Sequence: Difference between revisions
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Latest revision as of 21:36, 23 February 2025
Sequence is a term used in various branches of mathematics to describe an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and order is important.
Definition[edit]
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first n natural numbers (for a sequence of finite length n).
Types of Sequences[edit]
There are several types of sequences, including:
- Arithmetic sequence: A sequence in which each term after the first is obtained by adding a constant difference to the preceding term.
- Geometric sequence: A sequence in which each term after the first is obtained by multiplying the preceding term by a fixed, non-zero number called the common ratio.
- Fibonacci sequence: A sequence in which each number is the sum of the two preceding ones, usually starting with 0 and 1.
Applications[edit]
Sequences are used in a number of applications in mathematics and other fields. They are used in analysis for defining power series and the concept of limits. In computer science, they are used for data structures and algorithms, among other things.
