Chapter 4: Sequences and Mathematical Induction

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Chapter 4 Sequences and Mathematical Induction
4.1- Sequences . 4.2 , 4.3 Mathematical
Induction 4.4 Strong Mathematical induction and
WOP

• Instructor Hayk Melikya melikyan_at_nccu.edu

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Sequences
A sequence (informally) is a collection of
elements (objects or numbers usually infinite
number of) indexed by integers.
Examples
Each individual element ak is called a term,
where k is called an index
a1, a2, a3, , an,
General formula
1, 2, 3, 4, 1/2, 2/3,
3/4, 4/5, 1, -1, 1, -1,
1, -1/4, 1/9, -1/16,
Sequences can be computed using an explicit
formula ak k / (k 1) for k gt 0
Finding an explicit formula given initial terms
of the sequence
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Summation
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A Telescoping Sum
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Product
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Factorial
Factorial defines a product
How to estimate n!?
Turn product into a sum taking logs ln(n!)
ln(123 (n 1)n) ln 1 ln 2
ln(n 1) ln(n)
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Arithmetic Series
Given n numbers, a1, a2, , an with common
difference d, i.e. ai1 - ai d.
What is a simple closed form expression of the
sum?
Adding the equations together gives
Rearranging and remembering that an a1 (n -
1)d, we get
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Geometric Series
What is the closed form expression of Gn?
? xn1
Gn?xGn
1
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Infinite Geometric Series
Consider infinite sum (series)
for x lt 1
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Some Examples
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Principle of Mathematics induction
Let P(n) be a property that is defined for
integers n, and let a be a fixed integer Suppose
the following two statements are true 1. P(a)
is true. 2. for all integers k a, if P(k) is
true then P(k1) is true. Then the statement
P(n) is true for all integers n a

• Domino effect
• Inductive sets of integer.
• A subset of positive integers S is called
  inductive set
• if k ? S then k 1 ? S.
• Principle of mathematical induction states that
  if a ? S then all integers greater than or equal
  to a also are in S.

(chapter 4.2-4.4 of the book
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Method of Proof by Mathematical Induction
Consider a statement of the form For all
integers n a, a property P(n) is true. To
prove such a statement, perform the following two
steps Step1 (Basic step) Show tah the property
is true for n a ( P(a) is
true.) Step2 ( Inductive step) show that for
all integers k a, If property is true for n
k then it is true for n k1 ( ? k
a) (P(k)?P(k 1))
Let P(n) be the property n cent can be
obtained using 3 cent and 5 cent
coins. Proposition(4.2.1) P(n) is true for all
integers n 8.
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The Induction Rule when a 1
1 and (from n to n 1), proves 1, 2, 3,.
P(1), P(n)?P(n 1) (?m ? N) P(m)
Much easier to prove with P (n) as an assumption.
Very easy to prove
For any ngt1
Like domino effect
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Proof by Induction
Lets prove
Statements in green form a template for inductive
proofs. Proof (by induction on n) The induction
hypothesis, P(n), is
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Proof by Induction
Base Case (n 0)
Wait divide by zero bug! This is only true for
r ? 1
Theorem
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Proof by Induction
Induction Step Assume P(n) for some n ? 0 and
prove P(n 1)
Have P (n) by assumption So let r be any number
? 1, then from P (n) we have
How do we proceed?
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Proof by Induction
adding r n1 to both sides,
But since r ? 1 was arbitrary, we conclude (by
UG), that
which is P (n1). This completes the induction
proof.
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Summation
Try to prove
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Proving a Property
Base Case (n 1)
Induction Step Assume P(i) for some i ? 1 and
prove P(i 1)
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Proving an Inequality
Base Case (n 3)
Induction Step Assume P(i) for some i ? 3 and
prove P(i 1)
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Paradox
Proposition All horses are the same color.
Proof (by induction on n) Induction
hypothesis P(n) any set of n horses have
the same color Base case (n 1) true!
Inductive case Assume any n horses have the
same color. Prove that any n1 horses have the
same color.
Second set of n horses have the same color
First set of n horses have the same color
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Paradox
What is wrong?
n 1
Proof that P(n) ? P(n1) is false if n
1, because the two horse groups do not overlap.
Second set of n1 horses
First set of n1 horses
(But proof works for all n ? 1)
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Strong Induction
Prove P(1). Then prove P(n1) assuming all of
P(1), , P(n) (instead of just
P(n)). Conclude (?n.)P(n)
Strong induction
equivalent
1 ? 2, 2 ? 3, , n-1 ? n. So by the time we got
to n1, already know all of P(1), ,
P(n)
Ordinary induction
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Principle of Strong Mathematical Induction
Let P(n) be a property that is defined for
integers n, and let a and b be a fixed integers
with a b Suppose the following two statements
are true 1. P(a), P(a1), . . . And P(b) are
true. 2. for all integers k b, if P(i) is true
for all integers i from a through k then P(k1)
is true. Then the statement P(n) is true for all
integers n a
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Prime Products (revisited)
Every integer gt 1 is a product of primes.
Proof (by strong induction) Base case is
easy. Suppose the claim is true for all 2 lt i lt
n. Consider an integer n. In particular, n is not
prime.
So n km for integers k, m where n gt k,m
gt1. Since k,m smaller than n, By the induction
hypothesis, both k and m are product of primes k
p1? p2? ? ? ps m q1? q2? ? ? qt
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(No Transcript)
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Prime Products
Every integer gt 1 is a product of primes.

