Sequences and Recursive Patterns

Introduction

A sequence is an ordered list of terms. Each term has a position, so changing the order usually changes the sequence. You already meet sequences when you count by equal steps, double a quantity, arrange tiles in growing figures, or follow a rule that tells you what comes next.

A recursive pattern gives one or more starting terms and then explains how to create each new term from earlier terms. For example, start with 4 and add 3 each time: 4, 7, 10, 13, 16, ... . The starting value matters because the instruction "add 3" alone could describe many different sequences.

In this aiMOOC, you will learn to recognize, describe, compare, extend, graph, and create sequences. You will also learn why an explicit rule and a recursive rule can describe the same pattern in different ways. The course is designed for Grades 7–8 and connects arithmetic, algebra, geometry, data representation, and computational thinking.

The graph above is a Wikimedia Commons visualization of an arithmetic sequence. Notice that the terms are shown as separate points because a sequence is indexed by specific positions.

The video introduces the idea of finding rules in number patterns. While you watch, ask yourself: What operation changes one term into the next, and is that operation the same every time?

Learning Goals

By the end of the course, you should be able to identify a sequence from its terms, describe a recursive rule in words and symbols, find missing or later terms, distinguish additive from multiplicative patterns, compare recursive and explicit descriptions, organize a sequence in a table, graph sequence terms as discrete points, and explain your reasoning clearly.

A useful habit is to test a proposed rule against every given transition, not only the first one. Several rules can sometimes fit a short list, so good mathematical explanations state the rule and show why it works.

Understanding Sequences

Terms, Positions, and Notation

A sequence can be written as

2, 5, 8, 11, 14, ...

The first term is 2, the second term is 5, and so on. Mathematicians often use notation such as a₁, a₂, a₃, ... or a_1, a_2, a_3, ... to label the terms. The symbol a_n means "the term in position n."

A table makes the relationship between position and term easy to see:

Position n	Term a_n
1	2
2	5
3	8
4	11
5	14

The position numbers 1, 2, 3, 4, ... form the input values. The sequence terms form the output values. This connection prepares you for work with functions and coordinate graphs.

Look for Changes Between Terms

One of the fastest ways to investigate a sequence is to compare neighboring terms. For 6, 10, 14, 18, ... the change is +4 each time. For 81, 27, 9, 3, ... each term is one third of the previous term.

Do not assume that every pattern uses only addition. A sequence might use subtraction, multiplication, division, alternating operations, or a rule involving two previous terms. Some patterns depend on the position number rather than only on the previous term.

When a sequence is short, more than one rule may be possible. For example, many different formulas could begin 1, 2, 3. In school problems, you usually look for the simplest rule that fits all given information unless the problem states another condition.

Arithmetic Sequences

An arithmetic sequence changes by the same amount from one term to the next. This constant amount is called the common difference.

For the sequence

5, 8, 11, 14, 17, ...

the common difference is 3 because each term is 3 more than the previous term.

A recursive description is:

Start with 5. Add 3 to get each next term.

In symbols, this can be written as:

a_1 = 5 and a_n = a_{n-1} + 3 for later terms.

The part a_{n-1} means "the previous term." The recursive rule tells you how to move forward one step at a time.

This video focuses on arithmetic sequences. As you watch, compare the verbal idea "same change each time" with the algebraic idea of a constant common difference.

Increasing and Decreasing Arithmetic Patterns

The common difference can be positive, zero, or negative.

For 12, 17, 22, 27, ... the common difference is 5.

For 30, 24, 18, 12, ... the common difference is -6.

For 9, 9, 9, 9, ... the common difference is 0.

A decreasing sequence is not automatically geometric. You need to check whether the difference is constant, the ratio is constant, or another rule is operating.

From Recursive to Explicit Rules

A recursive rule is useful when you want to generate terms one after another. An explicit rule gives a term directly from its position.

For the arithmetic sequence 5, 8, 11, 14, ...:

Recursive rule: start at 5 and add 3 each time.

Explicit rule: a_n = 5 + 3(n - 1).

Why does this work? The first term needs zero additions of 3. The second term needs one addition. The nth term needs n - 1 additions.

For an arithmetic sequence with first term a_1 and common difference d, the general pattern is:

a_n = a_1 + (n - 1)d

You do not have to memorize this without understanding it. Rebuilding it from "first term plus repeated equal changes" is often more reliable.

Geometric Sequences

A geometric sequence is formed by multiplying each term by the same nonzero factor. That factor is called the common ratio.

For the sequence

3, 6, 12, 24, 48, ...

the common ratio is 2.

A recursive description is:

Start with 3. Multiply by 2 to get each next term.

In symbols:

a_1 = 3 and a_n = 2a_{n-1} for later terms.

The video introduces geometric sequences. Compare the graph of geometric growth with the more even step-by-step growth of an arithmetic sequence.

