The Significance Of The Dorado Sequence

In Candide by Voltaire, describes the transformation of the protagonist Candide, throughout the story. The author demonstrates the character development over the story starting with an innocent personality as a child who does not have responsibility to know, into a great man who experienced the life means. Basically Candide endures the human suffering to get his final destiny. During his crossing candide visited many cities which make him have different point of view about life experience such as El Dorado. It was a great place totally different from the rest of cities. In the story the language shows Candide’s progress towards maturity.

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When Candide was expulsed from the palace for his love to Ms. Cunegonde, he experimented a cruelty period of life. But it help him to face the philosophical view that all things in life are necessary for some greater good. As said his best friend and philosopher Pangloss. ,”He sees that everything does not happen for the best as the philosophers and metaphysician Pangloss had told him in the Baron’s castle.” (Philip Littell)Throughout Candide’s travels he develops a new philosophy of life, his eyes open to reality.
In the story El Dorado have a big role in Candide ideologies about life. People in El Dorado have different ways of thinking and priorities with Europe or visitors.” I simply can’t understand , said he, the passion you Europeans have for our yellow mud; but take all you want, and much good may it do you.”( Candide,408).El Dorado habitants know how the outsiders over appreciate money and gold. Also ,people from there ,know that materialistic things are important and indispensable for the visitors and most of the time the only way to meet happiness for a while they have those valuables were worthless for them. In constrat el Dorado is the notion of emotions such as love and care are freely chosen. People in El Dorado value the species and fellow human rather than having wars against each other for a ridiculous argument and ambition of possessing all the materialistic objects.
Moreover, El Dorado is a really simple and humble city, where people do not need extravagant things, they only have the essential things and there are no disagreements between them, poverty is nonexistent.” Cacambo it’s true my friend it again, the castle where I was born does not compare with the land where we now are; but Miss Cunegonde is not here.”(Candide, 408).In this quote I could perceive the reason of candide to leave el Dorado, it was Miss Cunengonde. Candide was really in love with her and he wanted to be with her. The love of both was not accept for their different social classes. Candide to get his change goes through many adventures and gradually matures into an experienced and practical man. “Finally, he decides to settle down and live by farming his own garden-this symbolizes his surrender to simple self-preservation and candide said that we must cultivate our garden”.(Voltaire 30,438) After a long and difficult struggle in which Candide is forced to overcome misfortune to find happiness, he concludes that everything is not as good as it seems the way Dr. Pangloss, his tutor had taught him. Another important point in El Dorado is that people shared their belongings and there is no such a feeling as avarice and envy in this place. Even though the residents of El Dorado do, they do not have an organized religion and do not believe in a religious persecution. None of the inhabitants they all believe the same thing. In order Voltaire remarks that unlike Europe and the rest of the world in El Dorado people are free to follow and express their own faith and not be afraid of future consequences of others disapproval. Furthermore, in El Dorado there are not courts or prisons because everyone acted with a good attitude toward each other. By the way, their system of education is well organized and advanced compare to the rest.
Unfortunately, the rest of the world is really different from El Dorado. In this part of the world we can see a lot of violence, corruption and misfortune that’s why El Dorado tries to keep away from the rest of the world, because habitants does not want to be infected with this style of life. Everyone wants to been in a community without those problems, because the environment plays a big role in the way we live and, also in the way we interact with the others. Instead of living in an infected world we should be looking for a better lifestyle.
” We only request of your majesty, Cacambo said, a few sheep loaded with provisions, some pebbles, and some of the mud of country.
The king laughed.
I simply can’t understand, said he, the passion you Europeans have for our yellow mud; but take all you want, and much good may it do you”.(candide, 408)
In addition, El Dorado don’t need of gold and jewels to been happy like the rest world, that the main priority is have more and more money to be “happy”. A great and important value from the habitants of El Dorado is that always they demonstrate love, respect and confidence to their children and community. One thing important that i learned of the book el dorado is that they always is important show love, respect and confidence to their children and community and the thing materialistic don’t need been the most important to them. In contrary, the rest world teach to children that materialistic things are important to reach the happy and I believe that is great problem that present the society today because they don’t found confidence, respect and love. Then they use drugs to gain attention from their parents, because their parents don’t have communications with them. Many teens use drugs because they are depressed or think drugs will help them escape their problems. El Dorado contrast to the rest of the world in many different ways one is that rest word is materialistic and the effects that cause a society materialistic are the use drugs, alcohol and crime. It is the same as our addiction to all the materialistic things in the world. In addition, rest world have become extremely addicted to the material world. People El Dorado is different because they are peace-loving people determined to live a life of simplicity, sheltering themselves from the outside world to better focus on the spiritual world. The most important for El Dorado community is that they want to feel better of follow with the values of their community.
“Candide listened attentively and believed innocently; for he thought Miss Cunegonde extremely beautiful, though he never had the courage to tell her so. He concluded that after the happiness of being born Baron of Thunder-ten-Tronckh, the second degree of happiness was to be Miss Cunegonde, the third that of seeing her every day, and the fourth that of hearing Master Pangloss, the greatest philosopher of the whole province, and consequently of the whole world” (candide 1, 378)
During his adventures he realize that things not always happen for the best, he understand that it just happen in his innocent mind. El dorado have a great important to one of some changes of Candide was his philosophy really optimistic mind” everything is for the best”. It was a phrase of his teacher Pangloss He taught that everything was for the best and Candide, having never heard any other philosophies, agrees blindly. In his amazing journey he finds that every event in the world has a reason, and whether there are positive or negative moments you have to live them.”

