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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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the idea of induction
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!

example induction proof
Example Induction Proof

Let’s prove:

(for r ≠ 1)

example induction proof1
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)

example induction proof2
Example Induction Proof

Base Case (n = 0):

1

OK!

example induction proof3
Example Induction Proof
  • Inductive Step: Assume P(n)for some n≥0 and prove P(n+1):
example induction proof4
Example Induction Proof

Now from induction

hypothesisP(n) we have

so add rn+1to both sides

example induction proof5
Example Induction Proof

adding rn+1to both sides,

This proves

P(n+1)

completing the

proof by induction.

an aside ellipsis
an aside: ellipsis

Means you

should see a pattern:

“” is an ellipsis.

  • Can lead to confusion (n= 0?)
  • Sum notation more precise