Sequence (ordered list of elements and mathematical concept)

Mathematical definition and notation

Mathematically, a sequence is most often defined as a function whose domain is the set of positive integers (or sometimes the non‑negative integers). If a function a assigns to each natural number n an element a(n), the ordered list (a(1), a(2), a(3), ...) is a sequence. Common notation for the n-th element is a_n or a(n). This functional viewpoint makes it straightforward to describe infinite sequences by giving a formula or a rule for a(n) rather than enumerating all terms.

Sequences are commonly classified as either finite or infinite. A finite sequence has a last term and can be written out completely, for example (1, 2, 3, 4, 5). An infinite sequence continues without end, such as the even numbers (2, 4, 6, 8, ...). To specify an infinite sequence one typically gives an explicit expression or a recurrence rule that determines each term from earlier terms; for instance, the rule a(n) = 2 × n defines the sequence of positive even integers.

Common types and properties

• Arithmetic sequence (progression): successive terms differ by a constant. Example: 3, 6, 9, 12, ...
• Geometric sequence: successive terms are multiplied by a constant ratio. Example: 2, 6, 18, 54, ...
• Recurrence relations: each term defined in terms of earlier ones, e.g. Fibonacci numbers.
• Monotonic, bounded, periodic, or random: qualitative properties often studied in analysis and probability.

Key distinctions include sequence versus set (order and repetition matter in sequences but not in sets) and sequence versus series (a series is a sum of sequence terms). A subsequence is obtained by selecting terms from the original sequence while preserving their order.

Construction, examples and notation

There are several practical ways to describe a sequence: list its elements explicitly (suitable for short finite sequences), give a formula for the nth term (for many infinite sequences), or provide a recurrence relation together with initial conditions. For example, the formula a(n) = 2 × n shows how to find any term — a(1) = 2, a(2) = 4, a(100) = 200 — without writing down intermediate terms. The symbol a_n or (a_n) is standard; sometimes parentheses or brackets indicate the ordered list.

History, applications and further notes

The idea of ordered progressions appears in ancient arithmetic and geometric problems, and over time the concept was formalized within arithmetic and analysis. Sequences are central in calculus and real analysis (for example, the notion of convergence depends on sequences), in discrete mathematics and combinatorics, and in computer science (arrays, queues and time series are treated as sequences). They are also used to model repeated events in science, such as genetic sequences in biology or sampled signals in engineering.

For more technical treatments, topics to explore include limits and convergence of sequences, subsequences and compactness, generating functions, and different classes of recurrence relations. Introductory resources and further readings are available in mathematical texts and online materials: rules and formulas, natural numbers and indexing, and practical examples of multiplication or scaling operations like "times n" are often illustrated in elementary accounts: examples of termwise operations.

Because sequences can be represented in multiple equivalent ways, choosing the most informative description depends on whether the sequence is finite or infinite, whether explicit computation of distant terms is needed, and which properties (e.g., monotonicity or boundedness) are most relevant.