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CONTENTS
TITLE PAGE
PART A
INTRODUCTION 3-8
THREE EXAMPLES OF CREATIONS 9-15
ESSAY ABOUT FIBONACCI SEQUENCES 16-17
PART B
INTEREST 19-20
PART C
REFLECTION 22
BIBLIOGRAPHIES 23
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RESOURCES 1
In mathematics, the Fibonacci numbers are the following sequence of numbers:
By definition, the first two Fibonacci numbers are 0 and 1, and each remaining
number is the sum of the previous two. Some sources omit the initial 0, instead
beginning the sequence with two 1s.
In mathematical terms, the sequence Fn of Fibonacci numbers is defined by the
recurrence relation
with seed values
The Fibonacci sequence is named after Leonardo of Pisa, who was known as
Fibonacci (a contraction of filius Bonaccio, "son of Bonaccio".) Fibonacci's 1202 book
Liber Abaci introduced the sequence to Western European mathematics, although
the sequence had been previously described in Indian mathematics.
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Origins
The Fibonacci sequence was well known in ancient India, where it was applied to the
metrical sciences (prosody), long before it was known in Europe. Developments
have been attributed to Pingala (200 BC), Virahanka (6th century AD), Gopla
(c.1135 AD), and Hemachandra (c.1150 AD).
The Fibonacci sequence is formed by adding S to a pattern of length n 1, or L to a
pattern of length n 2; and the prosodicists showed that the number of patterns of
length n is the sum of the two previous numbers in the sequence. Donald Knuth
reviews this work in The Art of Computer Programming.
In the West, the sequence was studied by Leonardo of Pisa, known as Fibonacci, in
his Liber Abaci (1202). He considers the growth of an idealised (biologically
unrealistic) rabbit population, assuming that:
y In the "zeroth" month, there is one pair of rabbits (additional pairs of
rabbits = 0).
y In the first month, the first pair begets anothe r pair (additional pairs of
rabbits = 1).
y In the second month, both pairs of rabbits have another pair, and the first pair
dies (additional pairs of rabbits = 1).
y In the third month, the second pair and the new two pairs have a total of three
new pairs, and the older second pair dies (additional pairs of rabbits = 2).
The laws of this are that each pair of rabbits has 2 pairs in its lifetime, and dies.
Let the population at month n be F(n). At this time, only rabbits who were alive at
month n 2 are fertile and produce offspring, so F(n 2) pairs are added to the
current population of F(n 1). Thus the total is F(n) = F(n 1) + F(n 2).
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List of Fibonacci numbers
The first 21 Fibonacci numbers (sequence A000045 in OEIS), also denoted as Fn, for
n = 0, 1, 2, ... ,20 are:
F
0
F
1
F
2
F
3
F
4
F
5
F
6F7 F8 F9
F1
0
F1
1F12 F13 F14 F15 F16 F17 F18 F19 F20
0 1 1 2 3 5 81
3
2
1
3
455 89
14
4
23
3
37
7
61
0
98
7
159
7
258
4
418
1
676
5
Using the recurrence relation, the sequence can also be extended to negative index
n. The result satisfies the equation
Thus the complete sequence is
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RESOURCES 2
The Fibonacci Sequence
edieval mathematician and businessman Fibonacci ( Leonardo Pisano)
posed the following problem in his treatise Liber Abaci (pub. 1202):
How many pairs of rabbits will be produced in a year, beginning with a single
pair, if in every month each pair bears a new pair which becomes productive from
the second month on?
It is easy to see that 1 pair will be produced the first month, and 1 pair also in the
second month (since the new pair produced in the first month is not yet mature), and
in the third month 2 pairs will be produced, one by the original pair and one by thepair which was produced in the first month. In the fourth month 3 pairs will be
produced, and in the fifth month 5 pairs. After this things expand rapidly, and we get
the following sequence of numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
This is an example of a recursive sequence, obeying the simple rule that to calculate
the next term one simply sums the preceding two:
F(1) = 1
F(2) = 1
F(n) = F(n 1) + F(n 2)
Thus 1 and 1 are 2, 1 and 2 are 3, 2 and 3 are 5, and so on.
