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0304-slides - CPSC 121 Models of Computation Instructor Bob...

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CPSC 121: Models of Computation Instructor: Bob Woodham [email protected] Department of Computer Science University of British Columbia Lecture Notes 2008/2009, Section 203 CPSC 121: Models of Computation
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Menu March 4, 2009 Topics: Mathematical Induction (cont’d) Examples Reading: Today: Epp 4.1 (as background) Epp 4.2–4.4 Reminders: Assignment 3 due Friday, March 13 (by 17:00) Midterm, Marked Assignment 2 available in tutorials — solutions posted to Sample Solutions area of the course WebCT site Lab 5 week of March 2–6 READ the WebCT Vista course announcements board CPSC 121: Models of Computation
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Example 1: Theorem: n N , a R - { 0 , 1 } n X i = 0 a i = a n + 1 - 1 a - 1 Proof (by mathematical induction): Let a be an arbitrary real number different from 0 and 1 Let P ( n ) : n X i = 0 a i = a n + 1 - 1 a - 1 Base step: Now, P ( 0 ) : 0 X i = 0 a 0 = 1 = a 0 + 1 - 1 a - 1 = a - 1 a - 1 = 1 ( a 6 = 0 , a 6 = 1 ) Therefore, P ( 0 ) is true CPSC 121: Models of Computation
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Example 1 (cont’d): Inductive step: For k N , we assume P ( k ) : k X i = 0 a i = a k + 1 - 1 a - 1 in order to prove P ( k + 1 ) : k + 1 X i = 0 a i = a k + 2 - 1 a - 1 k + 1 X i = 0 a i
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