Patterns and Sequences

Sequences describe growth, schedules, financial payments, algorithms, and discrete change. Moving among words, tables, formulas, and graphs prepares learners for functions.

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

Representations and interpretation

A position-value table displays the sequence as ordered pairs $(n,a_n)$. Arithmetic sequences form collinear points because output changes by a constant amount for each unit increase in position; geometric sequences curve on an ordinary coordinate graph.

Reasoning about variations

A few terms rarely determine a unique infinite rule: $1,2,3$ could continue $4$ or follow a more complicated pattern. A valid model should be supported by the stated context or rule family, not by visual guess alone.

Common mistakes

How to check your work

• Substitute $n=1$ and ensure the formula returns the first term.
• Compare successive outputs with the observed difference or ratio.
• Generate all provided terms from the proposed rule.

Practice

1. Find $a_{15}$ for $2,7,12,17,\ldots$.
2. Give an explicit rule for $3,6,12,24,\ldots$.
3. For $a_n=10-3n$, find the common difference.

Answers and brief solutions

Synthesis and transfer

A staircase built with two additional blocks on each level forms an arithmetic sequence; a recursive rule explains construction while an explicit rule predicts a distant level directly.

If the first staircase uses $4$ blocks and each new level adds $2$, then $a_n=4+2(n-1)$ predicts any level without building all earlier ones. The recursive form $a_n=a_{n-1}+2$ preserves the construction process, so the two formulas answer different practical needs. Constant first differences justify the linear explicit rule; a pattern with constant second differences would instead suggest a quadratic model. Testing the rule at $n=1$ catches the common mistake of writing $4+2n$, which starts at $6$. A useful sequence description includes its starting index because shifting the index changes the formula even when the listed values remain the same.

Related topics

• Relations
• Linear Relations
• Exponents

Teaching and accessibility note