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Good Essay About Fibonacci Sequence And The Golden Ratio

Type of paper: Essay

Topic: Ratio , Sequence , Mathematics , Nature , Sum , Rectangle , Body , Literature

Words: 1600

Published: 12/03/2021

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Introduction

In the realm of Mathematics, there are many concepts that can be applied to multiple fields of mathematics. One of these concepts is the “Fibonacci sequence” that was developed by Leonardo Pisano during the 13th century. The Fibonacci number and sequence proposed by him gives a clear indication that mathematics can be connected to many things that may seem unrelated to it. The Fibonacci sequence has remarkable characteristics; it can be applied to various fields like discrete mathematics, number theory and geometry. By definition, the numbers in the Fibonacci sequence starts with either 0 and 1, or 1and1, and each subsequent number is the sum of the previous two numbers. The Fibonacci sequence is the best example of any recursive sequence. This sequence appears in numerous mathematical problems. Fibonacci problem: Let us assume 2 rabbits (one male and one female) are born on 1st January. Assuming all the months have same number of days and rabbit starts producing young rabbits exactly after two months they were born i.e. after reaching two months age, each pair produces another pair (one female and one male), and then mixed pairs are produced every month thereafter and no rabbit dies in between. After one year, let us calculate the pairs of Rabbit produced?

Fibonacci Solution: Solution provided is the Sequence given by Fibonacci:

0,1,1,2,3,5,8,13,21,34,55,89,114 Thus, after completion of one year, pairs of rabbits produced will be 144. Thus the Fibonacci sequence and Fibonacci numbers can be defined recursively as: fn+2= fn + fn+1 where n>0 or n=0.

Applications of Fibonacci sequence

The Fibonacci sequence can be related to various items in the nature like: growing buds on the plants and trees, the growth of branches on trees, the spiral hexagon observed on the pineapple, the pinecone’s rows, the seed heads, the petals growing on various flowers such as sunflower, the starfish, shells observed on snails and nautilus, the spiral galaxies and the chambers of vegetables and fruits like apple, lemon, Chile etc.

Properties of Fibonacci sequence

Taking the Fibonacci numbers from odd indices, their sum is f2n i.e. from Fibonacci sequence f1,f2,f3,f4,f5.f2n-1, f2n , take numbers at odd indices and sum them, their sum will be equal to number to f2n. For example: 1+2+5+13= f8=21 which is correct. Taking the Fibonacci numbers from even indices, their sum is f2n+1 - 1 i.e. from Fibonacci sequence f1,f2,f3,f4,f5.f2n-1, f2n , take numbers at even indices and sum them, their sum will be equal to number to f2n+1 – 1. For example: 1 +3 + 8 = 12=f7 - 1. Golden Ratio: Sum of all (fn)/(fn-1) produces golden ratio. For example: 3/1+ 5/3 + 8/5 + 13/8 gives golden ratio ) / 2.

Golden Ratio

Golden section and Golden ratio were first mentioned by Euclid in Elements in 300 B.C. Euclid proposed the problem for the division of a line into golden section. He proposed that when a unit segment is divided into two lengths, then the ratio of the smaller part to the larger part of the whole segment is equal to the ratio of the larger part to the whole length of the segment. For example: Let the whole length of the segment is x+y, where x is the larger part and y is the smaller part, then golden ratio can be defined as: (x+y)/x=x/y= golden ratio represented by Greek letter phi ( or ). Its value is . Golden ratio is an irrational number. Golden ratio fascinated the Greeks a lot, as it appears frequently in the field of geometry. One of the best applications of the Golden ratio in the geometry can be, the proportion in the lines of the pentagram. Also in pentagon, when all the sides are equal, the diagonals when intersect divide each other in the golden ratio i.e. the ratio between the diagonal and a side is divine ratio. Many of the greatest architectural structures in the world are designed in such a way that they have golden ratio in their proportions. Many artists have used the golden ratio throughout the history for their works as found it really beneficial and pleasing.