Ratios and Repeated Multiplication

To test whether a nonzero sequence is geometric, divide each term by the previous term and check whether the ratio is constant.

For 80, 40, 20, 10, ... the ratio is one half.

For 5, 15, 45, 135, ... the ratio is 3.

Repeated multiplication can create growth or decay much faster than repeated addition. This difference becomes important in later studies of exponential growth.

An explicit rule for the geometric sequence 3, 6, 12, 24, ... is:

a_n = 3 · 2^(n - 1)

Here, the exponent records how many times the common ratio has been used.

Recursive Patterns

A recursive rule has two essential parts: initial information and a rule for producing later terms. If either part is missing, the sequence may not be determined.

Suppose the rule says "multiply the previous term by 2." Starting with 1 gives 1, 2, 4, 8, ... . Starting with 5 gives 5, 10, 20, 40, ... . The same operation creates different sequences because the starting terms differ.

This Khan Academy video compares explicit and recursive definitions. Some notation is more advanced than you need, but the central idea is useful: a recursive rule builds from earlier terms, while an explicit rule jumps directly to a chosen position.

Recursion with More Than One Previous Term

Some recursive sequences depend on two or more earlier terms. The Fibonacci sequence is a famous example. One common version begins:

0, 1, 1, 2, 3, 5, 8, 13, 21, ...

After the two starting terms, each new term is the sum of the previous two:

F_n = F_{n-1} + F_{n-2}

You need two starting terms because the rule looks back two positions. Some books start the Fibonacci sequence with 1, 1 instead of 0, 1; both conventions lead to the familiar continuing terms 1, 2, 3, 5, 8, ...

The video visualizes Fibonacci numbers and other patterns. Use it as a reminder that a sequence can be represented numerically, geometrically, or graphically.

A Visual Fibonacci Construction

A common visual construction places squares with side lengths 1, 1, 2, 3, 5, 8, ... next to one another and draws quarter-circle arcs through them. This creates a Fibonacci spiral approximation. It is a model built from Fibonacci-sized squares; it should not be confused with the exact logarithmic golden spiral.

Patterns related to Fibonacci numbers occur in some arrangements in nature, including spiral counts in certain plants. You should be careful with broad claims: not every natural spiral is a Fibonacci spiral, and seeing a spiral shape alone does not prove a Fibonacci relationship.

The sunflower image above provides a real-world prompt for investigation. Instead of simply saying "nature uses Fibonacci," count visible spiral families, record your method, and compare your observations with classmates.

Other Growing Patterns

Not every sequence is arithmetic or geometric. Consider the square numbers:

1, 4, 9, 16, 25, 36, ...

The first differences are 3, 5, 7, 9, 11, ... . These differences are not constant, so the sequence is not arithmetic. However, the differences themselves increase by a constant amount. Such patterns lead to quadratic sequences, which you will study more deeply in later algebra.

Another common type is an alternating pattern. For example:

2, 5, 3, 6, 4, 7, ...

One way to describe this is to alternate +3 and -2. When a rule alternates, write the repeating cycle clearly and test it across several transitions.

Shape Patterns and Tables

Sequences are not only lists of numbers. Suppose a growing tile figure uses 4 tiles in Stage 1, 7 tiles in Stage 2, 10 tiles in Stage 3, and 13 tiles in Stage 4. The tile counts form an arithmetic sequence with common difference 3.

A table can connect a diagram to algebra:

Stage	Number of tiles	Change from previous stage
1	4	Starting stage
2	7	Add 3
3	10	Add 3
4	13	Add 3

When you build or draw the figures, identify which tiles stay the same and which tiles are added at each step. That visual structure often explains the recursive rule.

Representing Sequences on a Graph

A sequence can be graphed by plotting each position and its term value as an ordered pair. For 2, 5, 8, 11, ... you plot:

(1, 2), (2, 5), (3, 8), (4, 11), ...

The horizontal coordinate shows the term position, and the vertical coordinate shows the term value. Because the sequence is defined only at its term positions, the graph is usually shown as separate points rather than one continuous line.

Arithmetic sequences produce points that line up along a straight line when position is plotted against term value. Geometric sequences generally create a curved pattern because equal multiplication does not create equal vertical changes.

Graphing helps you compare growth rates. Two sequences may begin with similar values but separate quickly if one grows by repeated addition and the other by repeated multiplication.

This video connects a geometric sequence in recursive form to its graph. Focus on how each term becomes one plotted point.

Solving Sequence Problems

A strong strategy is to move through four questions.

What changes? Compute differences, ratios, or another relationship between terms.

Is the change consistent? Test the rule across every transition you were given.

What information starts the sequence? State the first term, or the required first several terms.

Can you explain and verify? Generate several terms from your rule and compare them with the original sequence.

For example, consider 7, 12, 17, 22, ...

