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1 .Induction 2/24/12 1
2 .The Idea of Induction Color the integers ≥ 0 0 , 1 , 2 , 3 , 4 , 5 , … I tell you, 0 is red , & any int next to a red integer is red , then you know that all the ints are red ! 2/24/12 2
3 .Induction Rule 2/24/12 3
4 .Like Dominos…
5 .Example Induction Proof Let’s prove: (for r ≠ 1)
6 .Statements in magenta form a template for inductive proofs: Proof: (by induction on n ) The induction hypothesis, P ( n ) , is: Example Induction Proof (for r ≠ 1)
7 .Base Case ( n = 0 ) : Example Induction Proof 1 OK!
8 .Inductive Step: Assume P ( n ) for some n ≥ 0 and prove P ( n+1 ) : Example Induction Proof
9 .Now from induction hypothesis P ( n ) we have Example Induction Proof so add r n+1 to both sides
10 .adding r n+1 to both sides, Example Induction Proof This proves P ( n+1 ) completing the proof by induction.
11 .“ ” is an ellipsis . Can lead to confusion (n = 0 ?) Sum notation more precise Means you should see a pattern : an aside: ellipsis