Origin of Fibonacci

According to Dunlap (2003), the Italian mathematician Leonardo Pisano was born into a merchant family in Pisa around 1175 A.D. His father, Bonacci, held a diplomatic post in the city of Bugia in the North Africa and so Fibonacci was educated by the Mohammedans of the Barbary. Fibonacci travelled the Mediterranean region with his father during his childhood, which introduced him to innovative mathematical ideas and concepts from numerous countries. His travel fuelled his love for mathematics as he returned to Italy in 1200 A.D. to publish his book, Liber Abaci (Book of the Calculations). The Arabic system of numbers was introduced by this book to Europe and thus established Fibonacci as one of the most reputed mathematicians of his time. The book provided a solution to the problem involving the progeny of a single pair of rabbits which became the basis of the Fibonacci sequence (series). The Fibonacci sequence is thus defined by the formula: fn= fn-1+fn-2 where n>3 or n=3. Each Fibonacci number is obtained by adding two previous Fibonacci numbers together.

Other names for the Golden ratio

Golden Ratio is also known as Golden mean, Divine proportion, Golden section, extreme and mean ratio, golden section and golden number. The golden ratio is often used when taking ratios of distances in geometric figures like decagon, pentagram, pentagon and dodecahedron. The golden ratio is denoted by , or sometimes . For example there is a rectangle with sides in the ratio 1: x. can be defined as the unique number x such that when the original rectangle is partitioned into a new rectangle and a square with the new rectangle having the sides in the ratio 1: x. Such a rectangle is called golden rectangle. So, it can be defined as: /1=1/-1 giving using the quadratic equation and taking positive sign, the value of comes put be: . .

Human Body, Fibonacci sequence and the Golden Ratio

Human body illustrates the Divine proportion. For example, when we look at the proportions of our index finger, it is observed that each section of the index finger is larger than the proceeding one by Fibonacci numbers 2, 3, 5 and 8 also producing the Fibonacci ratio of 1.618 Figure 1 Proportions of the index finger

The ratio of the forearms to the hand is also phi.

Figure 2 Ratio of Forearms to hand

Golden Ratio in Nature

One of the most interesting features of both the Golden ratio and Fibonacci series is that they appear frequently in nature, Golden Ratio is one of the fundamental characteristics of the universe that are followed by many items in nature. For example: Flower petals arrangements. The number of petals in any flower constantly follows the Fibonacci sequence. For example lily which has three petals follows the Fibonacci sequence. Buttercups have five leaves. Each petal in flower is placed at 1.618034 per turn (out of 360-degree circle) which allows best exposure to sunlight and other factors. Figure 3 Petals arrangement The Fibonacci sequence is also found in the spiraling pattern produced by leaves on the plants’ stems. Animal bodies also exhibit Fibonacci sequence like on moving from navel to the floor and top of the head to the navel is the Divine ratio. Figure 4 Animal Body

Ultimately, the Fibonacci sequence applies to not only various parts of mathematics but also to many different aspects of nature and environment. It can be concluded that there is an abundance of the golden ratio, Fibonacci sequence, Fibonacci numbers and spirals in nature. They are found everywhere. Since, Fibonacci sequence can be found in numerous areas both inside and outside of the realm of the mathematics, research done on Fibonacci sequence and Golden ratio helps to understand its applications in areas like nature, human and animal bodies and art etc. The existence of this Fibonacci sequence is not coincidental; it has been properly established and maintained. The mathematical properties of the Fibonacci numbers can be explored even more in today’s mathematical curriculum. Research on Fibonacci numbers helps in exploring the existence of Fibonacci sequence in the aesthetic nature of God. The Fibonacci sequence and numbers are simply the example of God’s power and authority over mankind.

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Charran, S. R. (2011). The Equation of Life. The United States of America: LifeCode Series. Dunlap, R. (2003). The Golden Ratio and Fibonacci Numbers. Singapore: World scientific publishing. gizmodo.com. (2015). 15 Uncanny Examples of the Golden Ratio in Nature. Retrieved from io9.gizmodo.com: http://io9.gizmodo.com/5985588/15-uncanny-examples-of-the-golden-ratio-in-nature Leonesio, J. M. (2007). Fascinating Characteristics and Applications of the Fibonacci Sequence. Liberty University. Posamentier, A. S., & Lehmann, I. (2012). The Glorious Golden Ratio. New York: Prometheus Books.

