Sequence
Elementary
A sequence is a list of things in a particular order.
High school
A sequence is an ordered list of terms $a_1, a_2, a_3, \dots$ Both order and repetition matter: $(1, 2)$, $(2, 1)$, and $(1, 1, 2)$ are different sequences.
Undergraduate
A sequence is a function from $\nats$ (an infinite sequence) or from $\setdef{1, \dots, n}$ (a finite sequence, or $n$-tuple) to a set.
Examples
Examples
- The sequence $1, 2, 3, \dots$ Elementary
-
$(1, 2, 2)$
High school
$(1, 2, 2)$ is a sequence of length three, and so an indexed family, which is different from $(2, 1, 2)$ and from $(1, 2)$.
Non-examples
Things that may be mistaken for examples.
-
$\setdef{1, 2, 3}$
High school
A set has no order. A sequence lists its terms in order.
Distinctions involving this term
- Types of collections: Mathematics has several kinds of “a bunch of things,” which differ in whether order and repetition matter. A set ignores order and repetition. A multiset keeps repetition. A sequence keeps order and repetition. “Collection” is informal, and “group” is a technical term for a set with an operation.