The change is +5 each time, so the sequence is arithmetic. A recursive rule is "start with 7 and add 5." The next terms are 27 and 32. An explicit rule is a_n = 7 + 5(n - 1).

Now consider 2, 6, 18, 54, ...

The ratio is 3 each time, so the sequence is geometric. A recursive rule is "start with 2 and multiply by 3." The next term is 162.

Missing Terms

For a missing-term problem such as

14, 19, __, 29, 34

the visible differences suggest +5, so the missing term is 24.

For

4, 12, __, 108

the ratio suggests ×3, so the missing term is 36.

Always verify the completed sequence. A single guess based only on the first pair of terms is not enough.

Comparing Two Rules

Suppose Sequence A starts at 10 and adds 4 each step, while Sequence B starts at 2 and doubles each step.

Position	Sequence A	Sequence B
1	10	2
2	14	4
3	18	8
4	22	16
5	26	32

At first, Sequence A is larger. By Position 5, Sequence B is larger. This comparison shows why you should not judge long-term behavior from only one or two terms.

Recursive Thinking and Computing

Recursion is also an idea in computer science. A program can generate a sequence by storing earlier terms and repeatedly applying a rule. A simple loop can do the same job for many school-level patterns.

For example, a program could start with 5, display it, add 3, display the new value, and repeat. For Fibonacci numbers, the program needs to keep track of two previous values before creating the next one.

The mathematical skill is the same: clearly identify the starting data, the repeated rule, and the stopping condition. This is a useful bridge between algebra and algorithms.

Common Errors and How to Fix Them

Error: Giving only the operation. "Add 4" is incomplete as a recursive definition because it does not state a starting term. Fix it by saying, for example, "Start with 3 and add 4 each time."

Error: Checking only one step. A rule that explains 2 to 6 may fail for 6 to 11. Test every available transition.

Error: Confusing difference and ratio. For 3, 6, 12, 24, the differences change but the ratio remains 2. Use subtraction to test arithmetic patterns and division to test geometric patterns.

Error: Treating a sequence graph as continuous. A sequence usually has isolated term positions. Unless a context says otherwise, do not assume values between positions belong to the sequence.

Error: Assuming a pattern is unique. A short list can fit many possible rules. State the rule you are using and why it is reasonable in context.

Interactive Tasks

Quiz: Test Your Knowledge

What makes a sequence arithmetic? (The same amount is added or subtracted each step) (!The same number is squared each step) (!The terms must always increase) (!Every term must be even)

What information is required for a recursive rule? (A starting value and a rule for later terms) (!Only the final term) (!Only a graph title) (!A list with no rule)

Which rule describes 4 9 14 19? (Start with 4 and add 5 each time) (!Start with 4 and multiply by 5 each time) (!Start with 4 and subtract 5 each time) (!Start with 4 and add 4 each time)

Which sequence is geometric? (3 6 12 24) (!3 6 9 12) (!2 5 9 14) (!10 9 7 4)

What is the common difference in 18 13 8 3? (Negative five) (!Positive five) (!Negative three) (!Positive three)

What is the next term in 2 6 18 54? (162) (!108) (!72) (!58)

How is each new Fibonacci term formed after the starting terms? (Add the two previous terms) (!Multiply the two previous terms) (!Subtract one from the previous term) (!Double the first term)

Why are sequence graphs usually shown as separate points? (The sequence is defined at specific term positions) (!All sequences have negative values) (!Graphs cannot show addition) (!Every sequence has only four terms)

What is an advantage of an explicit rule? (It can give a chosen term directly from its position) (!It removes the need for a first term in every pattern) (!It guarantees that all sequences are arithmetic) (!It changes a geometric sequence into an arithmetic one)

Which action best checks a proposed sequence rule? (Test the rule across all given transitions) (!Use only the first two terms) (!Choose the rule with the longest wording) (!Ignore any term that does not fit)

Memory Game

Sequence	An ordered list of terms
Recursion	A method that builds later terms from earlier ones
Difference	The amount found by subtracting consecutive terms
Ratio	The factor found by dividing consecutive nonzero terms
Explicit	A rule that gives a term from its position
Fibonacci	A pattern in which each later term is the sum of the previous two

Drag and Drop

Match the correct terms.	Topic
Arithmetic pattern	Same additive change from term to term
Geometric pattern	Same multiplicative factor from term to term
Recursive rule	Uses earlier terms to produce later terms
Explicit rule	Uses position to find a term directly
Discrete graph	Shows separate points for sequence positions

...

Crossword Puzzle

Sequence	What is an ordered list of mathematical terms called?
Recursive	What word describes a rule that builds a new term from earlier terms?
Arithmetic	What kind of sequence has a constant difference?
Geometric	What kind of sequence has a constant ratio?
Difference	What do you call the constant additive change in an arithmetic sequence?
Fibonacci	Which famous sequence adds the previous two terms to make the next?

