**Dolciani Mathematical Expositions**

Volume: 27;
2003;
194 pp;
Hardcover

**Print ISBN: 978-0-88385-333-7
Product Code: DOL/27**

List Price: $58.00

AMS Member Price: $43.50

MAA Member Price: $43.50

**Electronic ISBN: 978-1-61444-208-0
Product Code: DOL/27.E**

List Price: $58.00

AMS Member Price: $43.50

MAA Member Price: $43.50

# Proofs that Really Count: The Art of Combinatorial Proof

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*Arthur T. Benjamin; Jennifer J. Quinn*

MAA Press: An Imprint of the American Mathematical Society

Recipient of the Mathematical Association of America's
Beckenbach Book Prize in 2006!

Mathematics is the science of patterns, and
mathematicians attempt to understand these patterns and discover new
ones using a variety of tools. In Proofs That Really Count,
award-winning math professors Arthur Benjamin and Jennifer Quinn
demonstrate that many number patterns, even very complex ones, can be
understood by simple counting arguments. The book emphasizes numbers
that are often not thought of as numbers that count: Fibonacci
Numbers, Lucas Numbers, Continued Fractions, and Harmonic Numbers, to
name a few. Numerous hints and references are given for all chapter
exercises and many chapters end with a list of identities in need of
combinatorial proof. The extensive appendix of identities will be a
valuable resource. This book should appeal to readers of all levels,
from high school math students to professional
mathematicians.

#### Reviews & Endorsements

'This book is written in an engaging, conversational style, and this reviewer found it enjoyable to read through (besides learning a few new things). Along the way, there are a few surprises, like the 'world's fastest proof by induction' and a magic trick. As a resource for teaching, and a handy basic reference, it will be a great addition to the library of anyone who uses combinatorial identities in their work.'

-- Society for Industrial and Applied Mathematics Review

# Table of Contents

## Proofs that Really Count: The Art of Combinatorial Proof

- cover cover11
- copyright page ii3
- title page iii4
- Foreword ix10
- Contents xiii14
- 1 Fibonacci Identities 116
- 2 Gibonacci and Lucas Identities 1732
- 3 Linear Recurrences 3550
- 4 Continued Fractions 4964
- 5 Binomial Identities 6378
- 6 Alternating Sign Binomial Identities 8196
- 7 Harmonic and Stirling Number Identities 91106
- 8 Number Theory 109124
- 9 Advanced Fibonacci & Lucas Identities 125140
- Some Hints and Solutions for Chapter Exercises 147162
- Appendix of Combinatorial Theorems 171186
- Appendix of Identities 173188
- Bibliography 187202
- Index 191206
- About the Authors 194209