lecture 4,5 mathematical induction and fibonacci sequences
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Lecture 4,5
Mathematical Induction and
Fibonacci Sequences
• 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?
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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
1. P(n0) is true.
2. For any positive integers k, k n0,3. if P(k) is true then P(k+1)is true.
If the above statements are true then the statement:
n N, such that n n0, P(n) is also true
What is Mathematical Induction?
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Claim: P(n) is true for all n Z+, for n n0
1. Basis– Show formula is true when n = n0
2. Inductive hypothesis– Assume formula is true for an arbitrary n = k
where, k Z+ and k n0
3. To Prove Claim– Show that formula is then true for k+1
Note: In fact we have to prove
1) P(n0) and2) P(k) P(k+1)
Steps in Proving by Induction
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Example 1• Prove that n2 n + 100 n 11
Solution
Let P(n) n2 n + 100 n 11
1. P(11) 112 11 + 100 121 111, true
2. Suppose predicate is true for n = k, i.e.
P(k) k2 k + 100, true k 11
3. Now it can be proved that
P(k+1) (k+1)2 (k+1) + 100, k2 + 2k +1 k +1 + 100 k2 + k 100 (by 1 and 2)
Hence P(k) P(K+1)
Proof by Induction
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Example 1• Prove that n2 n + 100 n 11SolutionInitially, base case
Solution set = {11} By, P(k) P(K+1) 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
Validity of Proof
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Reasoning of ProofExample 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., 1+3+5+…+(2k-1) = k2
• Note that: 1+3+5+…+(2k-1)+(2k+1) = k2+2k+1= (k+1)2
∴ P(k+1) true, by induction, P(n) is true for all n Z+
Another Proof 2
11
)1(2)12( nnnnniin
i
n
i
Another Easy Example
2
1
)12( nin
i
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Reasoning of ProofExample 3• Use mathematical Induction to prove that n < 2n for all n Z+
Proof• Let P(n) be the proposition that n < 2n
• Basis step : P(1) is true since 1 < 21 .• Inductive step : Assume that P(n) is true for a positive integer n = k,
i.e., k < 2k.• Now consider for P(k+1) : Since, k + 1 < 2k + 1 2k + 2k = 2.2k = 2k + 1 ∴ P(k+1) is true. It proves that P(n) is true for all n Z+.
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kH k
1...
3
1
2
11
21
2
nH n
The harmonic numbers Hk, k = 1, 2, 3, …, are
defined by
Use mathematical induction to show that
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,
whenever n is a nonnegative integer.
2/12
nH n
12/011120 HH
2/12
kH k
Example 4: Harmonic Numbers
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∴P(k+1) is true. Hence the statement is true for all n Z+.
kkkkkkkH222
1
22
1
12
1
2
1
3
1
2
11
12 1
12 2
1
22
1
12
1
kkkkH
12
1
22
1
12
1)
21(
kkk
k
kkkkkk
k
22
1
22
1
22
1)
21(
kk
kk
22
2)
21(
2
11
2
1
21
kk
Now consider
Example 4: Harmonic Numbers
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Fibonacci Sequences
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
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
Today Covered
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• 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 Fibonacci’s 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.
Fibonacci Sequence
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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, . . .
Fibonacci’s Problem
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Construction of Mathematical Model
end of month 2
end of month 1
end of month 4
end of month 3
end of month 6
end of month 5
F2 = 1
F3 = 2
F5 = 5
F6 = 8
F7 = 13. . . . . . . . .
F1 = 1
F0 = 0
F4 = 3
end of month 12
end of month 7
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• 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
Construction of Mathematical Model
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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 . . .
Computing Values using Mathematical Model
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21 kkk FFF
1 condition initialwith
2
10
21
FF
kFFF kkk
Theorem: The fibonacci sequence F0,F1, F2,…. Satisfies the recurrence relation
Find the explicit formula for this sequence.Solution:
Let tk is solution to this, then characteristic equation
The given fibonacci sequence
Explicit Formula Computing Fibonacci Numbers
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2
51 ,
2
51
2
411
21
tt
t
Fibonacci Sequence
For some real C and D fibonacci sequence satisfies the relation
0 1 0
F
2
51
2
51
0
0 2
51
2
51
0
0
00
0
FDC
DC
DCF
n
nDCF
nn
n
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1 2 12
51
2
51
2
51
2
51F
1 Now
1
1
FDC
DC
n
nn
D
52
51
5
1
2
51
5
1F
Hence5
1,
5
1C
get usly wesimultaneo 2 and 1 Solving
n
Fibonacci Sequence
Dr Nazir A. Zafar Advanced Algorithms Analysis and Design
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After simplifying we get
which is called the explicit formula for the Fibonacci sequence recurrence relation.
nn
2
51
5
1
2
51
5
1Fn
Fibonacci Sequence
5
1
5
1F
then 2
51 and
2
51Let
nnn
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Example: Compute F3
then 2
51 and
2
51 where
5
1
5
1F Since nn
n
33
3 2
51
5
1
2
51
5
1F
Verification of the Explicit Formula
8
555.1.35.1.31
5
1
8
555.1.35.1.31
5
1F Now,
22
3
555.1.35.1.315.8
1555.1.35.1.31
5.8
1F3
2555.1.35.1.31555.1.35.1.315.8
1F3
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Fibo-R(n)
if n = 0
then 0
if n = 1
then 1
else Fibo-R(n-1) + Fibo-R(n-2)
Recursive Algorithm Computing Fibonacci Numbers
Terminating conditions
Recursive calls
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• Least Cost: To find an asymptotic bound of computational cost of this algorithm.
Running Time of Recursive Fibonacci Algorithm
2 n )2()1(
2 if 1)(
nTnT
nnT
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Recursion Tree
Drawback in Recursive Algorithms
F(n)
F(n-1) F(n-2)
F(0) F(1)
F(n-2) F(n-3) F(n-3) F(n-4)
F(1) F(0)
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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, . . .
Generalization of Rabbits Problem
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Construction of Mathematical Model
F0 = 0
F1 = 1
F2 = 1
F3 = 3
F4 = 5
F5 = 11
F6 = 21
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• Total pairs at level k = Total pairs at level k-1 + Total pairs born at level k (1)
• SinceTotal 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
Construction of Mathematical Model
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• 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
Generalization
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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 . . .
Computing Values using Mathematical Model
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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, . . .
Another Generalization of Rabbits Problem
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Construction of Mathematical Model
F3 = 1
F4 = 2
F6 = 4
F8 = 9
F1 = 1
F0 = 0
F5 = 3
F2 = 1
F7 = 6
F9 = 13
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• 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
Construction of Mathematical Model
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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 . . .
Computing Values using Mathematical Model
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• 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
More Generalization
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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 Euclid’s algorithm.
Applications of Fibonacci Sequences
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