A sequence is an ordered set of objects. A sequence that goes on forever is called an infinite sequence, whereas one that does not is called a finite sequence. The sum of a sequence is called a series.
Sequences are usually denoted as
with being the term number and being the bounds of the series, and being the term, usually found with a function or rule describing them.
An example would be
Sequences described with the previous terms are called recursive sequences. For instance, the Fibonacci numbers can be described as
Two common types are arithmetic and geometric sequences. Arithmetic sequences have a given difference between each term. For example,
Geometric sequences take the form
Where is a common ratio.
Sets vs Sequences
Unlike a set, sequences allow repeats and the order matters.
- Let be a set
- Let be the set of natural numbers.
- Then, a mapping is called a sequence of elements of . The image of an element of under (that is) is denoted as , where .
An equivalent definition is an indexed family indexed by the natural numbers.
While the formal definition of a sequence is a treats a sequence as a function, in practice they are treated somewhat like a set with "order".