sequence

(noun)

A set of things next to each other in a set order; a series

Related Terms

  • finite
  • set
  • general term
  • series

(noun)

An ordered list of elements, possibly infinite in length.

Related Terms

  • finite
  • set
  • general term
  • series

Examples of sequence in the following 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.
  • Introduction to Sequences

    • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers $(2, 4, 6, \cdots )$.
    • 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.
    • Assume our sequence is $t_1, t_2, \dots $.
  • The General Term of a Sequence

    • Given terms in a sequence, it is often possible to find a formula for the general term of the sequence, if the formula is a polynomial.
    • 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:
  • 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

    • When discussing arithmetic sequences, you may have noticed that the difference between two consecutive terms in the sequence could be written in a general way:
    • 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:
    • Use a recursive formula to find specific terms of a sequence
  • Summing Terms in an Arithmetic Sequence

    • An arithmetic sequence which is finite has a specific formula for its sum.
    • For instance, the sequence $5, 7, 9, 11, 13, \cdots$ is an arithmetic progression with common difference of $2$.
    • The sum of the members of a finite arithmetic sequence is called an arithmetic series.
    • Even if one is dealing with an infinite sequence, the sum of that sequence can still be found up to any $n$th term with the same equation used in a finite arithmetic sequence.
    • Calculate the sum of an arithmetic sequence up to a certain number of terms
  • 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.
    • We can use a formula to find the sum of a finite number of terms in a sequence.
    • Therefore, by utilizing the common ratio and the first term of the sequence, we can sum the first $n$ terms.
    • 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
  • Sums and Series

    • The summation of all the terms of a sequence is called a series, and many formulae are available for easily calculating large series.
    • Summation is the operation of adding a sequence of numbers; the result is their sum or total.
    • For finite sequences of such elements, summation always produces a well-defined sum (possibly by virtue of the convention for empty sums).
    • 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.
  • Theoretical Probability

    • By the Fundamental Rule of Counting, the total number of possible sequences of choices is $5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 = 120$ sequences.
    • Each sequence is called a permutation (or ordering) of the five items.
    • By the Fundamental Rule of Counting, the total number of possible sequences of choices is a permutation of each of the items.
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