{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

lecture19bw

# lecture19bw - LECTURE 19 Mathematical Induction RECURSION...

This preview shows pages 1–4. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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
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.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 10

lecture19bw - LECTURE 19 Mathematical Induction RECURSION...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online