Course Outline
Course Overview
Sequences and series allow mathematicians to recognize patterns, describe change, and make predictions. From financial investments and population growth to computer science and natural phenomena, patterns are everywhere. In this course, students will learn how to identify, analyze, and model numerical patterns using sequences and series. Through arithmetic and geometric relationships, recursive thinking, and real-world applications, students will develop a deeper understanding of how mathematics can be used to describe growth, change, and prediction.
Unit 1: Understanding Sequences
Develop the foundational skills needed to recognize and describe mathematical patterns.
1.1 What Is a Sequence?
- Terms of a sequence
- Ordered patterns
- Sequence notation
1.2 Extending Numerical Patterns
- Identifying pattern rules
- Predicting future terms
- Recognizing trends
1.3 Recursive Rules
- Defining terms from previous terms
- Recursive notation
- Building recursive sequences
1.4 Explicit Rules
- Using term numbers
- Developing formulas
- Finding distant terms efficiently
1.5 Sequences in Nature and Society
- Population patterns
- Natural growth patterns
- Everyday applications
Unit 2: Arithmetic Sequences and Series
Explore patterns that change by a constant amount.
2.1 Arithmetic Sequences
- Common differences
- Generating terms
- Recognizing arithmetic patterns
2.2 The nth Term Formula
- Developing explicit formulas
- Using algebraic notation
- Predicting sequence values
2.3 Graphing Arithmetic Sequences
- Discrete graphs
- Linear patterns
- Connections to linear functions
2.4 Arithmetic Series
- Summing sequence terms
- Finite series
- Real-world applications
2.5 Sum Formulas for Arithmetic Series
- Using arithmetic series formulas
- Efficient calculations
- Problem-solving applications
Unit 3: Geometric Sequences and Series
Investigate patterns that grow or shrink by a constant factor.
3.1 Geometric Sequences
- Common ratios
- Generating terms
- Identifying geometric patterns
3.2 The nth Term Formula
- Exponential relationships
- Explicit formulas
- Predicting future values
3.3 Graphing Geometric Sequences
- Exponential growth
- Exponential decay
- Comparing arithmetic and geometric patterns
3.4 Geometric Series
- Finite geometric sums
- Repeated multiplication patterns
- Applications
3.5 Applications of Geometric Growth
- Population growth
- Compound interest
- Scientific modeling
Unit 4: Special Sequences and Mathematical Patterns
Explore famous mathematical sequences and discover connections across mathematics.
4.1 The Fibonacci Sequence
- Recursive growth patterns
- Historical background
- Natural occurrences
4.2 Pascal's Triangle
- Constructing Pascal's Triangle
- Number relationships
- Connections to probability
4.3 Triangular and Square Numbers
- Figurate numbers
- Visual models
- Pattern exploration
4.4 Recursive Models
- Population models
- Financial models
- Repeated processes
4.5 Patterns Across Mathematics
- Connections to algebra
- Connections to geometry
- Connections to probability
Unit 5: Applications of Sequences and Series
Apply sequence concepts to solve meaningful real-world problems.
5.1 Compound Interest and Investments
- Growth over time
- Savings plans
- Financial literacy
5.2 Population and Resource Models
- Growth and decay
- Long-term predictions
- Sustainability considerations
5.3 Motion and Repeated Processes
- Physics applications
- Iterative processes
- Technology applications
5.4 Mathematical Modeling Project
- Collecting and analyzing data
- Identifying patterns
- Building mathematical models
5.5 Capstone Sequences Investigation
- Open-ended exploration
- Pattern discovery
- Mathematical communication
Course Summary
By the end of this course, students will understand how sequences and series can be used to describe patterns, model change, and make predictions. They will be able to recognize arithmetic and geometric relationships, develop formulas, calculate series sums, and apply sequence concepts to real-world situations involving finance, science, technology, and nature.
Students will learn to:
- Identify and extend numerical and visual patterns.
- Write recursive and explicit formulas.
- Analyze arithmetic and geometric sequences.
- Calculate sums of arithmetic and geometric series.
- Graph and interpret sequence relationships.
- Model real-world situations involving growth and change.
- Explore famous mathematical patterns such as the Fibonacci sequence and Pascal’s Triangle.
- Apply mathematical reasoning to financial, scientific, and technological problems.
- Communicate mathematical ideas using appropriate notation and vocabulary.
This course provides an important bridge between algebra and advanced mathematics. By studying sequences and series, students strengthen their ability to recognize patterns, analyze relationships, and understand the mathematical structures that appear throughout science, economics, engineering, computer science, and higher-level mathematics.