Mathematical Induction II

1
Mathematical Induction II

• Lecture 20
• Section 4.3
• Wed, Feb 16, 2005

2
Putnam Question B-1 (1981)

• Find
• limn?? (1/n5) ?h 1..n ?k 1..n (5h4 18h2k2
  5k4).
• Solution
• Express ?h?k 5h4, ?h?k 18h2k2, and ?h?k 5k4 as
  polynomials in n.
• Simplify the polynomials.
• Divide by n5.
• Take the limit as n ? ?.
• The answer is -1.

3
Lets Play Find the Flaw

• Theorem For every positive integer n, in any
  set of n horses, all the horses are the same
  color.
• Proof
• Basic Step.
• When n 1, there is only one horse, so trivially
  they are (it is) all the same color.

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Find the Flaw

• Inductive Step
• Suppose that any set of k horses are all the same
  color.
• Consider a set of k 1 horses.
• Remove one of the horses from the set.
• The remaining set of k horses are all the same
  color.

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Find the Flaw

• Replace that horse and remove a different horse.
• Again, the remaining set of k horses are all the
  same color.
• Therefore, the two horses that were removed are
  the same color as the other horses in the set.
• Thus, the k 1 horses are all the same color.

6
Find the Flaw

• Thus, in any set of n horses, the horses are all
  the same color.

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Example

• Find a formula for
• 1 3 5 (2n 1).
• Clever solution
• 1 3 5 (2n 1)
• (1 2 3 2n) (2 4 6 2n)
• (1 2 3 2n) 2(1 2 3 n)
• (2n)(2n 1)/2 2(n(n 1)/2)
• n2.

8
Exercise

• Find a formula for
• 12 32 52 (2n 1)2.
• Then verify it using mathematical induction.