Pattern
A pattern is a sequence of numbers, shapes, or events that follow a consistent rule, allowing the next term to be predicted.
Definition
A pattern is a sequence that repeats or grows by following a consistent rule; once you find the rule, you can predict any term in the sequence. Common types include arithmetic patterns (constant difference), geometric patterns (constant ratio), and other algebraic rules, and the $n$th term can often be expressed as a formula, which is far more powerful than listing terms one by one because it lets you jump directly to any term. Formally, a sequence is a function $a: \mathbb{N} \to \mathbb{R}$ (or another set), and a pattern is a sequence defined by a recurrence relation (such as $a_{n+1} = a_n + d$) or a closed-form formula (such as $a_n = a_1 + (n-1)d$); generating functions encode entire sequences as coefficients of a power series, enabling powerful analysis via complex analysis and combinatorics.
Example
$2, 5, 8, 11, 14, \ldots$: each number is $3$ more than the one before, so the rule is "add $3$" and the next term would be $17$. The pattern $1, 4, 9, 16, 25$ follows the rule $a_n = n^2$, and $3, 6, 12, 24$ follows $a_n = 3 \cdot 2^{n-1}$ (geometric, ratio $2$). The Fibonacci sequence is defined by the recurrence $a_n = a_{n-1} + a_{n-2}$ with $a_1 = a_2 = 1$, and its closed form (Binet's formula) is $a_n = (\varphi^n - \psi^n)/\sqrt{5}$, where $\varphi = (1 + \sqrt{5})/2$ is the golden ratio.
Key Insight
Patterns are everywhere in nature and math, and recognizing and describing them precisely is the first step toward writing algebraic formulas. Generating functions transform sequence analysis into function analysis: the Fibonacci generating function $F(x) = x/(1 - x - x^2)$ encodes all Fibonacci numbers as coefficients, and partial fraction decomposition of $F(x)$ yields Binet's formula.