Sequences - Algorithms

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Sequences - Algorithms

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LATEST EDITIONAPRIL17, 2015AT15:43

Sequences - Algorithms

Proofreading of English byLaurence Weinstock

1 Sequences : general overview2

1.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Examples of sequences. . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Monotonicity of a sequence. . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Showing that a sequence is monotonic. . . . . . . . . . . . . . . . . 4

1.5 Graphing a sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Arithmetic sequences (review)6

2.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 How to recognize an arithmetic sequence. . . . . . . . . . . . . . . 6

2.3 Expression of the general term as a function ofn. . . . . . . . . . . 6

2.4 Sum of the firstnterms : finite series. . . . . . . . . . . . . . . . . . 6

3 Geometric sequences (review)7

3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 How to recognize a geometric sequence. . . . . . . . . . . . . . . . 7

3.3 Expression of the general term as a function ofn. . . . . . . . . . . 8

3.4 Sum of the firstnterms : finite series. . . . . . . . . . . . . . . . . . 8

3.5 Limit of a geometric sequence. . . . . . . . . . . . . . . . . . . . . . 8

4 Algorithms9

4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Writing conventions for algorithms. . . . . . . . . . . . . . . . . . . 9

4.3 Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3.2 Variable declaration. . . . . . . . . . . . . . . . . . . . . . . 10

4.4 Assigning a numeric value to a variable. . . . . . . . . . . . . . . . 10

4.5 Reading and writing a variable. . . . . . . . . . . . . . . . . . . . . 11

4.6 Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.7 Loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.7.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.7.2 The while loop. . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.7.3 Loop counter. . . . . . . . . . . . . . . . . . . . . . . . . . . 13

PAUL MILAN1TERMINALE S

1 Sequences : general overview

1.1 Definition

Definition 1 :A sequence(un)is a function whose domain isN(or possibly N-[[0,k]]) and whose codomain isR, that relates a real number writtenunto an integern. (un):NorN-[[0,k]]-→R n?-→un

Note :

•N-[[0,k]]is the setNminus the firstknatural numbers. •unis called the general term of the sequence(un). •Note the difference between the sequence(un)and the general termun •If a sequence is defined from the indexponwards, it is denoted(un)n?p

Examples :

•(un): 2; 5; 8; 11; 14; 17; ... an arithmetic sequence •(vn): 3; 6; 12; 24; 48; 96; ... a geometric sequence

1.2 Examples of sequences

a)Explicitly defined sequences:un=f(n): u n=1 n,n?N?,vn=⎷n-3,n?3 b)Recursive sequencesdefined by a recurrence relation and one or more star- ting values : •By one starting value :un+1=f(un)?u 0=4 u n+1=0.75un+2

To calculateun, givenn

Variables:N,Iintegers

Ureal number

Inputs and initialization

ReadN

4→U returns u0

Processing

forIfrom 1 toNdo

0,75U+2→Urecurrencerelation

end

Output: PrintU

N5102030

U7.05087.77477.98737.9999

The sequence seems to be increasing and

to converge to 8

0=1,u1=1

u n+2=un+1+un

To calculateun, givenn

Variables:N,Iintegers

U,V,Wreal numbers

Inputs and initialization

ReadN

1→V returns u0

1→U returns u1

Processing

forIfrom 2 toNdo

U+V→W recurrence relation

V→U

W→U?

the next term in the sequence is calculated end

Output: PrintU

N10152030

U89987109461346269

PAUL MILAN2TERMINALE S

1. SEQUENCES : GENERAL OVERVIEW

c) A sequence can be defined via another sequence or a sum of terms, ... Given(un), the sequence(vn)can be defined by :vn=un-4 w n=n∑ i=11 i=1+12+13+···+1n To determine an approximate value of a particular term(wn), the following program can be written :

For instance, to find the values forw5,

10,w50.

If one wants to find the exact result as

a fraction with the TI 82, type : "Disp W?Frac"

We find the following values :

•w5=13760?2.283 •w10?2.923,w50?4.499

Variables:N,Iintegers

Wreal number

Inputs and initialization

ReadN

0→W

Processing

forIgoing from 1 toNdo

W+1I→W

end

Output: PrintW

d) A sequence can also be defined by an explicit formula without specifying the value of any term. For instance, the sequence(dn)that relates thenth termdnto thenth decimal of the numberπ=3,141592... :d1=1,d2=4,d3=1,d4=5,d5=9, d

6=2 ...

1.3 Monotonicity of a sequence

Definition 2 :Let(un)be a numeric sequence. The sequence(un)is said to be : •strictlyincreasing(from indexk) if u n+1>unfor all integersn?k •strictlydecreasing(from indexk) if u n+1Note : There are sequences that are neither increasing nor decreasing:an= (-1)n The first terms of the sequence are not necessarily important for determining the behavior of a sequence. They can, however, give an indication as to the monoto- nicity of the sequence.

