Recursion Mastery

🏆 From Confusion to Clarity: Your Complete Journey into Recursive Mathematics

Are you preparing for the AMC 10, AMC 12, or AIME? Do recurrence relations leave you frustrated and confused? Have you memorized formulas for arithmetic and geometric sequences but freeze when faced with problems like "A sequence is defined by a1=1a1​=1 and an+1=3an+2an+1​=3an​+2. Find a100​"?

If so, Recursion Mastery is the book you've been waiting for.

Written by Rohan Kumar Singh, IIT Madras alumnus, International Olympiad Gold Medalist, and founder of the prestigious EduGlobal Institute, this book transforms one of mathematics' most intimidating topics into an intuitive, accessible, and even enjoyable subject.

🎯 What Makes This Book Different?

Most textbooks treat recursion as a collection of disconnected tricks—"try this method for this type, that method for that type." Recursion Mastery takes a radically different approach: it builds understanding from the ground up, showing you not just how to solve recurrences, but why the methods work.

Key Features:

✅ Comprehensive Coverage: From basic arithmetic progressions to advanced generating functions and AIME-level probability recursions

✅ Proven Pedagogy: Based on the award-winning EduGlobal Method—Diagnostics → Concept Sprints → Mastery Checks → Feedback Loops

✅ 500+ Practice Problems: Stratified by difficulty—AMC 10, AMC 12, AIME, and select USAMO-level challenges

✅ Complete Solutions: Every problem includes detailed step-by-step solutions with explanations

✅ Theoretical Depth: Rigorous proofs of all major theorems, including the characteristic equation, Binet's formula, and the Master Theorem

✅ Real-World Applications: Population dynamics, financial mathematics, algorithmic complexity, and more

✅ Competition Focus: Specifically designed for students targeting AMC 10, AMC 12, AIME, and other Olympiad-level competitions

📖 What You'll Learn

Part I: The Foundation

• What is a sequence? Explicit vs. recursive definitions

• The language of recurrences: order, linearity, homogeneity, constant coefficients

• Visualizing recursion with state diagrams

Part II: First-Order Recurrences

• Arithmetic and geometric progressions (the building blocks)

• The general first-order linear recurrence an=can−1+dan​=can−1​+d

• Three powerful methods: iteration, fixed point transformation, closed-form formula

• Applications: compound interest, population growth, loan payments

Part III: Higher-Order Linear Recurrences

• The characteristic equation method

• Distinct real roots, repeated roots, complex roots

• Fibonacci numbers and Binet's formula

• Extending to k-th order recurrences

Part IV: Non-Homogeneous Recurrences

• Structure: homogeneous + particular solutions

• Method of undetermined coefficients

• Handling special cases when forcing terms match roots

Part V: Combinatorial Recursions

• Tiling problems and domino tilings

• Lattice paths and grid walking

• Derangements: the hat-check problem

• Partitions and Bell numbers

Part VI: Generating Functions

• Ordinary generating functions (OGFs)

• Solving recurrences with generating functions

• Extracting coefficients for closed forms

Part VII: Advanced AIME Applications

• Recursion in number theory: Euclidean algorithm, GCD recurrences

• Recursion in probability: first-step analysis, gambler's ruin

• Recursion in geometry: fractals, gnomon figures, self-similarity

Part VIII: Special Techniques

• Non-linear recurrences and reduction to linear form

• Systems of recurrences and matrix methods

• Divide-and-conquer recurrences and the Master Theorem

Part IX: Practice and Mock Tests

• 100 AMC 10-level drill problems

• 100 AMC 12-level drill problems

• 75 AIME-level drill problems

• 3 full-length mock tests with complete solutions

👨‍🏫 About the Author

Rohan Kumar Singh is the Founder and Chief Mentor at EduGlobal Institute, a premier training ground for the world's most ambitious young mathematicians.

