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Unformatted text preview: 9.4 Proof by Mathematical Induction
Friday, November 04, 2011
6:47 AM Agenda:
1. Lesson: 9.4 Proof by Mathematical Induction Lesson: 9.4 Proof by Mathematical Induction
Goal: Prove summations by mathematical induction.
DOD Principals of Mathematical Induction
Let be a statement involving the positive integer . If:
is true (the base case)
2. Implying that is true shows that
is also true, for every positive
Then value, must be true for all positive integers . Our steps…
1. Show that is true (prove the base case) (this is usually trivial, but necessary!)
2. Assume is true. Using the fact that is assumed true, show that
also true. (this step is harder than 1., but with some algebraic manipulation, it is
• Think of Why Proofs by Induction?
• It is not enough to show that patterns hold. Do they hold up for every value of
the variable, or just the ones you tried? How do you know they will hold up for
every value of the variable.
Proofs by Induction can be thought of as showing that an infinite line of dominoes
will fall if you can prove these two cases.
• The first domino will fall.
Ch_9 Page 1 • The first domino will fall.
• Any domino will make another domino fall.
for the given . (substitute Prove by mathematical induction
1. Show that is true
2. Assume is true. Show that Ch_9 Page 2 for is also true. and simplify) Ch_9 Page 3 Ch_9 Page 4 ...
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