • Geometric Sequences

    • A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed non-zero number called the common ratio rrr.
    • For example, the sequence 2,6,18,54,⋯2, 6, 18, 54, \cdots2,6,18,54,⋯ is a geometric progression with common ratio 333.
    • Similarly 10,5,2.5,1.25,⋯10,5,2.5,1.25,\cdots10,5,2.5,1.25,⋯ is a geometric sequence with common ratio 12\frac{1}{2}​2​​1​​.
    • For instance: 1,−3,9,−27,81,−243,⋯1,-3,9,-27,81,-243, \cdots1,−3,9,−27,81,−243,⋯ is a geometric sequence with common ratio −3-3−3.
    • The behavior of a geometric sequence depends on the value of the common ratio.
  • 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+⋯64+ 32 + 16 + 8 + \cdots64+32+16+8+⋯
  • Applications of Geometric Series

    • 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.
    • Excluding the initial 1, this series is geometric with constant ratio r=49r = \frac{4}{9}r=​9​​4​​.
    • The first term of the geometric series is a=319=13a = 3 \frac{1}{9} = \frac{1}{3}a=3​9​​1​​=​3​​1​​, so the sum is
    • Apply geometric sequences and series to different physical and mathematical topics
  • 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≠1r\neq 1r≠1, the sum of the first nnn terms of a geometric series is:
    • Calculate the sum of the first nnn terms in a geometric sequence
  • 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

    • A geometric sequence follows the formula an=r⋅an−1.a_n=r\cdot a_{n-1}.a​n​​=r⋅a​n−1​​. This is another example of a recursive formula.
    • An applied example of a geometric sequence involves the spread of the flu virus.
    • Suppose each infected person will infect two more, such that the terms follow a geometric sequence.
    • Using this equation, the recursive equation for this geometric sequence is: an=2⋅an−1.a_n=2 \cdot a_{n-1}.a​n​​=2⋅a​n−1​​.
    • 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+biz=a+biz=a+bi is associated with the point (a,b)(a,b)(a,b) on the usual xyxyxy-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.
    • Note that (a+bi)(a−bi)=a2−abi+abi−b2i2=a2+b2.(a+bi)(a-bi)=a^2-abi+abi-b^2i^2=a^2+b^2.(a+bi)(a−bi)=a​2​​−abi+abi−b​2​​i​2​​=a​2​​+b​2​​. This number has a geometric significance.
  • 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.
  • 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.
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