The binomial theorem - the Australian Mathematical Sciences Institute

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The bionomial theorem – A guide for teachers (Years 11–12)Principal author: Dr Michael Evans AMSIPeter Brown, University of NSWAssociate Professor David Hunt, University of NSWDr Daniel Mathews, Monash UniversityEditor: Dr Jane Pitkethly, La Trobe UniversityIllustrations and web design: Catherine Tan, Michael ShawFull bibliographic details are available from Education Services Australia.Published by Education Services AustraliaPO Box 177Carlton South Vic 3053AustraliaTel: (03) 9207 9600Fax: (03) 9910 9800Email: info@esa.edu.auWebsite: www.esa.edu.au© 2013 Education Services Australia Ltd, except where indicated otherwise. You maycopy, distribute and adapt this material free of charge for non-commercial educationalpurposes, provided you retain all copyright notices and acknowledgements.This publication is funded by the Australian Government Department of Education,Employment and Workplace Relations.Supporting Australian Mathematics ProjectAustralian Mathematical Sciences InstituteBuilding 161The University of MelbourneVIC 3010Email: enquiries@amsi.org.auWebsite: www.amsi.org.au
Assumed knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6A look at Pascal’s triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Expansions and the notation ( nr). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8The binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Pascal’s triangle — the observations . . . . . . . . . . . . . . . . . . . . . . . . . . 15Applying the binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Proof of the binomial theorem by mathematical induction . . . . . . . . . . . . . 24Further identities and results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Links forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30A result using complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Series for e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30The generalised binomial theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31History and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Answers to exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
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