geometric

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 $r$.
  • For example, the sequence $2, 6, 18, 54, \cdots$ is a geometric progression with common ratio $3$.
  • Similarly $10,5,2.5,1.25,\cdots$ is a geometric sequence with common ratio $\frac{1}{2}$.
  • For instance: $1,-3,9,-27,81,-243, \cdots$ is a geometric sequence with common ratio $-3$.
  • 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 $64+ 32 + 16 + 8 + \cdots$

• 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 $r = \frac{4}{9}$.
  • The first term of the geometric series is $a = 3 \frac{1}{9} = \frac{1}{3}$, 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 $r\neq 1$, the sum of the first $n$ terms of a geometric series is:
  • Calculate the sum of the first $n$ terms in a geometric sequence

• 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 $a_n=r\cdot a_{n-1}.$ 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: $a_n=2 \cdot a_{n-1}.$
  • 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 $z=a+bi$ 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.
  • Note that $(a+bi)(a-bi)=a^2-abi+abi-b^2i^2=a^2+b^2.$ 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.