In my research of the Fibonacci Numbers, I have found that the Fibonacci numbers appear anywhere from leafs on plants, patterns of flowers, in fruits, some animals, even in the human body. Could this be nature’s numbering system?
For those who are unfamiliar with the Fibonacci numbers they are a series of numbers discovered by Leonardo Fibonacci in the 12th century in an experiment with rabbits. The order goes as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610 and so on. Starting with 1, each new number is simply the sum of the two before it. The ratio, which is called the Golden Ratio, between the numbers is 1.618034
The most exciting thing about the Fibonacci numbers is how it is portrayed in the human body. For example, the Fibonacci numbers can be seen in the human hand. We have 1 thumb on each hand, 2 bones in each thumb, 3 bones in each finger, 5 digits on each hand, and 8 fingers. You will also find that you have 1 nose, 2 eyes, 3 segments in each limb and the 5 fingers on each hand. Not to mention the Golden Ratio being found in the proportions and measurements of the human body. The ratio between the forearm and the hand. The ratio of the distance between the navel and the knee. The ratio of distance between the knee and the end of the foot. These are just a few examples that I found to be very interesting.
Aside from the body, the Fibonacci numbers are found in the majority of flowers. If you count the number of petals on a flower, you might find the total to be one of the Fibonacci numbers. Lilies and the iris flower have 3 petals. The buttercup and wild roses have 5 petals. Delphiniums have 8 petals. Ragworts and cineraria’s have 13 petals and some flowers even have 55 or 89 petals.
The Fibonacci numbers are not limited to flower petals. Some items found in nature that are connected to the Fibonacci numbers are, the patterns of a pinecone, the seed pattern of the sunflower, the pineapple, some vegetables, and the starfish,...
...Arithmetic Sequence is a sequence of numbers in which each succeeding term differs from the preceding term by the same amount. This amount is known as the common difference and can be found using a specific formula by substituting the numbers from the word problem into the equation. When you plug in all the information, you are able to find out the money that needs to be spent and saved in the following word problems.
35. A person hired a firm to build a CB radio tower. The firm charges $100 for labor for the first 10 feet. After that, the cost of the labor for each succeeding 10 feet is $125 more than the preceding 10 feet will cost $125, the next ten feet will cost $150 etc. How much will it cost to build a 90 foot tower?
an=a1+ (n1) d
a125=100+ (1251) (150)
a125=100+124(150)
a125=100+18600
a125=18700
sn =n (a1 + an) / 2
= 125 (100+18700) /2
=125(1880) /2
=62.5 (18800) =1175000
The cost to build a 90foot tower is $11,750.
37. A person deposited $500 in a savings account that pays 5% annual interest that is compound yearly. At the end of 10 years, how much money will be in the savings account?
S+ (0.5) S n=10
S+ (1+0.5) r=1.05
S (1.05) a1= 500(1.05) =525
an= a1(rn1)
a10=525(1.059)
a10=525(1.551328216)
a10=814.4473134
The balance in the savings account at the end of 10 years will be $814.44.
I chose to use the Arithmetic...
...Anatolia College 
Mathematics HL investigation

The Fibonacci sequence 
Christos Vassos

Introduction
In this investigation we are going to examine the Fibonacci sequence and investigate some of its aspects by forming conjectures and trying to prove them. Finally, we are going to reach a conclusion about the conjectures we have previously established.
Segment 1: The Fibonacci sequence
The Fibonaccisequence can be defined as the following recursive function:
Fn=un1+ un2
Where F0=0 and F1=1
Using the above we can find the first eight terms of the sequence. An example of calculations is given below:
F2=F1F0F2=1+0=1
We are able to calculate the rest of the terms the same way:
F0  F1  F2  F3  F4  F5  F6  F7 
0  1  1  2  3  5  8  13 
Segment 2: The Golden ratio
In order to define the golden ratio we need to examine the following sketch:
The line above is divided into two segments in such a way that ABAP=APPB
The ratio described above is called the golden ratio.
If we assume that AP=x units and PB=1 units we can derive the following expression:
x+1x=x1
By solving the equation x2x1=0 we find that: x=1+52
Segment 3: Conjecture of φn
In this segment we examine the following geometric sequence:
φ,φ2,φ3…
Since x=1+52 can simplify φ by replacing the value of x to the formula of the golden ratio we discussed...
...
This work MAT 126 Week 1 Assignment  Geometric and Arithmetic Sequence shows "Survey of Mathematical Methods" and contains solutions on the following problems:
First Problem: question 35 page 230
Second Problem: question 37 page 230
Mathematics  General Mathematics
Week One Written Assignment
Following completion of your readings, complete exercises 35 and 37 in the “Real World Applications” section on page 280 of Mathematics in Our World .
For each exercise, specify whether it involves an arithmetic sequence or a geometric sequence and use the proper formulas where applicable . Format your math work as shown in the Week One Assignment Guide and be concise in your reasoning. Plan the logic necessary to complete the exercise before you begin writing. For an example of the math required for this assignment, please review the Week One Assignment Guide .
The assignment must include ( a ) all math work required to answer the problems as well as ( b ) introduction and conclusion paragraphs.
Your introduction should include three to five sentences of general information about the topic at hand.
The body must contain a restatement of the problems and all math work, including the steps and formulas used to solve the problems.
Your conclusion must comprise a summary of the problems and the reason you selected a particular method to solve them. It would also be appropriate to...
...to calculate a certain term (number of months starting from January) the two previous terms must be known. These are then added together to give the desired month.
The table below shows the rabbit’s breeding numbers throughout the whole year.
