Pattern
Pre-AlgebraA 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.
Example
$2, 5, 8, 11, 14, \ldots$ Each number is $3$ more than the one before. The rule is "add $3$." The next term would be $17$.
Key Insight
Patterns are everywhere in nature and math. Recognizing and describing them precisely is the first step toward writing algebraic formulas.
Definition
A mathematical pattern is a sequence or arrangement governed by a consistent rule. Common types include arithmetic patterns (constant difference), geometric patterns (constant ratio), and other algebraic rules. The nth term can often be expressed as a formula.
Example
The pattern $1, 4, 9, 16, 25$ follows the rule $a_n = n^2$. The pattern $3, 6, 12, 24$ follows the rule $a_n = 3 \cdot 2^{n-1}$ (geometric, ratio $2$).
Key Insight
Finding a formula for the $n$th term is far more powerful than listing terms one by one. It allows you to jump directly to any term, like finding the $100$th term without computing all $99$ before it.
Definition
A sequence is a function $a: \mathbb{N} \to \mathbb{R}$ (or another set). A pattern is a sequence defined by a recurrence relation (e.g., $a_{n+1} = a_n + d$) or a closed-form formula (e.g., $a_n = a_1 + (n-1)d$). Generating functions encode entire sequences as coefficients of a power series, enabling powerful analysis via complex analysis and combinatorics.
Example
The Fibonacci sequence is defined by the recurrence $a_n = a_{n-1} + a_{n-2}$, with $a_1 = a_2 = 1$. 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
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.