Lecture 4,5 Mathematical Induction and Fibonacci Sequences

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Lecture 4,5 Mathematical Induction
and Fibonacci
Sequences
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What is Mathematical Induction?

• Mathematical induction is a powerful, yet
  straight-forward method of proving statements
  whose domain is a subset of the set of integers.
• Usually, a statement that is proven by induction
  is based on the set of natural numbers.
• This statement can often be thought of as a
  function of a number n, where n 1, 2, 3,. . .
• Proof by induction involves three main steps
• Proving the base of induction
• Forming the induction hypothesis
• Proving that the induction hypothesis holds true
  for all numbers in the domain.

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What is Mathematical Induction?

• Let P(n) be the predicate defined for any
  positive integers n, and let n0 be a fixed
  integer. Suppose the following two statements are
  true
• P(n0) is true.
• For any positive integers k, k ? n0,
• if P(k) is true then P(k1)is true.
• If the above statements are true then the
  statement
• ? n ? N, such that n ? n0, P(n) is also
  true

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Steps in Proving by Induction

• Claim P(n) is true for all n ? Z, for n ? n0
• Basis
• Show formula is true when n n0
• Inductive hypothesis
• Assume formula is true for an arbitrary n k
• where, k ? Z and k ? n0
• To Prove Claim
• Show that formula is then true for k1
• Note In fact we have to prove
• P(n0) and
• P(k) ? P(k1)

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Proof by Induction

• Example 1
• Prove that n2 ? n 100 ? n ? 11
• Solution
• Let P(n) ? n2 ? n 100 ? n ? 11
• P(11) ? 112 ? 11 100 ? 121 ? 111, true
• Suppose predicate is true for n k, i.e.
• P(k) ? k2 ? k 100, true k ? 11
• Now it can be proved that
• P(k1) ? (k1)2 ? (k1) 100,
• k2 2k 1 ? k 1 100 ? k2 k ? 100 (by 1 and
  2)
• Hence P(k) ? P(K1)

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Validity of Proof

• Example 1
• Prove that n2 ? n 100 ? n ? 11
• Solution
• Initially, base case
• Solution set 11
• By, P(k) ? P(K1) ? P(11) ? P(12), taking k 11
• Solution set 11, 12
• Similarly, P(12) ? P(13), taking k 12
• Solution set 11, 12, 13
• And, P(13) ? P(14), taking k 13
• Solution set 11, 12, 13, 14
• And so on

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Reasoning of Proof
Another Easy Example

• Example 2
• Use Mathematical Induction to prove that sum of
  the first n odd positive integers is n2.
  
• Proof
• Let P(n) denote the proposition that
• Basis step P(1) is true , since 1 12
• Inductive step Let P(k) is true for a positive
  integer k, i.e.,
  135(2k-1) k2
• Note that 135(2k-1)(2k1) k22k1
  (k1)2
• ? P(k1) true, by induction, P(n) is true
  for all n ? Z
• Another Proof

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Reasoning of Proof

• Example 3
• Use mathematical Induction to prove that
• n lt 2n for all n ? Z
• Proof
• Let P(n) be the proposition that n lt 2n
• Basis step P(1) is true since 1 lt 21 .
• Inductive step
• Assume that P(n) is true for a positive
  integer n k, i.e., k lt 2k.
• Now consider for P(k1)
• Since, k 1 lt 2k 1 ? 2k 2k 2.2k
  2k 1
• ? P(k1) is true.
• It proves that P(n) is true for all n ? Z.

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Example 4 Harmonic Numbers
The harmonic numbers Hk, k 1, 2, 3, , are
defined by Use
mathematical induction to show that
whenever n is a nonnegative integer.
Proof Let P(n) be the proposition that
Basis step P(0) is true,
since,
Inductive step Assume that P(k) is true for
some k,
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Example 4 Harmonic Numbers
Now consider
?P(k1) is true. Hence the statement is true for
all n ? Z.
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Fibonacci Sequences
Dr Nazir A. Zafar
Advanced Algorithms Analysis and Design
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Today Covered

• In this lecture we will cover the following
• Fibonacci Problem and its Sequence
• Construction of Mathematical Model
• Recursive Algorithms
• Generalizations of Rabbits Problem and
  Constructing its Mathematical Models
• Applications of Fibonacci Sequences

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Fibonacci Sequence

• By studying Fibonacci numbers and constructing
  Fibonacci sequence we can imagine how mathematics
  is connected to apparently unrelated things in
  this universe.
• Even though these numbers were introduced in 1202
  in Fibonaccis book Liber abaci.
• Fibonacci, who was born Leonardo da Pisa gave a
  problem in his book whose solution was the
  Fibonacci sequence as we will discuss it today.

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Fibonaccis Problem

• Statement
• Start with a pair of rabbits, one male and one
  female, born on January 1.
• Assume that all months are of equal length and
  that rabbits begin to produce two months after
  their own birth.
• After reaching age of two months, each pair
  produces another mixed pair, one male and one
  female, and then another mixed pair each month,
  and no rabbit dies.
• How many pairs of rabbits will there be after
  one year?
• Answer The Fibonacci Sequence!
• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, .
  . .

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Construction of Mathematical Model
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Construction of Mathematical Model

• Total pairs at level k Total pairs at level k-1
  Total pairs born at level k (1)
• Since
• Total pairs born at level k Total pairs at
  level k-2 (2)
• Hence by equation (1) and (2)
• Total pairs at level k Total pairs at level
  k-1 Total pairs at level k-2
• Now let us denote
• Fk Total pairs at level k
• Now our recursive mathematical model will become
• Fk Fk-1 Fk-2

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Computing Values using Mathematical Model

• Since Fk Fk-1 Fk-2 F0 0, F1 1
• F2 F1 F0 1 0 1
• F3 F2 F1 1 1 2
• F4 F3 F2 2 1 3
• F5 F4 F3 3 2 5
• F6 F5 F4 5 3 8
• F7 F6 F5 8 5 13
• F8 F7 F6 13 8 21
• F9 F8 F7 21 13 34
• F10 F9 F8 34 21 55
• F11 F10 F9 55 34 89
• F12 F11 F10 89 55 144 . . .

