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Unformatted text preview: CPSC 121 Lecture 21 March 2, 2009 Menu March 2, 2009 Topics: Mathematical Induction (Introduction) Reading: Today: Epp 4.1 (as background) Epp 4.2 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 26 READ the WebCT Vista course announcements board Mathematical Induction Recap We have considered theorems of the form x D,P ( x ) Q ( x ) for a variety of domains of discourse, D , and proof techniques (direct proof, proof by contradiction and proof by contrapositive) Now lets consider theorems specifically of the form n Z + ,P ( n ) NOTES: D is restricted to Z + , the set of positive integers Proof by contrapositive wont work Direct proofs and proofs by contradiction might work IMPORTANT: There is a another proof technique available to us, called mathematical induction Example 1: Motivating example... Lets find a formula for the sum of the first n positive odd integers Lets look at the first few cases... n = 1 sum = 1 n = 2 sum = 1 + 3 = 4 n = 3 sum = 1 + 3 + 5 = 9 n = 4 sum = 1 + 3 + 5 + 7 = 16 ...there appears to be a pattern! Conjecture: the sum of the first n positive odd integers is n 2 We would like to prove this conjecture How do we know for sure that we have the correct formula? Perhaps it will fail at say n = 101 or n = 212 ? Maybe the correct formula is n 4- 10 n 3 + 36 n 2- 50 n + 24 . This formula works for n = 1 , 2 , 3 , 4 ....
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