Geometric Sequences

Introduction

A geometric sequence is a number pattern in which each term is obtained by multiplying the previous term by the same constant factor. That factor is called the common ratio. Geometric sequences are an important bridge between sequences, ratios, exponents, and exponential functions.

For example, in the sequence 3, 6, 12, 24, 48, ... each term is twice the previous term, so the common ratio is 2. In the sequence 160, 80, 40, 20, ... each term is half the previous term, so the common ratio is $\frac{1}{2}$ .

This course is designed for Grades 9–10. You will learn to recognize geometric sequences, calculate their terms, write recursive and explicit rules, compare them with arithmetic sequences, graph them, and use them to model repeated percentage change.

The graph above represents a geometric sequence based on powers of 2. Notice how the terms are separate points: a sequence is defined at whole-number term positions.

Learning Goals

By the end of this course, you should be able to:

Recognize geometric sequences: Decide whether a pattern has a constant multiplicative ratio.
Find the common ratio: Calculate the factor connecting consecutive nonzero terms.
Write explicit rules: Use $a_{n} = a_{1} r^{n - 1}$ to find any term directly.
Write recursive rules: Describe each term using the previous term.
Interpret growth and decay: Explain how the size and sign of the common ratio affect a sequence.
Model situations: Represent compound growth, depreciation, repeated scaling, and similar processes.

Prerequisites

You should already be comfortable with integers, fractions, ratios, percentages, and basic exponents. A calculator or spreadsheet is useful for exploring longer sequences, but the main ideas can be understood by hand.

What Makes a Sequence Geometric?

A sequence is geometric when the same multiplication rule connects each term to the next. If the terms are nonzero, you can test this by dividing each term by the one before it. The quotients must be equal.

For the sequence 5, 15, 45, 135, ...:

$\frac{15}{5} = 3, \frac{45}{15} = 3, \frac{135}{45} = 3.$

Because every quotient is 3, the sequence is geometric with common ratio $r = 3$ .

Now compare 5, 8, 11, 14, ... . The terms increase by 3 each time, but the ratios are not constant. This is an arithmetic sequence, not a geometric one.

Multiplication, Not Addition

A geometric sequence is controlled by a constant ratio. An arithmetic sequence is controlled by a constant difference.

Feature	Arithmetic sequence	Geometric sequence
Repeated operation	Add or subtract a constant	Multiply by a constant
Key value	Common difference	Common ratio
Typical rule	$a_{n} = a_{1} + (n - 1) d$	$a_{n} = a_{1} r^{n - 1}$
Typical growth pattern	Linear	Exponential

The image compares linear and exponential growth. A geometric sequence with a positive ratio greater than 1 gives discrete points that follow an exponential growth pattern.

Finding the Common Ratio

For consecutive nonzero terms $a_{n}$ and $a_{n + 1}$ , calculate:

$r = \frac{a_{n + 1}}{a_{n}} .$

Example: 81, 27, 9, 3, ...

$r = \frac{27}{81} = \frac{1}{3} .$

Check another pair:

$\frac{9}{27} = \frac{1}{3} .$

The same ratio confirms the pattern.

A negative ratio causes signs to alternate. For example, 2, -6, 18, -54, ... has $r = - 3$ .

Recursive and Explicit Rules

There are two common ways to describe a geometric sequence.

Recursive Rule

A recursive rule tells you the first term and how to get each new term from the previous one:

$a_{1} = first term, a_{n} = r a_{n - 1} for n \geq 2.$

For 4, 12, 36, 108, ...:

$a_{1} = 4, a_{n} = 3 a_{n - 1} .$

This rule mirrors the process of generating the sequence, but to find a distant term you must usually pass through the earlier terms.

Explicit Rule

An explicit rule lets you calculate a term directly:

$a_{n} = a_{1} r^{n - 1} .$

The exponent is $n - 1$ because moving from the first term to the nth term requires $n - 1$ multiplications by $r$ .

For 4, 12, 36, 108, ...:

$a_{n} = 4 \cdot 3^{n - 1} .$

To find the sixth term:

$a_{6} = 4 \cdot 3^{5} = 4 \cdot 243 = 972.$

Why the Formula Works

Start with $a_{1}$ . Multiplying once gives $a_{2} = a_{1} r$ . Multiplying twice gives $a_{3} = a_{1} r^{2}$ . Continuing the pattern gives:

$a_{1}, a_{1} r, a_{1} r^{2}, a_{1} r^{3}, \dots$

Therefore the nth term is $a_{n} = a_{1} r^{n - 1}$ .

Working with Geometric Sequences

Example: Find a Later Term

Consider 7, 14, 28, 56, ...

