2011 Term 4 MAC Assigment

8/2/2019 2011 Term 4 MAC Assigment

1/6

TASK 1

1. F30 = 8320402. F14 = 3773. F146 = 1.45449 1030

TASK 2

1. L30 = 11498512. L14 = 5213. L146 = 2.01005 1030

TASK 3

(i)n

n

n F

FLim

1

= 1.61803

(ii)

When the value of A1 is increased, the ratio of A2

to A1 is effectively smaller but the next ratioA3 to A2 is evidently larger and in this forms a

cycle until the ratio approaches the limit of 1.62803.


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As the value of A2 increases, the ratio of A2 to A1 also grows while the

proceeding ratio of A3 to A2 is diminished and this forms a reverse cycle as the ratio approaches

the limit of 1.618034.

When A1 = A2, the ratio is slowly increasing and approaching

1.618034.

TASK 4

i)


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ii)

iii)

iv)

( )

TASK 5


4/6

TASK 6

TASK 8

N 1 2 3 4 5 6 7 8 9 10 11

F(n) 1 1 2 3 5 8 13 21 34 55 89

L(n) 2 1 3 4 7 11 18 29 47 76 123

Ln = Fn-1 + Fn+1

F2n = Ln Fn

TASK 9

12

= 1 = 11


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12

+ 12

= 2 = 12

12

+ 12

+ 22

= 6 = 23

12

+ 12

+ 22

+ 32

= 15 = 35

12 + 12 + 22 + 32 + 52 = 40 = 58

12

+ 12

+ 22

+ 32

+ 52

+82

= 104 = 813

12

+ 12

+ 22

+ 32

+ 52

+82

+132

= 273 = 1321

The sum of the squared values of consecutive terms in the Fibonacci sequence is equal to the

last number in the multiplication sequence multiplied by the proceeding number in the

Fibonacci sequence.

b) (F1)2+ (F2)

2+ (F3)

2+ (F4)

2+ (F5)

2+ (F6)

2+ .. + (Fn)

2= Fn Fn+1

c) C2 = C1 (A2)2

TASK 10

Frank Albert Benford (1883 1948) was an American electrical engineer and physicist best

known for his Benford's Law, a statistical statement about the occurrence of digits in lists of data.

Benford is also known for having devised, in 1937, an instrument for measuring the refractive

index of glass. He was also an expert in optical measurements, he published 109 papers in the

fields of optics and mathematics and was granted 20 patents on optical devices.

Benford's law, also called the first-digit law, states that in lists of numbers from many (but not

all) real-life sources of data, the leading digit is distributed in a specific, non-uniform way.

According to this law, the first digit is 1 about 30% of the time, and larger digits occur as the

leading digit with lower and lower frequency, to the point where 9 as a first digit occurs less

than 5% of the time. A set of numbers is said to satisfy Benford's law if the leading digit d (d

{1,, 9}) occurs with probability:

Benford was a research physicist at General Electric in the 1930s when he noticed something

unusual about a book of logarithmic tables. The first pages showed more wear than the last

pages, indicating that numbers beginning with the digit 1 were being looked up more often than

numbers beginning with 2 through 9. Benford seized upon this idea and spent years collecting

data to show that this pattern was widespread in nature. In 1938, Benford published his results,

citing more than 20,000 values such as atomic weights, numbers in magazine articles, baseball

statistics, and the areas of rivers.


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This counter-intuitive result has been found to apply to a wide variety of data sets, including

electricity bills, street addresses, stock prices, population numbers, death rates, lengths of rivers,

physical and mathematical constants, and processes described by power laws (which are very

common in nature). It tends to be most accurate when values are distributed across multiple

orders of magnitude.

To test the accuracy and the usefulness of this law, the initial digits of numbers from F14 to F146

were recorded and tabulated.

Here is a table of the initial digits as produced by the Fibonacci Calculator:

Initial digit frequencies of fib(n) for n from 14 to 146:

Digit 1 2 3 4 5 6 7 8 9

Frequency 40 23 16 13 11 8 8 8 6

Percent 30 17 12 10 8 6 6 6 5

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2011 Term 4 MAC Assigment