This simple, seemingly unremarkable recursive sequence has fascinated
mathematicians for centuries. Its properties illuminate an array of surprising topics,
from the aesthetic doctrines of the ancient Greeks to the growth patterns of plants
(not to mention populations of rabbits!).
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RESOURCES 3
THE FIBONACCI SEQUENCE, SPIRALS AND THE GOLDEN MEAN
The Fibonacci sequence exhibits a certain numerical pattern which originated as the
answer to an exercise in the first ever high school algebra text. This pattern turned
out to have an interest and importance far beyon d what its creator imagined. It can
be used to model or describe an amazing variety of phenomena, in mathematics and
science, art and nature. The mathematical ideas the Fibonacci sequence leads to,
such as the golden ratio, spirals and self - similar curves, have long been appreciated
for their charm and beauty, but no one can really explain why they are echoed so
clearly in the world of art and nature.
The story began in Pisa, Italy in the year 1202. Leonardo Pisano Bigollo was a
young man in his twenties, a member of an important trading family of Pisa. In his
travels throughout the Middle East, he was captivated by the mathematical ideas that
had come west from India through the Arabic countries. When he returned to Pisa he
published these ideas in a book on mathematics called Liber Abaci, which became a
landmark in Europe. Leonardo, who has since come to be known as Fibonacci,
became the most celebrated mathematician of the Middle Ages. His book was a
discourse on mathematical methods in commerce, but is now remembered mainly
for two contributions, one obviously important at the time and one seemingly
insignificant.
The important one: he brought to the attention of Europe the Hindu system for writing
numbers. European tradesmen and scholars were still cling ing to the use of the old
Roman numerals; modern mathematics would have been impossible without thischange to the Hindu system, which we call now Arabic notation, since it came west
through Arabic lands.
The other: hidden away in a list of brain-teasers , Fibonacci posed the following
question:
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2.EXAMPLE IN NATURE 1
Probably most of us have never taken the time to examine very carefully the number
or arrangement of petals on a flower. If we were to do so, several things would
become apparent. First, we would find that the number of petals on a flower is often
one of the Fibonacci numbers. One-petalled ...
white calla lily
and two-petalled flowers are not common.
euphorbia
Three petals are more common.
trillium
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There are hundreds of species, both wild and cultivated, with five petals.
columbine
Eight-petalled flowers are not so common as five-petalled, but there are quite a
number of well-known species with eight.
bloodroot
Thirteen, ...
black-eyed susan
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twenty-one and thirty-four petals are also quite common. The outer ring of ray floret s
in the daisy family illustrate the Fibonacci sequence extremely well. Daisies with 13,
21, 34, 55 or 89 petals are quite common.
shasta daisy with 21 petals
Ordinary field daisies have 34 petals ... a fact to be
taken in consideration when playing "she loves me, she loves me not". In saying that
daisies have 34 petals, one is generalizing about the species - but any individual
member of the species may deviate from this general pattern. There is more
likelihood of a possible under development than over -development, so that 33 is
more common than 35.
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EXAMPLE IN NATURE 2
With the scale patterns of pinecones, the seed patterns of sunflowers and even the
bumps on pineapples we have something rather different.
The seed-bearing scales of a pinecone are really modified leaves, crowded together
and in contact with a short stem. Here we do not find phyllotaxis as it occurs with
true leaves and suchlike. However, we can detect two prominent arrangements of
ascending spirals growing outward from the point where it is attached to the branch.
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In the pinecone pictured, eight spirals can be seen to be ascending up the cone in a
clockwise direction ...
while thirteen spirals ascend more steeply in a counterclockwise direction.
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EXAMPLE IN NATURE 3
Pineapple scales are also patterned into spirals and, ...
because they are roughly hexagonal in shape, three distinct sets of spirals may be
observed.
