Discrete Maths 2003 Lecture 23 3 Slides Pp

8/13/2019 Discrete Maths 2003 Lecture 23 3 Slides Pp

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Lecture 23, 11-September-200Discrete Mathematics 2003

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Sequences

Last lecture: We introduced sequences, whichare infinite lists of nos in a particular order

e.g. 1, 3, 5, 7, 9, 11, & 5, 10, 15, 20, 25, 30,

The nth term of a sequence is denoted by t(n) e.g. t(1) is the 1st term & t(7) is the 7th term

Sometimes we can write down an explicit rule forthe nth term of a sequence

e.g. For 1, 3, 5, 7, 9, 11, , it is t(n) = 2n 1

Such a formula provides a non-recursivedefinition of the sequence

2

Algorithms to Generate Sequences

A non-recursive defn can be used in an algorithmto obtain the first m terms of a sequence

An algorithm is a list of steps or instructions forperforming a task

The algorithm on the following slide generatesthe first m terms of 1, 3, 5, 7, 9, 11,

Note that the algorithm is written inpseudocode,a form of structured English which can be easilyconverted to code in the chosen programming

language

3

Algorithm for First m Terms of

1, 3, 5, 7, 9, 11,

1. Input m

2. Forn = 1 tom do (n is the index variable)

2.1. t 2n 1 (means assignment) 2.2. Output t

This uses the non-recursive defnt(n) = 2n 1

However, the sequence can be generated in adifferentway, based on the fact that the first termis 1, and each subsequent term is obtained byadding 2 to the previous term


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A Different Algorithm for First m

Terms of 1, 3, 5, 7, 9, 11,

1. Input m

2. t 1

3. Output t

4. Forn = 2 tom do

4.1. t t+ 2

4.2. Output t

A trace of this algorithm for m = 5 shows thatit does, indeed, output the first 5 terms of thesequence

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Recursive Definitions

The mathematical notation for the first term is 1,and each subsequent term is obtained by adding 2to the previous term is:

t(1) = 1

t(n) = t(n 1) + 2 (n > 1)

These 2 equations combine to give a recursivedefinition of the sequence

It is recursive because the formula for t(n)contains the previous term t(n 1) so theformula calls itself

t(1) = 1 gives a base (starting pt) for the recursion

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Recursive & Non-Recursive Defns

We thus have 2 ways of describing the sequence1, 3, 5, 7, 9, 11,

1: Recursive Definition

t(1) = 1 t(n) = t(n 1) + 2 (n > 1)

2: Non-Recursive Definition

t(n) = 2n 1

Either definition can be used to calculate terms ofthe sequence

Example: Calculate the 5th term using both defns


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More Examples

Example: Find recursive and non-recursivedefns for the sequence 3, 7, 11, 15, 19, 23,

Note that in the above example, the non-recursive formula is more difficult to find thanthe recursive formula, but is easier to use once ithas been found this is fairly typical

Example: Find the first 4 terms of the recursivelydefined sequence

t(1) = 6

t(n) = 2n + 2t(n 1) (n > 1)

Answer: 6, 16, 40, 96

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Example of an Algorithm

Example: Write an algorithm to output the firstm terms of the sequence in the previous example

Solution: The following algorithm is suitable:

1. Input m

2. t 6

3. Output t

4. Forn = 2 tom do

4.1. t 2n + 2t

4.2. Output t

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Iterative Algorithms

The algorithm on the previous slide is basedon a recursive definition of the sequence,

but is nota recursive algorithm. Instead, it is an iterative algorithm

The word iterate means to do somethingrepeatedly, as in a For-do loop

We will look at recursive algorithms later


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An 800-year old Problem

Example: The following

problem was posed by the Italianmathematician Leonardo Pisano

(Leonardo of Pisa) in 1202

A man put a pair of newborn rabbits in a place

surrounded on all sides by a wall. How many

pairs of rabbits will be present at later times if

it is supposed that every month each pair begets

a new pair which from the second month on

becomes productive?

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Solution to the Problem

TimeNo. of Non-Productive

Pairs

No. ofProductive

Pairs

TotalNo. of

Pairs

Initial 1 0 1

After 1 month 0 1 1

After 2 months 1 1 2








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More on the Rabbit Problem

The resulting sequence is1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,

This is known as the Fibonacci sequence(Fibonacci was another name for Leonardo)

It is obtained from the pattern 1st term is 1,2nd term is 1, then each term is the sum of thetwo previous terms

This leads to the recursive definition

t(1) = 1, t(2) = 1,

t(n) = t(n 2) + t(n 1) (n > 2)


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More on the Fibonacci Sequence

Its not so straightforward to obtain a non-

recursive defn of the Fibonacci sequence

In fact, the formula is

+=

nn

nt2

51

2

51

5

1)(

This goes to show that non-recursive

formulas can sometimes be quite difficult to

calculate and rather complicated in nature!

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Discrete Maths 2003 Lecture 23 3 Slides Pp