The first term is $a_{1} = 7$ and the ratio is $r = 2$ . The explicit rule is:

$a_{n} = 7 \cdot 2^{n - 1} .$

The ninth term is:

$a_{9} = 7 \cdot 2^{8} = 7 \cdot 256 = 1792.$

Example: Find a Missing Term

Suppose a geometric sequence has a positive common ratio and begins 4, __, 36, ... . Let the missing term be $x$ . Then the same ratio must work twice:

$\frac{x}{4} = \frac{36}{x} .$

So $x^{2} = 144$ . Because the common ratio is stated to be positive, $x = 12$ . The sequence begins 4, 12, 36, ... with $r = 3$ .

Without the condition that the ratio is positive, $x = - 12$ would also produce a geometric sequence: 4, -12, 36, ... with $r = - 3$ .

Example: Recover the Rule from Two Terms

Suppose $a_{3} = 12$ and $a_{6} = 96$ . Because three ratio steps separate the third and sixth terms:

$a_{6} = a_{3} r^{3} .$

So:

$96 = 12 r^{3}, r^{3} = 8, r = 2.$

Then $a_{3} = a_{1} r^{2}$ , so:

$12 = 4 a_{1}, a_{1} = 3.$

The explicit rule is $a_{n} = 3 \cdot 2^{n - 1}$ .

Example: Work Backward

If a geometric sequence has $a_{5} = 160$ and $r = 2$ , divide by 2 to move backward:

160, 80, 40, 20, 10.

Thus $a_{1} = 10$ . This is equivalent to solving:

$160 = a_{1} \cdot 2^{4} .$

How the Common Ratio Changes the Pattern

The value of $r$ controls both magnitude and sign.

Condition	Typical behavior	Example
$r > 1$	Positive terms grow in magnitude	2, 6, 18, 54, ...
$0 < r < 1$	Positive terms decay toward zero	80, 40, 20, 10, ...
$r = 1$	Terms stay constant	5, 5, 5, 5, ...
$r < 0$	Signs alternate	3, -6, 12, -24, ...
$\| r \| > 1$	Magnitudes grow	1, -2, 4, -8, ...
$0 < \| r \| < 1$	Magnitudes shrink toward zero	16, -8, 4, -2, ...

A useful habit is to separate two questions: What happens to the size of the terms? and What happens to their signs? The absolute value $| r |$ controls magnitude, while the sign of $r$ controls whether signs alternate.

Geometric Sequences and Exponential Functions

For a positive common ratio, the explicit rule for a geometric sequence has the same algebraic structure as an exponential function:

$a_{n} = a_{1} r^{n - 1} .$

The important difference is the domain. In a sequence, $n$ usually takes positive integer values. An exponential function can be defined for a continuous range of real inputs when its base is positive.

If you graph the terms of a geometric sequence, you get separate points. If you extend the same positive-base rule to real inputs, those points lie on an exponential curve.

This connection helps you move between discrete models and continuous-looking graphs.

Real-World Models

Geometric sequences appear whenever a quantity is repeatedly multiplied by the same factor over equal time intervals or stages.

Repeated Percentage Growth

If a quantity increases by 5% each period, it keeps 100% of its previous value and gains another 5%. The multiplier is therefore:

$1 + 0.05 = 1.05.$

Starting with 1000 units at time 0 gives:

1000, 1050, 1102.5, 1157.625, ...

Using time index $n = 0, 1, 2, \dots$ , the model is:

$A_{n} = 1000 (1.05)^{n} .$

This is the mathematics behind simplified models of compound growth.

Repeated Percentage Decrease

If a machine loses 20% of its value each year, it retains 80%:

$r = 1 - 0.20 = 0.80.$

A machine initially worth 15,000 units can be modeled by:

$V_{n} = 15000 (0.8)^{n} .$

After three years:

$V_{3} = 15000 (0.8)^{3} = 7680.$

This is a simplified depreciation model. Real prices may behave differently, so always check whether a constant percentage is reasonable for the situation.

Bouncing Ball Model

Suppose a ball reaches 70% of its previous maximum height after each bounce. If the first recorded height is 2 meters, the heights form:

2, 1.4, 0.98, 0.686, ...

The common ratio is $0.7$ . This type of experiment lets you test whether real measurements are close to an ideal geometric model.

Fractals and Repeated Construction

In the Sierpiński triangle, each stage can be built by keeping three smaller copies of the previous pattern. If stage 0 contains one kept triangle, the number of kept triangles follows:

1, 3, 9, 27, 81, ...

This is a geometric sequence with ratio 3.

The lengths in a fractal may shrink while the number of repeated pieces grows. Geometric sequences can describe either kind of repeated scaling.

Common Mistakes and How to Avoid Them

Mistake 1: Looking only at differences. A sequence such as 2, 6, 18, 54 does not have a constant difference, but it does have a constant ratio.

