Foundations of Sequences

Definition of a Sequence What Is a Sequence? A sequence is an ordered list of objects. The key thing that makes a sequence different from a set is that order matters, and the same element can appear multiple times at different positions. For example, the sequence $(1, 2, 2, 3)$ is different from the sequence $(1, 2, 3, 2)$, even though they contain the same elements. In a set, these would be identical. This ordering property is what makes sequences useful for mathematics—we often care not just about what elements we have, but in what order they appear. The Formal Definition Mathematically, we define a sequence more precisely as a function whose domain is an interval of integers. When we write a sequence, we specify: An index set: the set of allowable integers that tell us which positions exist (often $\mathbb{N} = \{0, 1, 2, 3, \ldots\}$ or $\mathbb{N} = \{1, 2, 3, \ldots\}$) A term at each position: the value assigned to each index This functional view is helpful because it clarifies that a sequence is a rule that assigns to each valid index some value. Notation for Sequences Since sequences are so common in mathematics, we have standard ways to write them. The $n$-th term is written as $an$, where $n$ is the index (the position). For example: $a1$ is the first term $a2$ is the second term $an$ is the $n$-th term We can denote an entire sequence in several ways: As a full list: $a1, a2, a3, \ldots$ (if infinite) With interval notation: $(an){n \in \mathbb{N}}$ means "the sequence where $an$ is the term at position $n$, and $n$ ranges over the natural numbers" With explicit bounds: $(an){n=1}^{10}$ means "the sequence where $n$ runs from $1$ to $10$" Example: The sequence of positive even numbers can be written as $(2n){n \in \mathbb{N}}$ (starting from $n=1$), which represents $2, 4, 6, 8, \ldots$ Length and Indexing A finite sequence has a definite first and last term. A finite sequence of length $n$ is sometimes called an $n$-tuple. For example, $(3, 5, 7, 9)$ is a finite sequence of length 4, or a 4-tuple. An infinite sequence (or singly infinite sequence) has a first term but no last term. It continues forever: $(2, 4, 6, 8, \ldots)$ A bi-infinite sequence extends in both directions and is indexed by all integers $\mathbb{Z}$. We write it as $(\ldots, a{-2}, a{-1}, a0, a1, a2, \ldots)$ The position of each term is called its rank or index. The starting index varies: sometimes we begin counting at $0$, sometimes at $1$, depending on context. Note: The empty sequence, written as $()$, is a finite sequence with no elements. Examples of Sequences The Fibonacci Sequence One of the most famous sequences in mathematics is the Fibonacci sequence: $0, 1, 1, 2, 3, 5, 8, 13, \ldots$ In this sequence, each term (after the first two) equals the sum of the two preceding terms. This sequence appears throughout nature—in spiral patterns of shells, flower petals, and tree branches. Function-Valued Sequences Not all sequences consist of numbers! A sequence can have other mathematical objects as terms, such as functions. For instance, the monomial basis is the sequence $(1, x, x^2, x^3, x^4, \ldots)$, where each term is a polynomial function. <extrainfo> This illustrates the generality of the sequence concept—anything can be a term as long as we can list them in order. </extrainfo> Defining Sequences by Recursion Recurrence Relations Often, we don't define a sequence by giving an explicit formula for the $n$-th term. Instead, we define it recursively using a recurrence relation (or recurrence formula). This is a rule that expresses each term in terms of one or more previous terms. For example, the Fibonacci sequence is defined by: $$Fn = F{n-1} + F{n-2}$$ This recurrence tells us that each term is the sum of the two previous terms. Initial Conditions Here's the crucial point: a recurrence relation alone is not enough to define a sequence. You must also specify initial conditions—enough terms at the start so that the recurrence can generate everything else. For Fibonacci, we specify: $$F0 = 0, \quad F1 = 1$$ Now we can compute all subsequent terms: $F2 = F1 + F0 = 1 + 0 = 1$ $F3 = F2 + F1 = 1 + 1 = 2$ $F4 = F3 + F2 = 2 + 1 = 3$ and so on... Without those initial values, the recurrence relation wouldn't uniquely determine the sequence. Linear Recurrences with Constant Coefficients A particularly important class of recurrence relations are linear recurrences with constant coefficients. These have the general form: $$an = c1 a{n-1} + c2 a{n-2} + \cdots + ck a{n-k}$$ where $c1, c2, \ldots, ck$ are fixed constants. This says: each term is a weighted sum of the previous $k$ terms. Examples: Fibonacci ($k=2$, $c1=1$, $c2=1$): $an = a{n-1} + a{n-2}$ A simple doubling sequence ($k=1$, $c1=2$): $an = 2a{n-1}$ (each term is twice the previous) These linear recurrences are particularly nice because they have well-developed theory for solving them, but that's beyond our current scope. Summary: Sequences are ordered lists where we can have repetitions. They're either finite or infinite, and can be defined either by explicit formulas or by recursive rules. The notation might seem tricky at first, but it becomes natural with practice—always remember that $an$ just means "the term at position $n$."