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Induction

Induction. 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 !. Induction Rule. Like Dominos…. Example Induction Proof. Let’s prove:.

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Induction

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  1. Induction

  2. The Idea of Induction Color the integers ≥ 0 0, 1, 2, 3, 4, 5, … I tell you, 0isred, & any int next toared integer isred, then you know that all the ints arered!

  3. Induction Rule

  4. Like Dominos…

  5. Example Induction Proof Let’s prove: (for r ≠ 1)

  6. Example Induction Proof Statements inmagenta form a template for inductive proofs: • Proof: (by induction on n) • The induction hypothesis, P(n), is: (for r ≠ 1)

  7. Example Induction Proof Base Case (n = 0): 1 OK!

  8. Example Induction Proof • Inductive Step: Assume P(n)for some n≥0 and prove P(n+1):

  9. Example Induction Proof Now from induction hypothesisP(n) we have so add rn+1to both sides

  10. Example Induction Proof adding rn+1to both sides, This proves P(n+1) completing the proof by induction.

  11. an aside: ellipsis Means you should see a pattern: “” is an ellipsis. • Can lead to confusion (n= 0?) • Sum notation more precise

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