• IIT Madras Alumnus

• International Olympiad Gold Medalist

• 2,400+ Students Mentored globally

• 460+ Olympiad Rankers (AMC 10/12, AIME, USAMO, UKMT)

• 700+ Perfect Scores (5/5) in AP Calculus BC and AP Physics C

• 320+ Student Research Papers published in recognized journals

Rohan's philosophy is simple: talent is not just born; it is engineered through structure, strategy, and deep conceptual understanding.

🌟 What Students Are Saying

"Before this book, I thought recurrence relations were impossible. Now I see them everywhere—and I know exactly how to solve them. The fixed point method alone is worth the price."
— Arjun M., AIME Qualifier

"Rohan sir doesn't just teach formulas; he teaches you how to think. The diagnostic checks caught gaps I didn't even know I had. This book is a masterpiece."
— Priya K., AMC 12 Perfect Scorer

"I've read multiple books on recurrences, but none explain the 'why' behind the characteristic equation like this one does. Finally, it all makes sense!"
— Daniel W., USAMO Participant

📋 Book Specifications

🎁 What's Inside

• 20 Theoretically Comprehensive Chapters

• 500+ Solved Examples with detailed explanations

• 500+ Practice Problems with complete solutions

• 50+ Diagnostic Checks to identify knowledge gaps

• 30+ Concept Sprint modules for rapid learning

• 20+ Mastery Checks for rigorous testing

• 15+ Feedback Loops addressing common mistakes

• 3 Full-Length Mock Tests with timing guidelines

• Complete Index for quick reference

• Formula Summary Sheets for each chapter

🚀 Who This Book Is For

✅ AMC 10/12 Aspirants: Master the recursion problems that appear in questions 16–25

✅ AIME Qualifiers: Develop the deep intuition needed for multi-step recursion problems

✅ Olympiad Hopefuls: Tackle advanced recurrences in number theory, combinatorics, and probability

✅ High School Students: Build a rock-solid foundation for college-level discrete mathematics

✅ Self-Learners: Progress at your own pace with clear explanations and abundant practice

✅ Teachers & Tutors: Access a complete curriculum for teaching recurrence relations

📈 Why Recursion Matters for Competitions

Recurrence relations appear consistently in every major mathematics competition:

• AMC 10/12: 2–3 problems per contest involving sequences, counting recursions, or probability trees

• AIME: At least one problem directly testing recursive thinking, often hidden in combinatorics or number theory

• USAMO: Deep theoretical problems requiring sophisticated recurrence techniques

Mastering recursion can mean the difference between qualifying and missing the cut.

💡 Key Techniques Covered

• Telescoping and summation methods

• Fixed point / equilibrium analysis

• Characteristic equations (all cases)

• Method of undetermined coefficients

• Generating functions

• First-step analysis for probability

• Matrix exponentiation

• Master Theorem for divide-and-conquer

• Reduction of non-linear to linear forms

📚 Table of Contents (Abridged)

Part I: The Foundation

• Chapter 1: What is Recursion?

• Chapter 2: Arithmetic and Geometric Progressions

• Chapter 3: First-Order Linear Recurrences

Part II: Higher-Order Recurrences

• Chapter 4: The Characteristic Equation

• Chapter 5: Non-Homogeneous Recurrences

• Chapter 6: Systems of Recurrences

Part III: Combinatorial Recursions

• Chapter 7: Tilings and Paths

• Chapter 8: Derangements and Permutations

• Chapter 9: Generating Functions

Part IV: Advanced Applications

• Chapter 10: Recursion in Number Theory

• Chapter 11: Recursion in Probability

• Chapter 12: Recursion in Geometry

Part V: Special Topics

• Chapter 13: Non-Linear Recurrences

• Chapter 14: Divide-and-Conquer Recurrences

• Chapter 15: Recurrences with Variable Coefficients

Part VI: Practice

• Chapter 16: AMC 10 Drills

• Chapter 17: AMC 12 Drills

• Chapter 18: AIME Drills

• Chapter 19: Mock Tests

• Chapter 20: Complete Solutions

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