Fibonacci Sequence

How Does the Fibonacci Sequence Relate to Nature and Other Math Processes?
Nature is all around us, and because I spend a lot of time outside I have been able to enjoy and observe all that nature has to offer. Due to the fact that I love science and discovering how everything around me functions and relates to everything else, I decided to investigate the relation that Fibonacci has with other math processes—as well as with the environment. I wanted to understand how plants know the best way to form their seeds or outer shell, and why some patterns may repeat in nature in different plants and organic materials. Thus, this exploration looks at two seemingly unrelated topics—Fibonacci and the golden ratio—both of which produce the same number, phi. While this could be mere coincidence, that possibility is negated when the fact that the number produced is irrational is introduced. It was this peculiar discovery, as well as the abundant appearances of Fibonacci in nature, that led me to choose this exploration topic.
To begin, I should start by identifying what initially sparked my curiosity in this subject: a pinecone. As with many other plants, as well as fruits and vegetables, pinecones display the golden ratio. In order to better understand what I am talking about I have included a picture of a pinecone similar to the one that I first inspected.
Labeled below is the noticeable spiral pattern on the pinecone. Counting the number of spirals in that direction produces the number eight, and in the other direction it produces the number thirteen while a third and tighter spiral produces twenty-one. These numbers are situational to the pinecone in the pictures, but the Fibonacci numbers as a whole are far more complex than they first appear to be.
To understand the importance of these numbers it is crucial to understand the fundamentals of the Fibonacci sequence itself. The sequence usually begins with the numbers 1, 1, 2, 3, 5, 8, 13 and follows an easily definable pattern.
1, 1, 2, 3, 5, 8, 13
Start with the number 5, or the nth number in the sequence. We’ll call it n. 5 equals the two numbers before it added together: 2 + 3. Or, in broader terms, a number in the sequence is the sum of the two numbers preceding it.
1, 1, 2, 3, 5, 8, 13n = n-1 + n-2
An interesting idea comes up at the mention of this formula though.
This ratio just so happens to equal a number often notated as, or phi.
> 1/11Phi is greater than one,
> 3/21.5Phi is greater than three halves,
> 8/51.6Phi is greater than eight fifths,
You’ll notice that each fraction listed above is made up of numbers from the original seven number sequence, in other words, each pair of Fibonacci numbers creates a ratio that gets closer and closer to phi as the numbers increase. This is better shown on a graph I created, displayed below.