Mistake 2: Using $r^{n}$ instead of $r^{n - 1}$ . The first term needs exponent 0 so that $a_{1} r^{0} = a_{1}$ .

Mistake 3: Forgetting negative ratios. A geometric sequence may alternate signs.

Mistake 4: Treating every exponential-looking context as exact. Real data can be approximately geometric without matching one fixed ratio perfectly.

Mistake 5: Confusing a sequence with a series. A sequence lists terms; a series adds terms.

Extension: Geometric Series

A geometric series is the sum of the terms of a geometric sequence. This topic is often studied after geometric sequences and is included here as an extension.

For the first $n$ terms of a geometric sequence with $r \neq 1$ :

$S_{n} = a_{1} \frac{1 - r^{n}}{1 - r} .$

One way to understand the formula is to write:

$S_{n} = a_{1} + a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n - 1}$

and then multiply by $r$ :

$r S_{n} = a_{1} r + a_{1} r^{2} + \dots + a_{1} r^{n} .$

Subtracting cancels the middle terms:

$(1 - r) S_{n} = a_{1} (1 - r^{n}) .$

When $| r | < 1$ , the terms shrink toward zero and the infinite geometric series has the sum:

$S_{\infty} = \frac{a_{1}}{1 - r} .$

The animation below shows partial sums approaching a limiting value when the common ratio has magnitude less than 1.

Worked Review

Use these examples to connect the main ideas.

Identify the ratio: In 6, 18, 54, 162, ... the ratio is 3 because each term is three times the previous term.
Write the rule: With $a_{1} = 6$ and $r = 3$ , the rule is $a_{n} = 6 \cdot 3^{n - 1}$ .
Find a term: $a_{5} = 6 \cdot 3^{4} = 486$ .
Interpret decay: In 200, 100, 50, 25, ... the ratio is $\frac{1}{2}$ , so the terms halve each step.
Interpret a negative ratio: In 5, -10, 20, -40, ... the ratio is -2, so signs alternate and magnitudes double.

Interactive Tasks

Quiz: Test Your Knowledge

Which statement best defines a geometric sequence? (Each term is found by multiplying the previous term by the same factor) (!Each term is found by adding the same amount) (!Every term must be positive) (!Every term must be an integer)

What is the common ratio of 4, 12, 36, 108? (3) (!2) (!4) (!8)

What is the common ratio of 80, 40, 20, 10? (One half) (!Two) (!Minus one half) (!Forty)

Which formula gives the nth term of a geometric sequence? (a sub n equals a sub one times r to the power n minus one) (!a sub n equals a sub one plus r times n) (!a sub n equals a sub one times n plus r) (!a sub n equals r plus n minus one)

For a geometric sequence with first term 5 and ratio 2, what is the fourth term? (40) (!20) (!30) (!80)

Which sequence has a negative common ratio? (3, -6, 12, -24) (!3, 6, 12, 24) (!24, 12, 6, 3) (!5, 10, 15, 20)

What usually happens to positive terms when the common ratio is between zero and one? (They decrease toward zero) (!They increase without bound) (!They alternate signs) (!They increase by a constant difference)

A value increases by 8 percent each year. Which multiplier models one year of growth? (1.08) (!0.08) (!0.92) (!8)

How does an arithmetic sequence differ from a geometric sequence? (It uses a constant difference instead of a constant ratio) (!It always decreases) (!It cannot contain fractions) (!It has no formula)

What is a geometric series? (The sum of terms from a geometric sequence) (!A graph of a geometric sequence) (!A geometric sequence with only positive terms) (!A sequence created by constant addition)

Memory Game

Common ratio	Fixed multiplier connecting consecutive terms
First term	Starting value of a sequence
Explicit formula	Rule that finds a term directly from its position
Recursive rule	Rule that uses an earlier term to generate the next
Exponential growth	Pattern in which magnitude repeatedly increases by a fixed factor
Exponential decay	Pattern in which magnitude repeatedly shrinks by a fixed factor

Drag and Drop

Match the correct terms.	Topic
Constant multiplication	Geometric sequence
Constant addition	Arithmetic sequence
Magnitude grows	Absolute ratio greater than one
Magnitude shrinks	Absolute ratio between zero and one
Signs alternate	Negative common ratio

...

Crossword Puzzle

Ratio	What fixed multiplier connects consecutive terms in a geometric sequence?
Sequence	What ordered list of values is studied in this course?
Recursive	What kind of rule uses the previous term to find the next?
Explicit	What kind of rule calculates a term directly from its position?
Growth	What word describes increasing magnitude when a positive ratio exceeds one?
Decay	What word describes shrinking magnitude when a positive ratio lies between zero and one?