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Explicit Formula Computing Fibonacci Numbers
Theorem The fibonacci sequence F0,F1, F2,.
Satisfies the recurrence relation
Find the explicit formula for this
sequence. Solution
The given fibonacci sequence
Let tk is solution to this, then characteristic
equation
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Fibonacci Sequence
For some real C and D fibonacci sequence
satisfies the relation
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Fibonacci Sequence
Dr Nazir A. Zafar
Advanced Algorithms Analysis and Design
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Fibonacci Sequence

• After simplifying we get
• which is called the explicit formula for the
  Fibonacci sequence recurrence relation.

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Verification of the Explicit Formula

• Example Compute F3

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Recursive Algorithm Computing Fibonacci Numbers

• Fibo-R(n)
• if n 0
• then 0
• if n 1
• then 1
• else Fibo-R(n-1) Fibo-R(n-2)

Terminating conditions
Recursive calls
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Running Time of Recursive Fibonacci Algorithm

• Least Cost To find an asymptotic bound of
  computational cost of this algorithm.

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Drawback in Recursive Algorithms

• Recursion Tree

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Generalization of Rabbits Problem

• Statement
• Start with a pair of rabbits, one male and one
  female, born on January 1.
• Assume that all months are of equal length and
  that rabbits begin to produce two months after
  their own birth.
• After reaching age of two months, each pair
  produces two other mixed pairs, two male and two
  female, and then two other mixed pair each month,
  and no rabbit dies.
• How many pairs of rabbits will there be after
  one year?
• Answer Generalization of Fibonacci Sequence!
• 0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, 683, .
  . .

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Construction of Mathematical Model
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Construction of Mathematical Model

• Total pairs at level k
• Total pairs at level k-1 Total pairs born at
  level k (1)
• Since
• Total pairs born at level k
• 2 x Total pairs at level k-2 (2)
• By (1) and (2), Total pairs at level k
• Total pairs at level k-1 2 x Total pairs at
  level k-2
• Now let us denote
• Fk Total pairs at level k
• Our recursive mathematical model
• Fk Fk-1 2.Fk-2
• General Model (m pairs production) Fk Fk-1
  m.Fk-2

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Generalization

• Recursive mathematical model
• (one pair production)
• Fk Fk-1 Fk-2
• Recursive mathematical model
• (two pairs production)
• Fk Fk-1 2.Fk-2
• Recursive mathematical model
• (m pairs production)
• Fk Fk-1 m.Fk-2

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Computing Values using Mathematical Model

• Since Fk Fk-1 2.Fk-2 F0 0, F1 1
• F2 F1 2.F0 1 0 1
• F3 F2 2.F1 1 2 3
• F4 F3 2.F2 3 2 5
• F5 F4 2.F3 5 6 11
• F6 F5 F4 11 10 21
• F7 F6 F5 21 22 43
• F8 F7 F6 43 42 85
• F9 F8 F7 85 86 171
• F10 F9 F8 171 170 341
• F11 F10 F9 341 342 683
• F12 F11 F10 683 682 1365 . . .

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Another Generalization of Rabbits Problem

• Statement
• Start with a different kind of pair of rabbits,
  one male and one female, born on January 1.
• Assume all months are of equal length and that
  rabbits begin to produce three months after their
  own birth.
• After reaching age of three months, each pair
  produces another mixed pairs, one male and other
  female, and then another mixed pair each month,
  and no rabbit dies.
• How many pairs of rabbits will there be after
  one year?
• Answer Generalization of Fibonacci Sequence!
• 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60,
  . . .

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Construction of Mathematical Model
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Construction of Mathematical Model

• Total pairs at level k
• Total pairs at level k-1 Total pairs born at
  level k (1)
• Since
• Total pairs born at level k Total pairs at
  level k-3 (2)
• By (1) and (2)
• Total pairs at level k
• Total pairs at level k-1 Total pairs at level
  k-3
• Now let us denote
• Fk Total pairs at level k
• This time mathematical model Fk Fk-1 Fk-3

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Computing Values using Mathematical Model

• Since Fk Fk-1 Fk-3 F0 0, F1 F2 1
• F3 F2 F0 1 0 1
• F4 F3 F1 1 1 2
• F5 F4 F2 2 1 3
• F6 F5 F3 3 1 4
• F7 F6 F4 4 2 6
• F8 F7 F5 6 3 9
• F9 F8 F6 9 4 13
• F10 F9 F7 13 6 19
• F11 F10 F8 19 9 28
• F12 F11 F9 28 13 41 . . .

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More Generalization

• Recursive mathematical model
• (one pair, production after three months)
• Fk Fk-1 Fk-3
• Recursive mathematical model
• (two pairs, production after three months)
• Fk Fk-1 2.Fk-3
• Recursive mathematical model
• (m pairs, production after three months)
• Fk Fk-1 m.Fk-3
• Recursive mathematical model
• (m pairs, production after n months)
• Fk Fk-1 m.Fk-n

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Applications of Fibonacci Sequences

• Fibonacci sequences
• Are used in trend analysis
• By some pseudorandom number generators
• Many plants show the Fibonacci numbers in the
  arrangements of the leaves around the stems.
• Seen in arrangement of seeds on flower heads
• Consecutive Fibonacci numbers give worst case
  behavior when used as inputs in Euclids
  algorithm.