The ratio created by these sequences as they approach phi is called the golden ratio. The golden ratio, however, is not as important to this study as the lesser known concept of the golden angle. Below is a representation of the golden ratio in relation to the golden angle, the smaller portion of the circle notated using alpha, or α.

α = 137.507764° 137.5°
The reason this conversion is necessary is because the golden angle is present in the next discussion topic: sunflowers. Or, more specifically, their seeds. Sunflowers are another great example of the appearance of Fibonacci in nature, and also led me to an interesting discovery.
In order to plot the distribution of a sunflower’s seeds we need an X and a Y coordinate pair. Using the square roots from an index numbered from one to one thousand and multiplying them by the cosine of the radian of the angle alpha gives us a formula to find x, dependent on the index number used. Y can be calculated with a very similar formula, using sine instead of cosine. The equations are listed in their entirety below.

When these formulas are used and input into Microsoft Excel they produce a graph similar to the following.

Wow! That graph bears a striking resemblance to the original Fibonacci spirals that appeared in the pinecones, and as mentioned earlier it is not mere coincidence.
While the use of the golden ratio is apparent, there is another aspect of it that I wish to address, the golden spiral. Its formulae are given by the following equations, and are readily apparent in nature as well (nautilus shells for example).

In these equations is the undetermined scaling factor and is the growth factor of the spiral. In the instance of the golden spiral, is equal to the operation below.

At first, these formulae appeared to be a strange smattering of numbers, and one I didn’t understand at all. However, upon noticing the appearance of a natural log in the formula for I made a connection to the letter , better known as Euler’s number, that is present in both the X and Y formulae. After thorough searches of many sources I discovered another math process that bares resemblance to the above formulae.

This is Euler’s formula. It becomes increasingly apparent that its resemblance is not coincidental when the formula is transformed into the final formula shown below.

While the visual similarities may be obvious when the formula is displayed as it is above, the importance of each variable can be clarified with simple explanations. is the arbitrary scaling factor, responsible for determining the scale of the spiral. dictates the rotation of the spiral, and remains constant. The in dictates the growth of the spiral, and the dictates the speed—together representing the speed of the growth of the spiral. More simply put, any given ordered pair can be found by multiplying the growth of the spiral by its rotation (as shown in the originally given formulae for finding said coordinates.)
What is produced, however, after inputting over two thousand pieces of data, derived from the coordinates calculated using the formulae above, into Microsoft Excel, is shown in the graph below.

After putting in the Fibonacci squares (using the original golden ratio) into the spiral its appearance and relation to Fibonacci become even clearer.

Very simply put, my investigation yielded the result that the Fibonacci sequence, the golden spiral, and Euler’s number are all related to one another in nature. The results are eye opening for me, as I am beginning to realize just how much of the world is made up of math—rather than my previous belief that everything natural occurred randomly. My exploration only stemmed into plants, and while that may only have practical use in fields such as botany, all three have great value in many fields. To begin with, Fibonacci appears in bee populations, proportions of the human body, formation of cells, and possibly more practically in code and the stock market. Any of these fields could present an interesting extension to my exploration, and because they all stem from Fibonacci they all have roots in combinatorics and number theory. The implications of this are staggering! Simply the thought that all of these vastly different fields are related to one another by one sequence of numbers discovered by Leonardo of Pisa, better known as Fibonacci himself, is baffling considering that he discovered them while looking at the breeding patterns of rabbits. There are so many other areas in nature that Fibonacci appears in, and I’m so excited that I have the opportunity to discover and study them now that I know more about them.
Works Cited
Azad, Kalid. “Intuitive Understanding Of Euler’s Formula.” Better Explained. N.p., n.d. Web. 23 Feb. 2015.>.
“Nature by Numbers.” Eterea. N.p., n.d. Web. 3 Feb. 2015.>.
Wolverson, Tim. “Plot a Fibonacci Spiral in Excel.” Reviews and Guides. WordPress, 08 Feb. 2014. Web. Jan.-Feb. 2015.>.
McDonald 1

Interlanguage & Developmental Sequence Analysis

Part 1: Developmental Sequence

Developmental Sequence of Focus: Question Formation



Developmental Stage


I can say whatever I want?



