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# Sequences

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by

Tweet## Andrew Shoemaker

on 19 March 2013#### Transcript of Sequences

an ordered set of mathematical objects. Sequences A sequence beginning with 1,1 where each subsequent number is the sum of the previous two.

1,1,2,3,5,8,13,21,34,55,89,144,233... Fibonacci Sequence A Cauchy Sequence is one that will converge in the real numbers.

Example: Babylonian Method of computing square roots.

(n+1)=(n+(2/n))/2

will converge to the square root of 2. Cauchy Sequence A sequence of numbers whose consecutive terms have a constant difference.

Add-Add Sequence Linear Sequence A sequence of numbers whose consecutive terms have a common ratio.

Multiply-Multiply Sequence Geometric Series A numerical sequence with a common second difference. Quadratic Sequence Add-Multiply Sequence Power Sequence A Farey sequence for any positive integer n is the set of fractions such that: Farey Sequence Multiply-Add Sequence Logarithmic Sequence The fraction is in lowest terms.

The denominator is less than or equal to n.

The fractions are arranged in increasing order. x f(x)

1 3

2 5

3 7

4 9

5 11

6 13

7 15

8 17 x f(x)

5 1

10 2

15 3

20 4

25 5

30 6

35 7

40 8 x f(x)

1 3

2 9

4 27

8 81

16 243

32 729

64 2187

128 6561 x f(x)

2 9

6 18

18 36

54 72

162 144

486 288

1458 576

4374 1152 x f(x)

1 5

2 20

3 37

4 56

5 77

6 100

7 125

8 152 x f(x)

1 5

2 13

3 25

4 41

5 61

6 85

7 113

8 145 x f(x)

1 6

2 12

3 24

4 48

5 96

6 192

7 384

8 768 x f(x)

1 1

4 3

7 9

10 27

13 81

16 243

19 729

21 2187 x f(x)

2 1

6 2

18 3

54 4

162 5

486 6

1458 7

4374 8 x f(x)

4 2

8 4

16 6

32 8

64 10

128 12

256 14

512 16

Full transcript1,1,2,3,5,8,13,21,34,55,89,144,233... Fibonacci Sequence A Cauchy Sequence is one that will converge in the real numbers.

Example: Babylonian Method of computing square roots.

(n+1)=(n+(2/n))/2

will converge to the square root of 2. Cauchy Sequence A sequence of numbers whose consecutive terms have a constant difference.

Add-Add Sequence Linear Sequence A sequence of numbers whose consecutive terms have a common ratio.

Multiply-Multiply Sequence Geometric Series A numerical sequence with a common second difference. Quadratic Sequence Add-Multiply Sequence Power Sequence A Farey sequence for any positive integer n is the set of fractions such that: Farey Sequence Multiply-Add Sequence Logarithmic Sequence The fraction is in lowest terms.

The denominator is less than or equal to n.

The fractions are arranged in increasing order. x f(x)

1 3

2 5

3 7

4 9

5 11

6 13

7 15

8 17 x f(x)

5 1

10 2

15 3

20 4

25 5

30 6

35 7

40 8 x f(x)

1 3

2 9

4 27

8 81

16 243

32 729

64 2187

128 6561 x f(x)

2 9

6 18

18 36

54 72

162 144

486 288

1458 576

4374 1152 x f(x)

1 5

2 20

3 37

4 56

5 77

6 100

7 125

8 152 x f(x)

1 5

2 13

3 25

4 41

5 61

6 85

7 113

8 145 x f(x)

1 6

2 12

3 24

4 48

5 96

6 192

7 384

8 768 x f(x)

1 1

4 3

7 9

10 27

13 81

16 243

19 729

21 2187 x f(x)

2 1

6 2

18 3

54 4

162 5

486 6

1458 7

4374 8 x f(x)

4 2

8 4

16 6

32 8

64 10

128 12

256 14

512 16