sequence

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.