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Lecture 4,5 Mathematical Induction

and Fibonacci

Sequences

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.

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

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)

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)

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

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

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.

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,

Example 4 Harmonic Numbers

Now consider

?P(k1) is true. Hence the statement is true for

all n ? Z.

Fibonacci Sequences

Dr Nazir A. Zafar

Advanced Algorithms Analysis and Design

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

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.

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, .

. .

Construction of Mathematical Model

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

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 . . .

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

Fibonacci Sequence

For some real C and D fibonacci sequence

satisfies the relation

Fibonacci Sequence

Dr Nazir A. Zafar

Advanced Algorithms Analysis and Design

Fibonacci Sequence

- After simplifying we get
- which is called the explicit formula for the

Fibonacci sequence recurrence relation.

Verification of the Explicit Formula

- Example Compute F3

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

Running Time of Recursive Fibonacci Algorithm

- Least Cost To find an asymptotic bound of

computational cost of this algorithm.

Drawback in Recursive Algorithms

- Recursion Tree

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, .

. .

Construction of Mathematical Model

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

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

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 . . .

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,

. . .

Construction of Mathematical Model

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

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 . . .

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

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.