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

  • Convergence of Series with Positive Terms

    • For a sequence $\{a_n\}$, where $a_n$ is a non-negative real number for every $n$, the sum $\sum_{n=0}^{\infty}a_n$ can either converge or diverge to $\infty$.
    • For a sequence $\{a_n\}$, where $a_n$ is a non-negative real number for every $n$, the sequence of partial sums
    • Therefore, it follows that a series $\sum_{n=0}^{\infty} a_n$ with non-negative terms converges if and only if the sequence $S_k$ of partial sums is bounded.
    • Visualization of the geometric sum in Example 2.

  • Introduction to negative binomial distribution

    • The geometric distribution describes the probability of observing the first success on the nth trial.
    • OR 10th sequence)
    • P(1st sequence OR 2nd sequence OR ...
    • OR 10th sequence) = P(1st sequence) + P(2nd sequence) +·+ P(10th sequence)
    • P(it takes Brian six tries to make four field goals) = [Number of possible sequences]×P(Single sequence)

  • Designating the Configuration of Chiral Centers

    • The assignment of these prefixes depends on the application of two rules: The Sequence Rule and The Viewing Rule.
    • The sequence rule is the same as that used for assigning E-Z prefixes to double bond stereoisomers.
    • Remembering the geometric implication of wedge and hatched bonds, an observer (the eye) notes whether a curved arrow drawn from the # 1 position to the # 2 location and then to the # 3 position turns in a clockwise or counter-clockwise manner.
    • The stereogenic carbon atom is colored magenta in each case, and the sequence priorities are shown as light blue numbers.
    • Rule # 3 of the sequence rules allows us to order these substituents.