Definition of Sequences

A sequence is one of the most fundamental objects in mathematics: an ordered list of numbers (or other mathematical objects) arranged according to a definite rule. Unlike a set, where order does not matter and repetition is ignored, a sequence cares deeply about order — the first term, the second term, the third term, and so on each occupy a specific, fixed position. For example, the list 2, 4, 6, 8, 10, … is a sequence of even positive integers, and swapping any two terms would produce a genuinely different sequence. Sequences appear everywhere in mathematics and its applications. The digits of π form a sequence, monthly bank-account balances form a sequence, and the number of bacteria in a culture sampled every hour forms a sequence. Understanding sequences is the essential first step toward understanding series — the sums of infinitely many terms — which in turn underpin calculus, physics, and engineering. Before you can ask whether an infinite sum converges to a finite value, you must first be comfortable reading, writing, and manipulating the sequence of terms being summed. In this article you will learn the precise mathematical definition of a sequence, how to describe sequences using explicit (closed-form) formulas and recursive formulas, and how to visualise a sequence on a number line or coordinate plane. You will also meet some classic families of sequences — arithmetic, geometric, and others — that recur throughout the rest of this unit on Infinite Series.

Key takeaways

Formal Definition
Explicit (Closed-Form) Formulas
Recursive (Recurrence) Formulas
Arithmetic and Geometric Sequences
Finite vs. Infinite Sequences and Limits

Formal Definition

Formally, a sequence is a function whose domain is the set of positive integers ℕ = {1, 2, 3, …} (sometimes including 0). Instead of writing f(1), f(2), f(3), … we write a₁, a₂, a₃, … and call these the terms of the sequence. The entire sequence is denoted {aₙ} or (aₙ), and the subscript n is called the index.

Explicit (Closed-Form) Formulas

An explicit formula gives aₙ directly in terms of n. For example, aₙ = 2n produces 2, 4, 6, 8, … — you can compute any term immediately without knowing the previous ones. Another example is aₙ = 1/n, giving 1, 1/2, 1/3, 1/4, … This sequence will later become crucial when studying convergence of series.

Recursive (Recurrence) Formulas

A recursive formula defines each term using one or more previous terms, together with one or more initial conditions. The famous Fibonacci sequence is defined by F₁ = 1, F₂ = 1, and Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 3, producing 1, 1, 2, 3, 5, 8, 13, 21, … Recursive definitions are powerful but require you to compute terms in order.

Arithmetic and Geometric Sequences

Two families appear constantly. An arithmetic sequence has a constant difference d between consecutive terms: aₙ = a₁ + (n−1)d. For example, 3, 7, 11, 15, … has a₁ = 3 and d = 4. A geometric sequence has a constant ratio r between consecutive terms: aₙ = a₁ · rⁿ⁻¹. For example, 5, 10, 20, 40, … has a₁ = 5 and r = 2. Geometric sequences are especially important for infinite series because their sums can be computed exactly.

Finite vs. Infinite Sequences and Limits

A finite sequence has a last term; an infinite sequence continues without end. For infinite sequences, we often ask: does the sequence converge? A sequence {aₙ} converges to a limit L if, as n grows without bound, the terms aₙ get arbitrarily close to L. We write lim(n→∞) aₙ = L. If no such L exists, the sequence diverges. For instance, aₙ = 1/n converges to 0, while aₙ = (−1)ⁿ diverges because it oscillates between −1 and 1 forever.

The Key Equation

a_n = a_1 + (n-1)d \quad \text{and} \quad a_n = a_1 \cdot r^{\,n-1}

a_n — nth term (dimensionless (or same units as a₁)): The value of the sequence at position n; what we are trying to find.
a_1 — first term (dimensionless): The starting value of the sequence, at index n = 1.
d — common difference (dimensionless): The fixed amount added to each term to get the next term in an arithmetic sequence. Can be positive, negative, or zero.
r — common ratio (dimensionless): The fixed number each term is multiplied by to get the next term in a geometric sequence. If |r| < 1 the terms shrink toward zero.
n — index (positive integer): The position of the term in the sequence, starting at n = 1 (or sometimes n = 0).

Worked Example

Problem. A sequence is defined by the explicit formula aₙ = (3n − 1) / (n + 2). (a) Write out the first five terms. (b) Determine whether the sequence converges, and if so, find its limit.

a_1=\tfrac{2}{3},\; a_2=\tfrac{5}{4},\; a_3=\tfrac{8}{5},\; a_4=\tfrac{11}{6},\; a_5=2; \quad \lim_{n\to\infty}a_n = 3

Answer. The first five terms are 2/3, 5/4, 8/5, 11/6, 2. The sequence converges to 3.

Figure

Diagram

Origins · 1 minAugustin-Louis Cauchy and how this was discovered

Augustin-Louis Cauchy

1789–1857 · French

Cauchy was the first mathematician to place the theory of sequences and series on rigorous foundations. He introduced the precise epsilon–delta definition of a limit and the concept now called a Cauchy sequence — a sequence in which the terms become arbitrarily close to one another — which is the modern criterion for convergence. His 1821 textbook Cours d'analyse essentially created the modern subject of mathematical analysis. Cauchy was extraordinarily prolific, publishing 789 papers — so many that the Paris Academy of Sciences eventually imposed a page limit on submissions specifically because of him.

Arithmetic Sequences — Arithmetic sequences are the simplest family of sequences, characterised by a constant common difference, and provide the foundation for understanding linear growth and arithmetic series.
Geometric Sequences — Geometric sequences, with their constant ratios, lead directly to geometric series whose infinite sums can converge — the key entry point into the theory of infinite series.
Convergence and Divergence of Series — Understanding whether a sequence's terms approach a limit is the prerequisite for deciding whether the corresponding infinite series (sum of terms) converges or diverges.
Sigma Notation — Sigma notation provides the compact language for writing the sum of sequence terms, bridging the concept of a sequence with that of a series.
Domain and Range of Functions — Because a sequence is formally a function with domain ℕ, a solid understanding of functions, domains, and ranges deepens comprehension of what a sequence really is.