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Fibonacci sequence

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• Story of Mathematics - Leonardo Fibonacci
• Art in Context - Fibonacci Sequence in Art – Using the Fibonacci Theory in Art
• Whitman College - The Fibonacci Numbers
• LiveScience - What is the Fibonacci Sequence?
• National Center for Biotechnology Information - PubMed Central - Fibonacci numbers: A population dynamics perspective
• Temple University - Department of Mathematics - The fibonacci sequence, spirals and the golden mean
• Fibonacci series - Student Encyclopedia (Ages 11 and up)

Fibonacci sequence , the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …, each of which, after the second, is the sum of the two previous numbers; that is, the n th Fibonacci number F n = F n − 1 + F n − 2 . The sequence was noted by the medieval Italian mathematician Fibonacci (Leonardo Pisano) in his Liber abaci (1202; “Book of the Abacus”), which also popularized Hindu-Arabic numerals and the decimal number system in Europe. Fibonacci introduced the sequence in the context of the problem of how many pairs of rabbits there would be in an enclosed area if every month a pair produced a new pair and rabbit pairs could produce another pair beginning in their second month. The numbers of the sequence occur throughout nature, such as in the spirals of sunflower heads and snail shells. The ratios between successive terms of the sequence tend to the golden ratio φ = (1 + Square root of √ 5 )/2 or 1.6180…. For information on the interesting properties and uses of the Fibonacci numbers, see number games: Fibonacci numbers .

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The Golden Spiral Unveiled: The Beauty of The Fibonacci Sequence

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What is the Fibonacci sequence?

From its origins to its significance, almost every popular notion about the famous Fibonacci sequence is wrong.

• Who discovered it?
• Why is it important?
• Are there real-life examples?

The Fibonacci sequence is a series of numbers in which each number is the sum of the two that precede it. Starting at 0 and 1, the first 10 numbers of the sequence look like this: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on forever. The Fibonacci sequence can be described using a mathematical equation: Xn+2= Xn+1 + Xn

People claim there are many special properties about the numerical sequence, such as the fact that it is “nature’s secret code” for building perfect structures, like the Great Pyramid at Giza or the iconic seashell that likely graced the cover of your school mathematics textbook. But much of that is incorrect and the true history of the series is a bit more down-to-earth.

Who discovered the Fibonacci sequence?

The first thing to know is that the sequence is not originally Fibonacci's, who in fact never went by that name. The Italian mathematician who we call Leonardo Fibonacci was born around 1170, and originally known as Leonardo of Pisa, said Keith Devlin, a mathematician at Stanford University. 

Only in the 19th century did historians come up with the nickname Fibonacci (roughly meaning, "son of the Bonacci clan"), to distinguish the mathematician from another famous Leonardo of Pisa , Devlin said. 

Keith Devlin is an emeritus mathematician at Stanford University, a co-founder and executive director emeritus of the Stanford H-STAR institute, a co-founder of the Stanford mediaX research network, and a senior researcher emeritus at the Center for the Study of Language and Information. He is a World Economic Forum fellow, a fellow of the American Association for the Advancement of Science, and a fellow of the American Mathematical Society. He has written 33 books and over 80  research articles.

Read more: Large Numbers that Define the Universe

Leonardo of Pisa did not actually discover the sequence, said Devlin, who is also the author of "Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World," (Princeton University Press, 2017). Ancient Sanskrit texts that used the Hindu-Arabic numeral system first mention it in 200 B.C. predating Leonardo of Pisa by centuries.

"It's been around forever," Devlin told Live Science.

However, in 1202 Leonardo of Pisa published the massive tome "Liber Abaci," a mathematics "cookbook for how to do calculations," Devlin said.  Written for tradesmen, "Liber Abaci" laid out Hindu-Arabic arithmetic useful for tracking profits, losses, remaining loan balances and so on, he added.

In one place in the book, Leonardo of Pisa introduces the sequence with a problem involving  rabbits . The problem goes as follows: Start with a male and a female rabbit. After a month, they mature and produce a litter with another male and female rabbit. A month later, those rabbits reproduce and out comes — you guessed it — another male and female, who also can mate after a month. (Ignore the wildly improbable biology here.) After a year, how many rabbits would you have? 