The Mathematical recursive formula that represents this is:
Where: Tn= The desired month (January1, February2, March3, and so on) and where Tn>3
It can be clearly seen from the graph that the pattern/structure is exponential. This is due to the previous numbers being added in succession with the next, resulting in the ‘gap’ between each number to increase.
The trend in which the numbers follow is called a Fibonacci sequence and is often found in nature as well.
Many instances in which the Fibonacci Series is present in nature are that a lot of flowers and cone shaped structures have the number of petals as one of the Fibonacci numbers. However some plants such as the sneezewort plant (as seen left) can be seen demonstrating the Fibonacci pattern in succession. It happens on both the number of stems and number of leaves.
Another appearance of the Fibonacci Series in nature is that a lot of flowers and cone shaped structures have the number of petals as one of the Fibonacci numbers. This includes the pineapple shown to the left. The number of spirals going in each direction is a Fibonacci number. For example, there are 13...
...Sequences and Convergence
Let x1 , x2 , ..., xn , ... denote an infinite sequence of elements of a metric space
(S, d). We use {xn }∞
n=1 (or simply {xn }) to denote such a sequence.
Definition 1 Consider x0 ∈ S. We say that the sequence {xn } converges to x0
when n tends to infinity iff: For all > 0, there exists N ∈ N such that for all
n > N , d(xn , x0 ) <
We denote this convergence by lim xn = x0 or simply xn −→ x0 .
n→∞
Example 2 Consider the sequence {xn } in R, defined by xn = n1 . Then xn −→
0.
The way to prove this is standard: fix > 0. We need to find N ∈ N such that
for all n > N , d(xn , 0) < . We have d(xn , 0) = xn − 0 =  n1 
So it is enough that n1 < , or equivalently n > 1 . So choosing N > 1 we know
that for all n > N , d(xn , 0) < .
The fact that we define the concept of convergence does not imply that every
sequence converges. This is illustrated in the next two examples. Let’s begin
with a remark about what it means for a sequence {xn } not to converge to x0 .
Remark: To know what the nonconvergence of a sequence means, we need
to write the negation of the definition of convergence. That reduces to: There
exists > 0, such that for all N ∈ N, there exists n > N such that d(xn , x0 ) ≥ .
For the ones of you familiar with propositional logic, notice that convergence to
x0 can be written as
(∀ > 0)(∃N ∈ N)(∀n > N )d(xn , x0 ) <
Its negation...
...years, particularly that of the Fibonacci sequence and the Golden Ratio. In Debussy’s Nocturne, composed in 1892, I look into the use of the Fibonacci sequence and the Golden Ratio. Previously it has been noted that composers used the Fibonacci sequence and the Golden Ratio in terms of form, however in my analysis I look into the use of it in terms of notation as well. I will explore how the idea of Sonata form is used along with the Mathematical Model of the Fibonacci sequence. It is however important to mention that as this is one of Debussy’s earlier works, the extent that the ratio and sequence are explored are not as elaborate as some of his later works. I will explore the Harmonic analysis of the piece to create a better understanding of where and how structure is used by Debussy. Debussy was a perfectionist and would only give perfected scores to the printers, as such it is impossible to prove whether or not the use of the sequences were intended or not, however considering that some of his contemporaries in other arts were very much involved with the idea of the Golden Ratio it does seem plausible that it was intended. The fact remains, though, that the use of the sequences and ratio are still evidently there and can be analysed; as this essay will show.
The Fibonacci sequence is a system of numbers that equate to the two preceding numbers. In terms of...
...Job sequence modeling using Genetic Algorithms
Dr.S.N.Sivanandam
Professor & Head
M.Kannan
Senior Lecturer
Department of Computer Science & Engineering,
P.S.G.College of Technology,
Coimbatore641 004
Abstract
This paper presents a Genetic algorithm (GA) based procedure for finding an optimum job sequence for N jobs / M machines problem based on minimum elapsed time. The search space is so large that the Genetic algorithms outperform the conventional procedures in solving optimization problems. In this paper we propose a Bell shaped sequence for N jobs / M machines problem. The optimum sequence resemble a Bell shaped sequence (Normal or Gaussian distribution like curve). In the sense that the maximum total processing time of a job M machines lie in the middle of the sequence, next maximum lie in the right side (or left side) and the next maximum lie in the left side (or right) and so on and an example is illustrated. By including the Bell shaped sequence in the proposed GA procedure, the convergence of optimum sequence is faster since the final optimum sequence almost always resemble a Bell shaped sequence. Various test cases are discussed in Appendix.
Keywords: Job sequencing, Genetic algorithms, Optimization, NPhard, Two Job point crossover, Mutation, Mirror image, Bell shaped sequence, Normal...
...A sequence shot involves both a long take and sophisticated camera movement; it is sometimes called by the French term planséquence. The use of the sequence shot allows for realistic and dramatically significant background and middle ground activity. Actors range about the set transacting their business while the camera shifts focus from one plane of depth to another and back again. Significant offframe action is often followed with a moving camera, characteristically through a series of pans within a single continuous shot. An example of this is the first scene in the jury room of 12 Angry Men, where the jurors are getting settled into the room. In a film script, a shooting sequence is a part of the script consisting of a single unified action and which can be shot in one place, at one time, with essentially the same cast throughout. A shooting sequence can be part of a scene, an entire scene, or several scenes in a script.
During preproduction, a script is lined, meaning that a line is drawn between each of the shooting sequences in the script and each important element in the sequence is highlighted.
Good video stories need strong individual shots. Great video stories present those shots in a sequence that complements the parts and creates a much greater whole.
Shooting and editing effective sequences are essential video storytelling...