PAUL MILAN3TERMINALE S

1.4 Showing that a sequence is monotonic

Law 1 :In order to show that a sequence is monotonic : •Determine the sign of the differenceun+1-un If the difference is positive (resp. negative) from indexkonwards, then the sequence is increasing (resp. decreasing) forn?k •If all the terms of the sequence are strictly positive from indexkonwards, then we can compare the ratioun+1 unto 1 If the ratio is greater than 1 (resp less than 1) from indexkonwards, the se- quence is increasing (resp. decreasing) forn?k •If the sequence is defined explicitly, then we can determine whetherthe func- tionfis monotonic inR+ •Using mathematical induction (see next chapter).

Examples :

•Show that the sequence(un)defined for allnby :un=n2-nis increasing.

First, determine the difference :un+1-un

u n+1-un= (n+1)2-(n+1)-(n2-n) =n2+2n+1-n-1-n2+n =2n Seeing as 2n?0 for alln?N, we haveun+1-un?0. The sequence(un)is increasing from index 0. •Show that the sequence(un)defined for alln?N?by :un=2nnis increasing. Given that for alln?N?,un>0, it is possible to compare the ratioun+1 unto 1 : u n+1 un=2 n+1 n+1 2n n= 2n+1 n+1×n2n=2nn+1 Seeing asn?1, by addingnto each side of the inequality, we find that

2n?n+1, hence :2n

n+1?1

As?n?1un+1

un?1, the sequence(un)is increasing from index 1. •Show that the sequence(un)defined for alln?2 by :un=2n+1n-1is decreasing. Let us determine whether the functionfdefined byf(x) =2x+1 x-1and whose domain isI= [2 ;+∞[is monotonic. This function is differentiable onI, so f ?(x) =2(x-1)-(2x+1) (x-1)2=-3(x-1)2thereforef?(x)<0x?I The functionfis decreasing onI, so the sequence(un)is decreasing.

PAUL MILAN4TERMINALE S

1. SEQUENCES : GENERAL OVERVIEW

1.5 Graphing a sequence

To visualize a sequence defined by the recurrence relationun+1=f(un), we can plot the graph of the functionfand the line with the equationy=x. The line is used to carry over the terms of the sequence onto thex-axis.

Consider the sequence(un)defined by :

?u 0=0.1 u n+1=2un(1-un)

We obtain the following representation

of the sequence after plotting the graph C fof the functionfdefined by : f(x) =2x(1-x) 0.5 0.5

Ou0u1u2u3u

4u 1u 2uquotesdbs_dbs4.pdfusesText_7

[PDF] Sequences - Algorithms

LATEST EDITIONAPRIL17, 2015AT15:43

Sequences - Algorithms

Proofreading of English byLaurence Weinstock

Contents

1 Sequences : general overview2

1.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Examples of sequences. . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Monotonicity of a sequence. . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Showing that a sequence is monotonic. . . . . . . . . . . . . . . . . 4

1.5 Graphing a sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Arithmetic sequences (review)6

2.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 How to recognize an arithmetic sequence. . . . . . . . . . . . . . . 6

2.3 Expression of the general term as a function ofn. . . . . . . . . . . 6

2.4 Sum of the firstnterms : finite series. . . . . . . . . . . . . . . . . . 6

3 Geometric sequences (review)7

3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 How to recognize a geometric sequence. . . . . . . . . . . . . . . . 7

3.3 Expression of the general term as a function ofn. . . . . . . . . . . 8

3.4 Sum of the firstnterms : finite series. . . . . . . . . . . . . . . . . . 8

3.5 Limit of a geometric sequence. . . . . . . . . . . . . . . . . . . . . . 8

4 Algorithms9

4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.2 Writing conventions for algorithms. . . . . . . . . . . . . . . . . . . 9

4.3 Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.3.2 Variable declaration. . . . . . . . . . . . . . . . . . . . . . . 10

4.4 Assigning a numeric value to a variable. . . . . . . . . . . . . . . . 10

4.5 Reading and writing a variable. . . . . . . . . . . . . . . . . . . . . 11

4.6 Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.7 Loops. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.7.1 Definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.7.2 The while loop. . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.7.3 Loop counter. . . . . . . . . . . . . . . . . . . . . . . . . . . 13

PAUL MILAN1TERMINALE S

CONTENTS

1 Sequences : general overview

1.1 Definition

Note :

Examples :

1.2 Examples of sequences

To calculateun, givenn

Variables:N,Iintegers

Ureal number

Inputs and initialization

4→U returns u0

Processing

0,75U+2→Urecurrencerelation

Output: PrintU

N5102030

U7.05087.77477.98737.9999

The sequence seems to be increasing and

0=1,u1=1

To calculateun, givenn

Variables:N,Iintegers

U,V,Wreal numbers

Inputs and initialization

1→V returns u0

1→U returns u1

Processing

U+V→W recurrence relation

V→U

W→U?

Output: PrintU

N10152030

U89987109461346269

PAUL MILAN2TERMINALE S

1. SEQUENCES : GENERAL OVERVIEW

For instance, to find the values forw5,

10,w50.

If one wants to find the exact result as

We find the following values :

Variables:N,Iintegers

Wreal number

Inputs and initialization

0→W

Processing

W+1I→W

Output: PrintW