geometric sequence

(noun)

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. Also known as a geometric progression.

Examples of geometric sequence in the following topics:

  • Geometric Sequences

    • The $n$th term of a geometric sequence with initial value $a$ and common ratio $r$ is given by
    • Generally, to check whether a given sequence is geometric, one simply checks whether successive entries in the sequence all have the same ratio.
    • The common ratio of a geometric series may be negative, resulting in an alternating sequence.
    • 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.
  • 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.
    • Use a recursive formula to find specific terms of a sequence
  • 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.
    • We can use a formula to find the sum of a finite number of terms in a sequence.
    • Find the sum of the first five terms of the geometric sequence $\left(6, 18, 54, 162, \cdots \right)$.
    • Calculate the sum of the first $n$ terms in a geometric sequence
  • Introduction to Sequences

    • Finite sequences are sometimes known as strings or words and infinite sequences as streams.
    • Finite sequences include the empty sequence $( \quad )$ that has no elements.
    • These are called recursive sequences.
    • A geometric sequence is a list in which each number is generated by multiplying a constant by the previous number.
    • The explicit definition of a geometric sequence is obtained in a similar way.
  • The General Term of a Sequence

    • Given several terms in a sequence, it is sometimes possible to find a formula for the general term of the sequence.
    • Then the sequence looks like:
    • Then the sequence would look like:
    • The second sequence of differences is:
    • For example, the geometric sequence $2, 4, 8, 16,\dots$ is given by the general term $2^n$.
  • Sums and Series

    • Summation is the operation of adding a sequence of numbers; the result is their sum or total.
    • If you add up all the terms of an arithmetic sequence (a sequence in which every entry is the previous entry plus a constant), you have an arithmetic 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.
    • Once again, pause to convince yourself that this will work on all geometric series, but only on geometric series.
    • So the total number of people infected follows a geometric series.
  • 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
  • Arithmetic Sequences

    • An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.
    • An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant.
    • For instance, the sequence $5, 7, 9, 11, 13, \cdots$ is an arithmetic sequence with common difference of $2$.
    • The behavior of the arithmetic sequence depends on the common difference $d$.
    • Calculate the nth term of an arithmetic sequence and describe the properties of arithmetic sequences
  • Sequences of Mathematical Statements

    • In mathematics, a sequence is an ordered list of objects, or elements.
    • Unlike a set, order matters in sequences, and exactly the same elements can appear multiple times at different positions in the sequence.
    • A sequence is a discrete function.
    • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2,4,6, \cdots )$.
    • Sequences of statements are necessary for mathematical induction.
  • 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 $64+ 32 + 16 + 8 + \cdots$
Subjects
  • Accounting
  • Algebra
  • Art History
  • Biology
  • Business
  • Calculus
  • Chemistry
  • Communications
  • Economics
  • Finance
  • Management
  • Marketing
  • Microbiology
  • Physics
  • Physiology
  • Political Science
  • Psychology
  • Sociology
  • Statistics
  • U.S. History
  • World History
  • Writing

Except where noted, content and user contributions on this site are licensed under CC BY-SA 4.0 with attribution required.