What’s the first question?



Miss what’s your favorite place to go?



What’s your favorite animal?



Where do you live?



Where are you from?



Who’s your favorite person?



What’s your favorite place to eat?



What’s your favorite song?



What’s your favorite color?





















































Part 2: Interlanguage Analysis


Form of Focus: -d/-ed morpheme; past tense marker


Incorrect Form in Context


Correct Form in Context


When I move [muv] to the U.S. two and a half years ago.


The lesson I learned is thatis [ɪz]. Better moving schools


The third one is when I ignore [ɪgnɔr] someone.


The lesson I learned is that you can learn how to survive [sərvaɪv]without someone.


It was the. Worst thing that happened to me.


The lesson I learned is that you have to ignore the people who always have to say something about you.

Part 3: Discussion of Findings


 Milly is the subject of these analyses. She is one of my former students. Milly is a 13-year-old girl in Springfield, Massachusetts. She has lived in Springfield, after moving from Puerto Rico, for three and a half years. She started learning English at the same time she moved to the United States. She is a rising 9th grader who has ESL services. After the ACCESS exam this year, she was placed at a Level 2 for the second year in a row; however, she appears to have more of an intermediate language proficiency. The following analyses appear to support this fact.  In terms of her personality and approach to learning, Milly is extremely outgoing, and she is usually one of the first students to volunteer to answer a question in class. She cares a lot about her academics and how she appears to others. She can be self-conscious about her learning, and when she makes mistakes she likes to try to pretend she knew that was wrong in the first place. However, making mistakes does not stop her from continuing to practice her language skills.

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 For these analyses, there are two major areas of focus. My initial area of focus is on developmental sequences broadly and question formation more specifically. My second area of focus is on interlanguage. I perform an interlanguage analysis with a focus on the morphemes -ed and -d; both are used to mark past tense. I am able to analyze both of these focus areas due to Milly’s participation in a casual conversation with me that her mother allowed me to transcribe and analyze.

Developmental Analysis: Question Formation

 In order to discuss the significance of these findings, it is important to first give a brief context for the developmental sequence of question formation. Tarone and Swierzbin note the order for question formation has been largely researched and the numbers that represent each stage have been created by linguistic researchers Pienemann, Johnston, and Brindley. The same researchers claim that Stage 1 and Stage 2 questions, though they occur initially in the question formation process, are known to “… continue to appear in the speech of highly proficient learners (and native speakers too).” They then clarify that not all of the early question forms continue occurring, and the ones that are actually ungrammatical are phased out of learners’ language by the time they reach a higher level of proficiency (2009, p. 46).

 The findings from this analysis are interesting and initially seem very straight forward. Milly should be at an intermediate stage in her language acquisition, and the questions she asks appear to reflect that. The most frequent question stage seen from this transcript is 4c, which is right in the middle of the question stages. That matches the intermediate stage Milly’s oral language is at according to the ACCESS exam. However, upon closer inspection, Milly appears to be reusing the same question format. She shows a strong acquisition of wh- questions with copula, but that appears to be where her comfort level stops when asking questions.

 One thing that I discovered about Milly based on this transcript was that she has entered question stage 5b. Before this specific “interview”, I had not heard Milly ask questions with a “do” operator. This shows that Milly might be ready to move on to more complex question structures, and that is an important thing to know as her teacher.

Interlanguage Analysis

 This transcript provides a clear example of Milly’s interlanguage. Tarone and Swierzbin explain that, “… The learner’s IL can only be observed when he or she is focused on the meaning of the message” (2009, p. 12). That is the case in this conversation and interview due to the strength of my teacher-student relationship with my former student, Milly. Additionally, this conversation didn’t take place as part of a graded assignment, which might have lead to Milly focusing more so on grammatical accuracy versus commuting meaning.

