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# Sequences

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by

## Andrew Shoemaker

on 19 March 2013

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#### 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
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