Patterns and Sequences

Intuition and core definition

A sequence is an ordered list whose terms follow a rule. A pattern is the regularity used to generate or relate those terms. An explicit rule finds a term directly from its position; a recursive rule gives starting term(s) and tells how to produce later terms.

Notation, language, and conditions

$a_n$ denotes the term at position $n$. An arithmetic sequence has constant difference $d$ and formula $a_n=a_1+(n-1)d$. A geometric sequence has constant ratio $r$ and formula $a_n=a_1r^{n-1}$. Position usually begins at $n=1$ unless stated otherwise.

Why this idea matters

A sequence encodes ordered change, and identifying whether differences, ratios, or another rule remain stable determines an efficient model.

A dependable method

1. List term numbers beside values and compute first differences or ratios.
2. If differences are constant, use an arithmetic model; if ratios are constant, consider geometric.
3. Determine the initial term and write an explicit or recursive rule with its domain.
4. Test the rule against several given terms, not only the next one.
5. Use the rule to predict and explain the requested term.

Worked example

Common mistakes