The answer, it turns out, is 144 — and the formula used to get to that answer is what's now known as the Fibonacci sequence.

Read more: 9 equations that changed the world  

"Liber Abaci" first introduced the sequence to the Western world. But after a few scant paragraphs on breeding rabbits, Leonardo of Pisa never mentioned the sequence again. In fact, it was mostly forgotten until the 19th century, when mathematicians worked out more about the sequence's mathematical properties. In 1877, French mathematician Édouard Lucas officially named the rabbit problem "the Fibonacci sequence," Devlin said.

Why is the Fibonacci sequence important?

Other than being a neat teaching tool, the Fibonacci sequence shows up in a few places in nature. However, it's not some secret code that governs the architecture of the universe, Devlin said.

It's true that the Fibonacci sequence is tightly connected to what's now known as  the golden ratio, phi , an irrational number that has a great deal of its own dubious lore. The ratio of successive numbers in the Fibonacci sequence gets ever closer to the golden ratio, which is 1.6180339887498948482...  

Read more: The 9 most massive numbers in existence

The golden ratio manages to capture some types of plant growth, Devlin said. For instance, the spiral arrangement of leaves or petals on some plants follows the golden ratio. Pinecones exhibit a golden spiral, as do the seeds in a sunflower, according to " Phyllotaxis: A Systemic Study in Plant Morphogenesis " (Cambridge University Press, 1994). But there are just as many plants that do not follow this rule.

"It's not 'God's only rule' for growing things, let's put it that way," Devlin said.

Are there real-life examples of the Fibonacci sequence?

Perhaps the most famous example of all, the seashell known as the nautilus, does not in fact grow new cells according to the Fibonacci sequence, he added. When people start to draw connections to the human body , art and architecture, links to the Fibonacci sequence go from tenuous to downright fictional.

"It would take a large book to document all the misinformation about the golden ratio, much of which is simply the repetition of the same errors by different authors," George Markowsky, a mathematician who was then at the University of Maine, wrote in a 1992 paper in the College Mathematics Journal.

Much of this misinformation can be attributed to an 1855 book by the German psychologist Adolf Zeising called "Aesthetic Research." Zeising claimed the proportions of the human body were based on the golden ratio. In subsequent years, the golden ratio sprouted "golden rectangles," "golden triangles" and all sorts of theories about where these iconic dimensions crop up. 

Since then, people have said the golden ratio can be found in the dimensions of the Pyramid at Giza, the Parthenon,  Leonardo da Vinci 's "Vitruvian Man" and a bevy of Renaissance buildings. Overarching claims about the ratio being "uniquely pleasing" to the human eye have been stated uncritically, Devlin said. All these claims, when they're tested, are measurably false, he added.

"We're good pattern recognizers. We can see a pattern regardless of whether it's there or not," Devlin said. "It's all just wishful thinking."

Editor's note: Adam Mann contributed to this article .

Originally published on Live Science .

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Tia is the managing editor and was previously a senior writer for Live Science. Her work has appeared in Scientific American, Wired.com and other outlets. She holds a master's degree in bioengineering from the University of Washington, a graduate certificate in science writing from UC Santa Cruz and a bachelor's degree in mechanical engineering from the University of Texas at Austin. Tia was part of a team at the Milwaukee Journal Sentinel that published the Empty Cradles series on preterm births, which won multiple awards, including the 2012 Casey Medal for Meritorious Journalism.

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A Report On The Fibonacci Sequence

Table of contents, golden ratio and fibonacci sequence.

• Divide (on a calculator) 1 by any number
• Repeat steps 1 and 2 to get closer and closer to the actual value of the golden ratio.

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Fibonacci Sequence – Definition, Formula, Examples

What is the fibonacci sequence, fibonacci sequence formula, fibonacci sequence properties, solved examples on fibonacci sequence, practice problems on fibonacci sequence, frequently asked questions on fibonacci sequence.

In mathematics, we define the sequence as an ordered list of numbers that follow a particular pattern. The numbers that are present in the sequence are also known as the terms. 

One such interesting sequence is given below. What do you observe? Can you identify the pattern?

This sequence is known as the ‘Fibonacci sequence’ in math. If you look closely at the numbers, you can see that each number is the sum of two previous numbers. 