One set of 5 spirals ascends at a shallow angle to the right, ...
a second set of 8 spirals ascends more steeply to the left, ...
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and the third set of 13 spirals ascends very steeply to the right.
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ESSAY
Fibonacci sequence is the sequence of numbers, 1, 1, 2, 3, 5, 8, 13, . . . , in which
each successive number is equal to the sum of the two preceding numbers.The
Fibonacci sequences are the sequence of numbers defined by the linear
recurrence equation
with . As a result of the definition,it is conventional to define .
The Fibonacci numbers for , 2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ...
Fibonacci numbers can be viewed as a particular case of the Fibonacci polynomials
with . Fibonacci numbers are implemented in Mathematica as
Fibonacci[n].
The Fibonacci sequence makes its appearance in other ways within mathematics as
well. For example, it appears as sums of oblique diagonals in Pascals triangle:
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1.SIMPLE INTEREST
Simple Interest Made Simple!
y The formula for simple interest is I = Prt
y In order to determine any of the variables (I, P, r, t) you only need the other 3
y When you know three of the four values, here's how you calculate the
unknown.
o To find principal (P) P = I/rt
o To find the interest rate (r) r = I/Pto To find the period of time ( t) t = I/Pr
The easy way to remember the above is to look at it as Simple Interest -
Triangular Forumlas
IIIII
Prt
Think of I at the top of the triangle and P rtat the bottom. The formulas above
are easy to remember when you keep the triangle in your mind as your visual.
A word about time. When the time is 6 months. It will be entered a .6 If you
have specific days such as 215 then you will divide the days by 365 which
gives you .58 for time. Don't forget to count the days of the month properly -
30 days in September, April, June, November and except for Feb. all the rest
have 31!
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2.COMPOUND INTEREST
Today, calculators will do the computational work for you, however, here's a
breakdown of how to calculate compound interest:
Compound interest is interest that is paid on both the principal and also on any
interest from past years. Its often used when someone reinvests any interest they
gained back into the original investment. For example, if I got 15% interest on my
$1000 investment, the first year and I reinvested the money back into the original
investment, then in the second year, I would get 15% interest on $1000 and the $150
I reinvested. Over time, compound interest will make much more money than simple
interest. The formula used to calculate compound interest is:
M = P( 1 + i )n
M is the final amount including the principal.
P is the principal amount.
i is the rate of interest per year.
n is the number of years invested.
Applying the Formula
Let's say that I have $1000.00 to invest for 3 years at rate of 5% compound interest.
M = 1000 (1 + 0.05) 3 = $1157.62.
You can see that my $1000.00 is worth $1157.62.
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REFLECTION
First I would like to say thank you to Allah S.W.T because I can finish my
basic mathematic task before the final date to send the task. I also would like to
thank the lecturer and also my tons of friends who give me great help and support.
This basic mathematics task make me know deeper and know more about
mathematics and the way to solve the mathematics problems such as which involve
the Fibonaccis and simple also compound interest.
I also get know about mathematics are around me in the nature of Gods
creation. By doing this tasks, I have learnt about the most distinguished
mathematicians of the middle ages who is Leonardo of Pisa(1170-1250). . I also
found that the structure of pascals triangle is very interesting and useful .
I learnt how to use the formula to calculate compound interest, total instalment
price, finance charge. It will help me in the future when it is about problems in solving
money and all related to it in my daily life.
I also can cooperate better with my friends when it is about teamworking
condition to solve problems, my critical thinking skills are growing and expanding and
it sure give me and my friends help so then we can keep going better and better.
Positif thinking is generated from our hard working and teamwork. We now believe
that we can do other task better because of this ba sic mathematics experience task.
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BIBLIOGRAPHIES
y http://www.math.temple.edu/~reich/Fib/fibo.html
y http://en.wikipedia.org/wiki/Fibonacci_sequence
y http://www.mathacademy.com/pr/prime/articles/fibonac/
y http://www.about.com