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Infinite Surds Coursework

Extracts from this document...

Introduction

Petri Alexia

12T1

Standard Maths

Infinite Surds Coursework

The following expression is an example of an infinite surd.

image00.png

Find the formula for an+1 in terms of a

a1 = image01.png

a2 = image12.pnga2 = image22.png

a3 = image31.pnga3 = image35.png

an+1 = image36.png

an= image37.png

Calculate the decimal values of the first ten terms of the sequence

a1 = 1.414213562373100

a2 = 1.553773974030040

a3 = 1.598053182478620

a4 = 1.611847754125250

a5 = 1.616121206508120

a6 = 1.617442798527390

a7 = 1.617851290609670

a8 = 1.617977530934740

a9 = 1.618016542231490

a10 = 1.618028597470230


Using technology, plot the relation between n and an. Describe what you notice.

21

1.618033988736670

22

1.618033988745810

23

1.618033988748630

24

1.618033988749500

25

1.618033988749770

26

1.618033988749860

27

1.618033988749880

28

1.618033988749890

29

1.618033988749890

30

1.618033988749890

31

1.618033988749890

32

1.618033988749890

33

1.618033988749890

34

1.618033988749890

35

1.618033988749890

36

1.618033988749890

37

1.618033988749890

38

1.618033988749890

39

1.618033988749890

40

1.618033988749890

n

an

1

1.414213562373100

2

1.553773974030040

3

1.598053182478620

4

1.611847754125250

5

1.616121206508120

6

1.617442798527390

7

1.617851290609670

8

1.617977530934740

9

1.618016542231490

10

1.618028597470230

11

1.618032322752000

12

1.618033473928150

13

1.618033829661220

14

1.618033939588790

15

1.618033973558280

16

1.618033984055430

17

1.618033987299220

18

1.618033988301610

19

1.618033988611370

20

1.618033988707090


By plotting the relation between n and an, one notices that as n increases, anincreases. However this increase is not proportional to the increase of n, an

...read more.

Middle

an. Describe what you notice.

21

2.0000000000

22

2.0000000000

23

2.0000000000

24

2.0000000000

25

2.0000000000

26

2.0000000000

27

2.0000000000

28

2.0000000000

29

2.0000000000

30

2.0000000000

31

2.0000000000

32

2.0000000000

33

2.0000000000

34

2.0000000000

35

2.0000000000

36

2.0000000000

37

2.0000000000

38

2.0000000000

39

2.0000000000

40

2.0000000000

n

an

1

1.847759065

2

1.9615705608

3

1.9903694533

4

1.9975909124

5

1.9993976374

6

1.9998494037

7

1.9999623506

8

1.9999905876

9

1.9999976469

10

1.9999994117

11

1.9999998529

12

1.9999999632

13

1.9999999908

14

1.9999999977

15

1.9999999994

16

1.9999999999

17

2.0000000000

18

2.0000000000

19

2.0000000000

20

2.0000000000


image09.png

By plotting the relation between n and an, one notices that as n increases, anincreases. However this increase is not proportional to the increase of n, anseems to be increasing towards 2.

Once n reaches 17 anceases to increase, remaining stable at 2.

This suggests that as n becomes very large an – an+1 = 0

As such, we can conclude that the exact value for this infinite surd is 2.

Consider the general infinite surd :

image10.png

Find the formula for an+1 in terms of a

a1 = image11.png

a2 = image13.pnga2 = image14.png

a3 = image15.pnga3 = image16.png

image17.png

x

...read more.

Conclusion

>20

19

1368

342

6

120

30

20

1520

380

7

168

42

21

1680

420

8

224

56

22

1848

462

9

288

72

23

2024

506

10

360

90

24

2208

552

11

440

110

25

2400

600

12

528

132

26

2600

650

13

624

156

27

2808

702

14

728

182

28

3024

756


For image28.png to be an integer, image29.png has to be an odd perfect square.

Values of odd perfect squares

57

Values of image29.png

3249

  1

59

1

3481

3

61

9

3721

5

63

25

3969

7

65

49

4225

9

67

81

4489

11

69

121

4761

13

71

169

5041

15

73

225

5329

17

75

289

5625

19

77

361

5929

21

79

441

6241

23

81

529

6561

25

83

625

6889

27

85

729

7225

29

87

841

7569

31

89

961

7921

33

91

1089

8281

35

93

1225

8649

37

95

1369

9025

39

97

1521

9409

41

99

1681

9801

43

101

1849

10201

45

103

2025

10609

47

105

2209

11025

49

107

2401

11449

51

109

2601

11881

53

111

2809

12321

55

113

3025

12769

One can notice that the value of the odd perfect square is a series with image30.png

image32.png

Therefore:

image33.png

y+1 is always an integer

The general statement that represents all the values of k for which the expression is an integer is image34.png with n any integer.

The general statement image34.png is valid for all integers and therefore has no limitations.

You can see from the steps above the process I took to find the general statement.

Page  of

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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