After doing the interlanguage analysis, one thing became clear very quickly. Milly had more instances of correct past tense use with -ed/-d than without. While this initially shows that she has a solid acquisition of past tense markers, I realized that there was a pattern upon closer inspection. A majority of her correct uses of past tense with the -ed/-d ending are within the same sentence construction: “The lesson I learned is…” This sentence construction is familiar to me because it’s the same sentence starter I gave my students when they had to complete their memoir final project. It appears that Milly memorized this sentence starter and has learned how to use it in the correct context. While that is a great development, and it reflects that sentence starters are a great way for students to acquire correct grammatical constructions, it is not really a true representation of a rule that Milly is applying in her interlanguage.

 Another pattern I uncovered from this transcript is that when Milly would lead utterances with the phrase “When I…”, she would fail to use the necessary past tense markers or morphemes. In Spanish, these instances would use the imperfect tense whereas her correct uses of the past tense would be the preterite tense in Spanish. This could mean that she hasn’t yet made the connection that verbs in the imperfect tense in Spanish also translate to the English past tense. It would require more study to make any definitive claims. 


 This transcript made me realize that Milly’s acquisition of the past tense is not as strong as I had previously believed. In other instances in the conversation, she correctly used irregular past tense forms. This makes me think that it is less likely that Milly has truly learned the grammatical rules that govern linguistic systems and that she has instead memorized which words she should use with the past tense. As a language educator, I would work to teach Milly the actual grammatical rules that govern the past tense in an effort to increase her metacognition and her self-monitoring of her language production.

 Additionally, as a language educator, I would support this learner by pushing her to use and practice more complex question types. She exhibits a lot of comfort with wh- questions, and from the one more advanced question she asked, it appears that she is ready to move on to later stages. I would model more complex questions for her frequently in class before providing her with more “interview” opportunities to practice her questioning skills.


 One key limitation of performing error-based analyses is that, with the focus on incorrect forms, it becomes easier to overlook correct uses of language produced by the target learner. Tarone and Swierzbin further clarify the benefits of being able to look at the full picture of what a learner can do by saying, “… If we can systematically identify those instances where a learner gets a form right, then we may have gained some pedagogical leverage in figuring out how to extend that accuracy to more problematic instances” (2009, p. 29). In other words, it is easier for an educator to help learners recreate their success when they have worked to identify the success.

Works Cited:

Tarone, E., & Swierzbin, B. (2009). Exploring learner language. Oxford: Oxford University Press.


Livesay: Okay, Milly. You need to think about a time when you did something fun with a friend.

Milly: Okay. One with Jessica. She’s my bestie

Livesay: Okay, that’s a good start. When were you last with Jessica?

Milly: Um. I’m not sure… Saturday

Livesay: What did you do?

Milly: We eat, dancing, and using my phone

Livesay: And where were you?

Milly: At my house. We were at my house

Livesay: How did you feel?

Milly: Happy. And ex-ci-ted

Livesay: How long were you together?

Milly: A few hours I think

Livesay: Okay let’s move on. I’m going to ask you some more questions now. All of these questions are about you, so there’s not a right or a wrong answer

Milly: I can say whatever I want?

Livesay: As long as it answers the question yes

Milly: Okay. What’s the first question?

Livesay: What is the best thing that has ever happened to you?

Milly: Pfft that one is easy. Having a big sister

Livesay: Oh that’s very sweet you should tell her that

Milly: Also having you as a teacher

Livesay: (Laughter) Okay let’s move on. What is the worst thing that has ever happened to you?

Milly: Oh. The worst thing was when I lost [lɔsd] my dog in Puerto Rico.

Livesay: Oh I’m so sorry, that is pretty bad.

Milly: I was. Sad

Livesay: That makes sense. Okay. The next question is what event has changed your life?

Milly: Pffft that’s easy. When I move [muv] to the U.S. two and a half years ago.