• So
• n k? m p1? p2? ? ? ps? q1? q2? ? ? qt
• is a prime product.
• ? This completes the proof of the induction
  step.

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Postage by Strong Induction
Available stamps
5
3
What amount can you form?
Theorem Can form any amount ? 8
Prove by strong induction on n gt 0. P(n) can
form (n 7).
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Postage by Strong Induction
Base case (n 1) (1 7)
Inductive Step assume (m 7) for 1 ? m ? n,
then prove ((n 1) 7)
cases n 1 1, 9 n 1 2, 10
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Postage by Strong Induction
case n 1 ? 3 let m n ? 2. now n ? m
? 0, so by induction hypothesis have
(n 1)8
(n ?2)8
Were done!
In fact, use at most two 5-cent stamps!
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Postage by Strong Induction
Given an unlimited supply of 5 cent and 7 cent
stamps, what postages are possible?
Theorem For all n gt 24, it is possible to
produce n cents of postage from 5 and 7 stamps.
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Well Ordering Principle
Every nonempty set ofnonnegative integers has a
least element.
Familiar? Now you mention it, Yes. Obvious?
Yes. Trivial? Yes. But watch out
Every nonempty set of nonnegative rationals has a
least element.
NO!
Every nonempty set of nonnegative integers has a
least element.
NO!
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Well Ordering Principle
Theorem is irrational
Proof suppose
can always find such m, n without common factors
why always?
By WOP, ? minimum m s.t.
where m0 is minimum.
so
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Well Ordering Principle
but if m0, n0 had common factor c gt 1, then
and
contradicting minimality of m0

• The well ordering principle is usually used in
  proof by contradiction.
• Assume the statement is not true, so there is a
  counterexample.
• Choose the smallest counterexample, and find a
  even smaller counterexample.
• Conclude that a counterexample does not exist.

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Well Ordering Principle in Proofs

• To prove ?n??. P(n) using WOP
• Define the set of counterexamples
• C n ?? P(n)
• 2. Assume C is not empty.
• 3. By WOP, have minimum element m0 ? C.
• 4. Reach a contradiction (somehow)
• usually by finding a member of C that is lt m0 .
• 5. Conclude no counterexamples exist. QED

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Induction (Proving equation)

• For any integer ngt2,
• Proof
• We prove by induction on n .
• Let P(n) be the proposition that

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• Base case, n2
• So P(2) is true.
• Inductive step
• Suppose that P(n) is true for some ngt2. So,

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• Then, for ngt2,
• By induction, P(n) is true for all integers ngt2.

By the inductive hypothesis
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Induction (Divisibility)

• For any integer ngt1, is
  divisible by 6
• Proof
• We prove by induction on n .
• Base case, n1
• is divisible by 6.
• So it is true for n1.

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• Inductive step Suppose that for some ngt1,
  is divisible by 6
• Then,
• Either n1 or n2 is even, so the last term is
  divisible by 6.
• Therefore
  is divisible by 6.
• By induction, is divisible by
  6 for all integers ngt1

By the inductive assumption
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Induction (Proving inequality)

• For any integer ngt4,
• Proof We prove by induction on n .
• Base case, n4
• So the claim is true for n4.

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• Inductive step
• Assume that for some
  ngt4
• Then,
• By induction, for all
  integer ngt4.

By the inductive hypothesis
By assumption, ngt2
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Chapter 4 (Sections 4.1, 4.2, 4.3, 4.4)

• 1. (4.1) Expand a sequence/sum/product from
  sequence/sum/product notation, and vice versa
• 2. (4.1) Rewrite a sum by separating off and
  adding on the last term
• 3. (4.1) Know the definition of a factorial.
• 4. (4.2) Know the method of proof by induction
• 5. (4.2) Prove formulas for the sums of sequences
  using induction
• 6. (4.3) Use induction to prove inequalities
• 7. (4.3) Use induction to prove results with
  divisibility
• 8. (4.4) Use strong induction on recursively
  defined sequences.
• Practice Problems
• (Section 4.1) 1, 2, 10, 11, 19-22, 34, 35, 43, 46
• (Section 4.2) 4, 6, 10, (11-15)
• (Section 4.3) 8-10, 16-20
• (Section 4.4) 1-8, 14, 17