lecture19bw - LECTURE 19 Mathematical Induction RECURSION...

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1 LECTURE 19 Mathematical Induction RECURSION E7, Fall 2008, M. Frenklach 1 EXAMPLE: 1 + 2 + … ( ) 1 1 2 ... 2 kk k + ++ + = [] ( ) 2 12 1 k ⎡⎤ =+ + + + ⎣⎦ ±²²²²³²²²²´ 2 MATHEMATICAL INDUCTION Given statements P 1 , P 2 , … IF: 1 P is true 1. P 1 is true 2. P k is true P k+1 is true THEN: every P n is true 3
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2 MATHEMATICAL INDUCTION: Proof 1. P 1 is true 2. P k is true P k+1 is true assumptions: consider: { } 123 1 P,P,P, ,P,P , kk + …… assume: all these are false but this contradicts assumption 2 therefore all P must be true 4 MATHEMATICAL INDUCTION: Example ( ) 1 1 P 1 2 ... 2 k k i ik = + ≡= + + + = () 111 1 + = Base Case, k = 1 : 2 Inductive Step: ( ) 1 1 P 2 12 P 2 k k + + = ++ = Assuming Prove 5 ( ) 1 1 1 2 ... 1 P1 1 1 k i k k k k + = =+ + + + + =+ + + PROOF: See if P k +1 is true, exploiting that P k is true P k is true” ( ) ( ) [] 2 (1 ) ( 1 ) 22 11 1 1 P m k k mm = =+ += ⎡⎤ + + ⎣⎦ == = k +1 = m P k +1 is true” 6
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3 So, using a “proof by induction”, we have now established that the identity USING INDUCTION 1 k kk + holds for all integers k 1.
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This note was uploaded on 11/01/2009 for the course ENGLISH 7 taught by Professor Sengupta during the Spring '09 term at University of California, Berkeley.

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lecture19bw - LECTURE 19 Mathematical Induction RECURSION...

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