Livesay: That does make sense. Okay. Who is your favorite person?

Milly: My. Best Friend because she is one of the person [pɜrsən] that is always there for me in good and. Bad times. And we always go to places together like next week we are going to the movies.

Livesay: That’s great! Is there anyone else you want to talk about?

Milly: Oh. Yes. My sister because she is the. Best and I go with her everywhere. And she get me out of the trouble. And she is always there for me in good and. Bad times like my. Best Friend They are the only Persons that are always there for me.

Livesay: Okay, Milly, now I’m going to ask some harder questions. It’s okay if you need time to think before answering. First… What was an important life event for you? And what lesson did you learn?

Milly: My first one is going to different schools. That was important because of the different language. The lesson I learned is that is [ɪz]. Better moving schools and state [steɪt] because you can learn more and get a better job.

Livesay: Great answer. What is another important life event for you?

Milly: The second event is losing someone. The lesson I learned is that you can learn how to survive [sərvaɪv] without someone.

Livesay: That does sound very important.

Milly: It was the. Worst thing that happened to me.

Livesay: It does sound very bad. Can you think of any other important life events?

Milly: The third one is when I ignore [ɪgnɔr] someone. The lesson I learned is that you have to ignore the people who always have to say something about you.

Livesay: What do you mean by that?

Milly: You know the people who always have something to say and it [ɪt] always mean.

Livesay: I hope you don’t have anyone in your life who talks about you behind your back anymore.

Milly: No not anymore they were too fake

Livesay: Okay Milly now you can ask me questions about my life

Milly: Miss what’s your favorite place to go?

Livesay: My favorite place to go is my mom’s house because I love her and I miss her.

Milly: What’s your favorite animal?

Livesay: My favorite animal is a dolphin.

Milly: Where do you live?

Livesay: I live in Worcester, Massachusetts.

Milly: Where are you from?

Livesay: I am from Honolulu, Hawaii and I grew up in San Diego, California.

Milly: Who’s your favorite person?

Livesay: My favorite person is my sister.

Milly: What’s your favorite place to eat?

Livesay: Chick Fil A.

Milly: What’s your favorite song?

Livesay: Something Good Can Work by Two Door Cinema Club.

Milly: What’s your favorite color?

Livesay: My favorite color is teal.

An Explanation of the Fibonacci Sequence

In this paper, I will examine the properties of Fibonacci Sequence and some identities related to Fibonacci sequence. I will also touch the applications of Fibonacci numbers, recurrence and closed forms of Fibonacci sequence.
1 Introduction
2 Fibonacci Numbers in Pascal’s Triangle
3 Cassini’s Identity
4 Difference of Square of Fibonacci Numbers
5 Sum of Odd Fibonacci Numbers
6 Golden Ratio
7 Golden Spiral
8 Fibonacci Recurrence Relation
9 Fibonacci Closed Form
10 Conclusion

Fibonacci numbers were discovered by Leonardo Pisano. His nickname is Fibonacci. The Fibonacci Sequence is the series of numbers:
Each number in the sequence is sum of the previous two terms.
u1 = 1 u2 = 1 un = un−1+un−2,n > 2
Fibonacci numbers first used in ‘rabbit problem’. At the beginning, Fibonacci had one male and one female rabbits. He supposed that, every month. Next month more babies (male and female) were born. The next month these babies were grown and the first pair had two more babies (again male and female) . The next month the two adult pairs each have a pair of baby rabbits and the babies from last month mature. Fibonacci asked how many rabbits a single can produce after a year with this breeding process (rabbits never die, every month each adult pair produces a mixed pair of baby rabbits who mature the next month). Fibonacci asked how many would be formed in a year. Following the Fibonacci sequence perfectly the rabbits reproduction was determined 144 rabbits (Sinha,2017).