Let’s learn the definition, formula, properties, and some interesting facts.

Fibonacci Sequence Definition

The Fibonacci sequence is an infinite sequence that starts with 0 and 1 and continues in such a way that each number is the sum of the previous two numbers .

The numbers in the Fibonacci sequence are also known as Fibonacci numbers. The following image shows the examples of fibonacci numbers and explains their pattern.

The Fibonacci Sequence is given as:

0, 1, 1, 2, 3, 5, 8, 13, 21, …

Here, we obtain the third term “1” by adding the first and second term. $(0 + 1 = 1)$.

Similarly, we obtain “2” by adding the second and third term. $(1 + 1 = 2)$.

So, the term after 21 will be the sum of 13 and 21, i.e., $13 + 21 = 34$. Therefore, the next term in the sequence is 34. 

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The Fibonacci sequence of numbers, say “Fn” where the suffix n denotes the order or rank of term, is defined by 

Initial term: $F_{0} = 0$ 

First term: $F_{1} = 1$ 

These two terms together are known as the kick-off part.

Formula for next terms: $F_{n} = F_{n \;-\; 1} + F_{n \;-\; 2}$ 

This is known as a recursive formula since we repeat the same rule to get the next terms. 

$F_{2} = F_{1} + F_{0}$

$F_{3} = F_{2} + F_{1}$

$F_{4} = F_{3} + F_{2}$

$F_{5} = F_{4} + F_{3′}$

Note: The sequence starts with 0 (not 1). So, the 4 th term of the sequence will be $F_{3}$. 

The kick-off part is $F_{0} = 0$ and $F_{1} = 1$

The recursive relation part is $F_{n} = F_{n\;-\;1} + F_{n\;-\;2}$

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Fibonacci Sequence List

The list of fibonacci numbers (up to F 20 ) is given below:

The properties of the Fibonacci sequence are given as follows:

• Fibonacci numbers are related to the Golden ratio. In mathematics, two quantities are said to be in golden ratio if their ratio is equal to the ratio of their sum to the larger of the two quantities. It is commonly found in nature, and also in mathematics, design, geometry, etc. The ratio of any two successive Fibonacci numbers is close to the Golden ratio, which is approximately equal to ​​1.618034. Any Fibonacci number can be calculated using the golden ratio.
•  The ratio of consecutive Fibonacci terms is given below:

By multiplying the previous Fibonacci Number using the golden ratio, i.e., 1.618034, we get the approximated Fibonacci number. 

For example, 8 is the sixth term in the sequence. To find the seventh term, we multiply 8 by the golden ratio.

$8 \times 1.618034… = 12.94427… \approx 13$.

• The terms of the Fibonacci sequence below zero can also be calculated by: 

$F_{-n} = (\;-\;1)^{n+1}\; F_{n}$.

For example: $F_{-3} = (\;-\;1)^{3+1} \times 2 = 2$. 

• There is an interesting pattern in the sequence. 

Every 3rd number in the sequence starting from 2 is a multiple of 2. 

Examples: $F_{3} = 2,\; F_{6} = 8$

Every 4th number in the sequence starting from 3 is a multiple of 3.

Examples: $F_{4} = 3,\; F_{8} = 21$

Every 5th number starting from 5 is a multiple of 5; and so on.

• The sum of n terms of Fibonacci Sequence is given by:

$\sum_{i=0}^{n}\;F_{i} = F_{n+2}\;-\;F_{2} = F_{n+2}\;-\; 1$, where Fn is the n th Fibonacci number. 

For example, the sum of the first 5 terms of sequence $= \sum_{i=0}^{4}\;F_{i} = F_{4+2}\;-\;F_{2}$

$= F_{6}\;-\;1$

$= 8 \;–\; 1$ 

Fibonacci Spiral

Fibonacci spiral is a geometric pattern or a spiral formed with squares having sides representing the numbers in the Fibonacci sequence. The squares fit together due to the pattern in which Fibonacci numbers occur and thus form a spiral.

Connecting the corners of the squares, the spiral is drawn. The larger the numbers are in the Fibonacci sequence, the closet the ratio becomes to the golden ratio. 

In this Fibonacci spiral, every two consecutive terms represent the length and breadth of a rectangle. The following are the ratios of every two successive terms of the Fibonacci sequence.