LearningApps

Cloze Text

A mathematical

is an ordered list of terms. An arithmetic sequence has a constant

between consecutive terms. A geometric sequence has a constant

between consecutive nonzero terms. A recursive description needs at least one

value as well as a rule for later terms. An explicit rule uses the term's

to calculate its value directly. In the Fibonacci sequence, each later term is the

of the previous two terms. When a sequence is graphed, its terms are usually shown as

points. A good rule should be tested against

given transition.

Open-Ended Tasks

Easy

Sequence Hunt: Find three sequences in your school day, home, games, schedules, or designs; record at least six terms for each and explain what makes each example a sequence.
Pattern Poster: Create a one-page poster that shows one arithmetic pattern with numbers, arrows, a recursive rule, and a small graph.
Human Sequence: Work with classmates to act out a recursive rule in which each person receives a value, applies the rule, and passes the next value along; record where mistakes are most likely to happen.
Photo Pattern: Take or create a photo of a growing visual pattern, label four stages, count the objects in each stage, and describe the number sequence in clear English.

Standard

Sequence Interview: Interview a family member, teacher, craftsperson, coder, musician, or athlete about a repeated pattern used in their work or hobby; translate the pattern into a sequence and explain where the model fits or fails.
Arithmetic and Geometric Video: Produce a two-minute teaching video that compares an additive pattern with a multiplicative pattern using your own examples, tables, and spoken explanations.
Recursive Spreadsheet: Use a spreadsheet to generate at least twenty terms of a recursive sequence, graph the results, and explain how changing the starting value changes the pattern.
Fibonacci Investigation: Examine a pine cone, sunflower image, leaf arrangement, or another safe natural object; count visible structures carefully, document your method with images or sketches, and explain whether the evidence supports a Fibonacci-related claim.

Advanced

Rule Comparison Project: Create two sequences that begin with similar values but follow different recursive rules; find when one overtakes the other and justify your conclusion with a table, graph, and algebra.
Multiple Rules Challenge: Start with a short list of four terms and invent two different mathematically valid rules that reproduce those terms; extend both rules and explain why a short pattern does not always determine a unique future.
Algorithm Design: Write pseudocode or a simple program that generates an arithmetic, geometric, or Fibonacci-type sequence; test it with several starting values and explain the role of initialization and repetition.
Mathematics Field Visit: Visit a library, science museum, technology center, garden, architecture site, or mathematics display where repeated patterns can be observed; collect notes or permitted images and create a report that separates measured evidence from assumptions.

Learning Assessment

Model a Savings Plan: A student saves the same extra amount each week; build a recursive rule and an explicit rule for the total, explain the meaning of each part, and compare which rule is more efficient for finding Week 20.
Compare Growth Rules: Given one arithmetic and one geometric sequence with different starting values, determine when their order changes, support your conclusion with a table or graph, and explain why the faster-growing rule eventually dominates in your example.
Diagnose a Mistake: A classmate says that 4, 8, 12, 16 is geometric because the numbers increase regularly; identify the error, show the correct classification, and describe a reliable test for future problems.
Design from a Diagram: Create or analyze a growing tile figure, record the first five term values, write a recursive rule, and justify how the visible growth in the figure matches the numerical rule.
Translate Representations: Start from a recursive rule and produce a term list, a table, a graph, and a verbal explanation; then identify what information is easiest to see in each representation.
Evaluate a Pattern Claim: Investigate a claim that a real object follows a Fibonacci or other recursive pattern, collect measurable evidence, and write a conclusion that clearly distinguishes observation from interpretation.

Evidence of Learning

Important knowledge includes understanding sequence position, recursive rules, starting values, common differences, common ratios, explicit rules, Fibonacci-type recursions, and discrete sequence graphs.

Important skills include calculating differences and ratios, extending patterns, writing and interpreting recursive rules, converting between words, tables, formulas, and graphs, checking a rule against evidence, and explaining why a classification is valid.

Useful products include a correctly labeled sequence table, a graph with discrete points, a visual pattern model, a short teaching video, a spreadsheet or program that generates terms, and a written comparison of two sequence rules.

Strong transfer is shown when you can recognize sequence structures in a new context, decide which mathematical model is reasonable, use technology appropriately, and state the limits of your model instead of forcing every pattern into an arithmetic, geometric, or Fibonacci category.

OERs on the Topic

The English Wikipedia article on mathematical sequences provides a broad reference for the idea of ordered terms, notation, and many types of sequences.

Linked Learning Areas

Sequences connect arithmetic patterns to algebraic rules, graphs, functions, geometric designs, data tables, and algorithms. Understanding how a repeated process creates later values prepares you for linear functions, exponential growth, mathematical modeling, and programming.

Sequences and Recursive Patterns

aiMOOC Projects