Fig.1 rabbit problem diagram

The Fibonacci Numbers are also applied in Pascal’s Triangle. Entry is sum of the two numbers either side of it, but in the row above. Diagonal sums in Pascal’s Triangle are the Fibonacci numbers.
An interesting property of Pascal’s Triangle is that its diagonals sum to the Fibonacci sequence, as shown in the picture below:

Fig.2 diagonal sums in Pascal’s triangle
The sum of the entries in the nth diagonal of Pascal’s triangle is equal to the nth Fibonacci numbers for all positive integers n. Suppose ∑dn sum of the numbers at nth diagonal and fn = nth Fibonacci number, n ≥ 0.
Principle of Mathematical Induction will be used for proof.
For n = 0, ∑d0 = 0, f0 = 0 (The result is true for n = 0) Assume that ∑dk = fk fk+1 = fk+ fk−1, Property of Fibonacci Sequence f(k) = ∑dk and f(k−1) = ∑dk−1, inductive hypothesis
Therefore, ∑dk+1 = ∑dk+∑dk−1
F12+F22 = 12

Cassini(1625-1712) is the astronomer who found Cassini Identity. Cassini’s identity is a mathematical identity for Fibonacci sequence.
The history of science is silent as to why Cassini took such a great interest in Fibonacci numbers. Most likely it was simply a hobby of the Great astronomer. At that time many serious scientist took a great interest in Fibonacci numbers and the golden mean. These mathematical objects were also a hobby of Cassini’s contemporary, Kepler(Stahbov,2012).
Proof by induction
Let’s check that if the identity works for n=1.
For n = 1,2×1−12 = 1 = 11+1
Assume that the identity works for some n=k,
We want to show that also
To prove this we will establish
Here is how:
= Fk+1Fk−Fk−FkFk−1−Fk+1Fk−1
= Fk+1Fk−2FkFk

The difference of the square of two consecutive Fibonacci numbers is equal to the product of two Fibonacci numbers.
F FnFn−1
Use the following identities.
F FnFn−1
Fn+1 = Fn+Fn−1
an = F2n−1 bn = 2FnFn−1
a2n = b2n+c2n
Consider separately
= (Fn+Fn−1)4−4(Fn+Fn−1)FnFn−1
After simplifying this by using identities
We can show that FFnFn−1)2 = (FnFn−1)2
Hence F(FnFn−1)2 So, F FnFn−1.

The sum of odd Fibonacci numbers is also a Fibonacci number.
F1+F3+F5+…+F2n−1 = F2n,F0 = 0,F1 = 1
Proof by induction. base case: for n=1 F1 = 1 = F2 This proves that the base case is true.
inductive step: Lets assume that it is true for some n ≥ 1.
F1+F3+F5+…+F2n−1 = F2n
We need to prove that it is also true for n+1.
F1+F3+F5+…+F2(n+1)−1 = (F1+F3+F5+…+F2n−1)+F2(n+1)−1
= F2n+F2(n+1)−1
= F2n+F2n+1
= F2n+2
= F2(n+1)
This proves that it is true for n+1. Hence it is true for all n ≥ 1.

Any Fibonacci number divided by the previous Fibonacci number has a quotient of approximately 1.618034…
FFn−n1 ≈ 1.618034(φ)

Fig.3 ratio of consecutive Fibonacci numbers
The Golden Ratio is also found all throughout the physical universe. Weather patterns, Whirlpool have almost the same form like the golden spiral. Even the Sea Wave sometimes shows almost the same spiral pattern. The three rings of Saturn are designed naturally based on the Golden Ratio. The Galaxy, Milky Way, also has the spiral pattern almost like golden spiral. Relative planetary distances of the Solar System also have the golden ratio properties. (2011)

There is a good change of variable analysis in textbook ‘An Introduction to the Analysis of Algorithms’ (2013).
an = 1/(1+an−1) (2013)
Let’s write some terms.
a0 = 1
a1 = =
a2 = 1+1+1 = 1+2 =
When we continue, we will see the pattern to continued fractions continues.
a3 = 1+1+1 1 = 1+3 =
a4 = 1+1 = 1+5 =
1+ 1
The form an = bbn−n1 is certainly suggested:
substituting this equation into the recurrences gives
bbn−1 = 1/(1+ bbnn−−21) for n > 1 with b0 = b1 = 1.
Dividing both sides by bn−1gives
bn−1+1bn−2 for n > 1 with b0 = b1 = 1,
which implies that bn = Fn+1, the Fibonacci sequence. This argument generalizes to give a way to express general classes of “continues fraction” representations as solutions to recurrences (2013).

