slides06 - Mathematical Induction CMSC 250 1 Logic of...

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1 CMSC 250 Mathematical Induction
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2 CMSC 250 Logic of Induction Induction principle: P(1) ( 2200 n ≥ 1)[P( n ) P( n +1)] ( 2200 n ≥ 1)[P( n )]
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3 CMSC 250 Logic of Induction Induction principle: P(1) ( 2200 n ≥ 2)[P( n -1) P( n )] ( 2200 n ≥ 1)[P( n )]
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4 CMSC 250 Description Inductive proofs must have: Base case: P (1) Usually easy Inductive hypothesis: Assume P ( n -1) Inductive step: Prove P ( n -1) P ( n )
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5 CMSC 250 Example For all n ≥1 + = = n i n n i 1 2 ) 1 ( Proof in class
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6 CMSC 250 Another example For all n ≥1 - = - = 1 2 2 1 0 n n k k Proof in class
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7 CMSC 250 Another example- Geometric Series - - = - = 1 1 1 0 r r r n n k k Proof in class For all n ≥1 and real r ≠1
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8 CMSC 250 Another example- a divisibility property )] 3 (mod [ 3 n n Proof in class
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9 CMSC 250 A sequence example Theorem Let Then [ ] 2 ) 1 ( n a n n = 2200 1 1 = a [ ] ) 1 2 ( ) 2 ( 1 - + = 2200 - n a a n n n Proof in class
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10 CMSC 250 An example with an inequality Prove this statement: Base case ( n = 3): 7 1 6 1 ) 3 ( 2 : = + = + LHS 8 2 : 3 = RHS RHS LHS < [ ] n n Z n 2 1 2 ) ( 3 < + 2200 Proof in class
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11 CMSC 250 Another example with an inequality Proof in class For all n ≥0 and real x≥0
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This note was uploaded on 12/04/2011 for the course CMSC 250 taught by Professor Staff during the Spring '08 term at Maryland.

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slides06 - Mathematical Induction CMSC 250 1 Logic of...

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