LearningApps

Cloze Text

A geometric sequence is created by repeatedly multiplying by the same

. The first term is usually written as

. The explicit formula uses an exponent of

. A recursive rule calculates a new term from the

. When a positive common ratio is greater than one, the sequence shows

. When a positive common ratio lies between zero and one, the terms move toward

. A negative common ratio makes the signs

. Repeated percentage growth can be modeled by converting the percentage change into a

. A geometric series is formed by

the terms of a geometric sequence.

Open-Ended Tasks

Easy

Sequence Photo Hunt: Find or create three visual patterns that could represent repeated multiplication, photograph or sketch them, and explain the possible common ratio in each case.
Rule Card: Design a one-page reference card showing the meaning of first term, common ratio, recursive rule, and explicit rule with one original example.
Pattern Interview: Ask a classmate to invent a four-term geometric sequence, identify its common ratio, and write a short explanation of how you checked it.
Mini Video Explanation: Record a one-minute video teaching how to decide whether a sequence is geometric, using one example and one non-example.

Standard

Bouncing Ball Experiment: Drop a ball from a measured height, record several rebound heights, calculate consecutive ratios, and discuss whether a geometric model is reasonable.
Savings Model: Create a spreadsheet for an imaginary account that grows by a fixed percentage each year, then graph the values and explain how the common ratio appears in the model.
Depreciation Interview: Interview an adult or local business worker about how an item such as a phone, bicycle, or machine loses value, then compare their description with a fixed-percentage geometric model.
Sierpinski Poster: Construct several stages of a Sierpiński triangle, record the number of kept triangles at each stage, and explain the resulting geometric sequence.

Advanced

Model Comparison Project: Collect a small real dataset that changes over equal time intervals, compare an arithmetic model with a geometric model, and justify which model better represents the data.
Reverse Engineering Challenge: Create a geometric sequence from two nonconsecutive terms, show how to recover possible common ratios, and explain when more than one real solution can occur.
Geometric Series Investigation: Use diagrams or a spreadsheet to explore partial sums for several ratios with magnitude below and above one, then explain which cases appear to converge.
Mathematical Communication Project: Produce a three-minute tutorial, infographic, or article that connects geometric sequences to exponential functions and one real-world application, including assumptions and limitations.

Learning Assessment

Model Selection: Given several real-world scenarios, decide which can reasonably be represented by a geometric sequence and justify each choice by identifying or rejecting a constant multiplicative factor.
Error Analysis: Analyze a worked solution that incorrectly uses $a_{n} = a_{1} r^{n}$ , explain the indexing error, and correct the solution.
Multiple Representations: Translate one geometric sequence among a table, recursive rule, explicit rule, graph, and verbal description, explaining how the common ratio appears in each representation.
Parameter Reasoning: Compare two geometric sequences with different first terms and ratios, determine when one becomes larger than the other, and support your conclusion with calculations or a graph.
Data Modeling: Use measured or supplied data to estimate a common ratio, construct a geometric model, test predictions, and discuss why real observations may not fit the model exactly.
Transfer Problem: Design and solve a new problem involving repeated percentage change, clearly define the starting value, time step, multiplier, formula, and interpretation of the result.

Evidence of Learning

Strong evidence of learning includes more than recalling a formula. You should be able to show connected knowledge, accurate procedures, mathematical communication, and transfer to unfamiliar situations.

Conceptual understanding: You can explain why a constant ratio defines a geometric sequence and how this differs from a constant difference.
Procedural fluency: You can calculate common ratios, missing terms, explicit rules, recursive rules, and selected terms accurately.
Representation: You can connect sequences with tables, graphs, equations, and verbal descriptions.
Mathematical modeling: You can identify repeated percentage change, select a meaningful time step, build a geometric model, and interpret its parameters.
Reasoning: You can explain the effect of positive, fractional, and negative ratios on magnitude and sign.
Product creation: You can produce a clear graph, poster, spreadsheet, report, experiment record, or video that uses geometric sequences correctly.
Transfer: You can apply the ideas to a new situation and state assumptions or limitations of the model.

OERs on the Topic

The English Wikipedia article on geometric progressions provides background, formulas, properties, and links to related topics:

For further openly accessible study, you can also use:

OpenStax: Geometric Sequences: A textbook explanation with examples and exercises.
OpenStax Algebra 1: Define a Geometric Sequence by the nth Term: Practice with recursive and explicit rules.
Wikimedia Commons: Geometric series media: Openly licensed diagrams and animations related to geometric sequences and series.

Linked Learning Areas

The main ideas connect naturally to ratios, exponent laws, functions, algebraic modeling, finance, and fractals.

Geometric Sequences

aiMOOC Projects

Diesen aiMOOC als Battle spielen: Mathematik, Klasse 9-10, G-Niveau

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