Golden spiral is derived from golden ratio.The growth factor of golden spiral is φ , the golden ratio.

Fig.4 golden spiral
To draw Golden Spiral, draw squares by using Fibonacci sequence. This works by drawing squares which have the side lengths of Fibonacci numbers and combining them.
Golden spiral is used in real life in many areas like art, beauty, clothing, architecture,logos etc.
The Golden ratio is prevalent in Da Vinci’s The Annunciation, Madonna with Child and Saints, The Mona Lisa and St. Jerome. He was famous for using the Golden ratio in his works. The Mona Lisa, a well known portrait of a woman with a coy smile, is embedded with Golden rectangles (2018).

Fig.5 The Mona Lisa

We can find recurrence formula of Fibonacci sequence by using its definition.
Fn = Fn−1+Fn−2 xn = xn−1+xn−2
We will divide each term by xn−2
x2 = x+1 x2−x−1 = 0
We will solve the quadratic equation by using quadratic formula
x x
Fn n
F0 = 0,F1 = 1
c c
Fibonacci Recurrence Relation

Let G(z) be closed form of Fibonacci sequence.
G(z) = F0+F1z+F2z2+F3z3+F4z4+…
Now let’s find zG(z) and z2G(z) by multiplying G(z) by z and z2.
zG(z) = F0z+F1z2+F2z3+F3z4+F4z5+… z2G(z) = F0z2+F1z3+F2z4+F3z5+F4z6+…
Now we will find the difference of G(z)−zG(z)−z2G(z)
(1−z−z2)G(z) = F0+(F1−F0)z+(F2−F1−F0)z2+(F3−F2−F1)z3+…
We know that F1 = F2 = 1
F2−F1 = 0.
Also since for n > 2 we have that
Fn = Fn−1+Fn−2 then
F3−F2−F1 = 0,F4−F3−F2 = 0,…,
Fn+1−Fn−Fn−1 = 0.
(1−z−z2)G(z) = z
This will give us the closed form of Fibonacci sequence as:
G(z) = 1−zz−z2

In conclusion, studying Fibonacci numbers was great experience. It was interesting and joyful to see the harmony of numbers and the beauty they made. I searched the history of Fibonacci numbers, some identities related to Fibonacci numbers and the applications of Fibonacci numbers. Seeing the applications of Fibonacci numbers was eye opening.
1. Sinha, Sudipta, 2017, The Fibonacci Numbers and Its Amazing Applications, International Journal of Engineering Science Invention, Volume 6, Issue6
2. Stakhov, Alexey, 2012, A generalization of the Cassini Formula, The International Clubof the Golden Section
3. Akhtaruzzaman, Md , Shafie, Amir A. ,2011, Geometrical Substantiation of Phi, theGolden Ratio and the Baroque of Nature, Architecture, Design and Engineering International Journal of Arts
4. Thapa ,Gyan Bahadur , Thapa , Rena, 2018, Journal of the Institute of Engineering, TheRelation of Golden Ratio, Mathematics and Aesthetics
5. Sedgewick, R., & Flajolet, P. (2013). An introduction to the analysis of Algorithms, (Second Edition). Addision-Wesley Professional.
6. Wiki page Golden Ratio.
7. Fig.1 Knott, Ron, 2016, Fibonacci Numbers and Nature.
8. Fig.2
9. Fig.3 Ancient Architecture(ratio of consecutive Fibonacci numbers).
10. Fig.4. Wikimedia commons.
11. Fig.5. The Mona Lisa