The book's objectives are to expose students to analyzing and formulating various patterns such as linear, quadratic, geometric, piecewise, alternating, summation-type, product-type, recursive and periodic patterns. The book will present various patterns graphically and analytically and show the connections between them. Graphical presentations include patterns at same scale, patterns at diminishing scale and alternating patterns.
The book's goals are to train and expand students' analytical skills by presenting numerous repetitive-type problems that will lead to formulating results inductively and to the proof by induction method. These will start with formulating basic sequences and piecewise functions and transition to properties of Pascal's Triangle that are horizontally and diagonally oriented and formulating solutions to recursive sequences. The book will start with relatively straight forward problems and gradually transition to more challenging problems and open-ended research questions.
The book's aims are to prepare students to establish a base of recognition and formulation of patterns that will navigate to study further mathematics such as Calculus, Discrete Mathematics, Matrix Algebra, Abstract Algebra, Difference Equations, and to potential research projects. The primary aims out of all are to make mathematics accessible and multidisciplinary for students with different backgrounds and from various disciplines.
Contents:
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Introduction to Patterns:
- Geometrical Arrangements
- Piecewise Functions
- Analytical Formulations
- Recursive Sequences
- Piece-wise Sequences
- Periodic Cycles
- Exercises
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Geometrical Configurations:
- Patterns at Same Scale
- Patterns at Different Scales
- Alternating and Piecewise Patterns
- Summation of Areas
- Exercises
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Sequences, Products and Summations:
- Linear Sequences
- Quadratic Sequences
- Summation-Type Sequences
- Geometric Sequences
- Product-Type Sequences
- Alternating and Piecewise Sequences
- Exercises
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Summations and Proof by Induction:
- Linear and Geometric Summations
- Proof by Induction
- Exercises
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Traits of Pascal's Triangle:
- Horizontal Identities
- Diagonal Identities
- Binomial Expansion
- Exercises
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Recursive Relations:
- Formulating a Recursive Relation
- Obtaining an Explicit Solution
- Non-Autonomous Recursive Sequences
- Exercises
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Periodic Traits:
- Autonomous Recursive Sequences
- Multiplicative Formof Eq. (7.1)
- Additive Form of Eq. (7.1)
- Additive and Multiplicative Forms of Eq. (7.1)
- Special Case of Eq. (7.1)
- Exercises
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Answers to Chapter Exercises:
- Answers to Chapter 1 Exercises
- Answers to Chapter 2 Exercises
- Answers to Chapter 3 Exercises
- Answers to Chapter 4 Exercises
- Answers to Chapter 5 Exercises
- Answers to Chapter 6 Exercises
- Answers to Chapter 7 Exercises
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Appendices:
- Right Triangles
- Isosceles Triangle
- Equilateral Triangle
- Area of Figures
- Patterns (Sequences)
- Alternating Patterns (Sequences)
- Summation Properties
- Finite Summations
- Laws of Exponents
- Factoring Methods
- Binomial Expansion
Readership: High school and undergraduate levels in mathematics for Non-STEM disciplines, Pattern Recognition, Discrete Mathematics, Difference Equations.
Key Features:
- The book provides several repetitive-type guided examples that navigate to deeper fundamentals and to solving more challenging problems
- These practice problems lead to formulating patterns, to inductive reasoning and to proof by induction
- The book introduces the inductive approach to grasp the concepts
- The book presents each concept graphically and analytically and the connection between the two methods
- The book also emphasizes the similarities and contrasts between assorted patterns and widens the readers' analytical skills and welcomes them to unfold more mathematical essences