This spiral can also be observed in nature. Take a look!

Applications of the Fibonacci Sequence in Real Life

The Fibonacci sequence has many applications due to its unique pattern and relation with the golden ratio.

• Computer algorithms, coding theories, security coding, and data structures use Fibonacci numbers.
• Many things in nature follow the Fibonacci sequence. Examples: Branching in trees, arrangement of leaves, pine cone structure.
• It is also used in numerous fields of science that includes high-energy physical science, quantum mechanics, cryptography, etc.
• Finance, trading, and stock markets use the Fibonacci pattern for technical analysis.

In this article, we learned about the Fibonacci sequence, which is a sequence in which a term is the sum of the terms before that. Let’s solve a few examples for better understanding!

1. Find the 11th term of the Fibonacci series if the 9th and 10th terms are 34 and 55 respectively.

Solution: 11th term $=$ 9 th term + 10 th term, i.e.,

$F_{11} = F_{10} + F_{9} = 55 + 34 = 89$

2. Find the Fibonacci number when $n = 7$ , using the recursive relation.

Solution: The formula to calculate the Fibonacci number is: $F_{n} = F_{n\;-\;1} + F_{n\;-\;2}$

Taking $F_{0} = 0$ and $F_{1} = 1$

$F_{2} = 0 + 1 = 1,\; F_{3} = 1 + 1 = 2,\; F_{4} = 2 + 1 = 3,\; F_{5} = 3 + 2 = 5,\; F_{6} = 5 +3 = 8$

$F_{7} = 8 + 5 = 13$

3. Find the sum of the first 6 Fibonacci terms.

Solution: The formula to calculate the sum of the first 6 Fibonacci terms is:

$\sum_{i=0}^{n}\;F_{i} = F_{n+2}\;-\;F_{2} = F_{n+2}\;-\;1$, where $F_{n}$ is the n th Fibonacci number

$\sum_{i=0}^{5}\;F_{i} = F_{5+2}\;-\;F_{2} = F_{7}\;-\;1 = 13\;-\;1 = 12$

4. Find the missing terms in the Fibonacci sequence: 

0, 1, 1, 2, 3, 5, 8, 13, 21, ?, ?, …

Solution: 

The next term after 21 will be $21 + 13 = 34$.

The next term after 34 is $34 + 21 = 55$

5. Find the value of 14th and 15th terms in the Fibonacci sequence if the 12th and 13th terms are 144 and 233 respectively. 

Solution: 14 th term $=$ 13 th term + 12 th term $= 233 + 144 = 377$

15 th term $=$ 14 th term $+$ 13 th term $= 377 + 233 = 610$

Fibonacci Sequence

Attend this quiz & Test your knowledge.

The first term of the Fibonacci sequence is

If the $16^{th}$ term in the fibonacci series is 610. find the next term of the series., which of the following is the approximate value of the golden ratio, what is the value of $f_{-\;4}$ if $f_{4} = 3$, what is the sum of the first 7 terms of the fibonacci sequence.

What are the examples of the Fibonacci series in nature?

The Fibonacci series can be spotted in the nature around us in different forms. It can be found in the spirals of the petals of certain flowers such as in the flower heads of sunflowers. We can also find them in the branches of trees.

What is the 100 th term of the Fibonacci series?

The 100 th term in a Fibonacci series is 354,224,848,179,261,915,075. Using the recursion formula, the 100 th term is the sum of the 98 th and 99 th terms.

What is the importance of the Fibonacci series?

The Fibonacci series is important because it is related to the golden ratio and Pascal’s triangle. Except for the initial numbers, the numbers in the series have a pattern that each number $\approx 1.618$ times its previous number. The value becomes closer to the golden ratio as the number of terms in the Fibonacci series increases.

How is the Fibonacci sequence related to humans?

The number of bones of your finger (from knuckles to wrist) are based on the Fibonacci sequence. Facial proportions are also based on the golden ratio. Human eye finds any object featuring the golden ratio appealing and beautiful.

What fruits show a Fibonacci pattern?

Fruits like the pineapple, banana, persimmon, apple and others exhibit patterns that follow the Fibonacci sequence.

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