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Fibonacci Sequence

The Fibonacci sequence is a type series where each number is the sum of the two that precede it. It starts from 0 and 1 usually. The Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The numbers in the Fibonacci sequence are also called Fibonacci numbers . In Maths, the sequence is defined as an ordered list of numbers that follow a specific pattern. The numbers present in the sequence are called the terms. The different types of sequences are arithmetic sequence, geometric sequence, harmonic sequence and Fibonacci sequence. In this article, we will discuss the Fibonacci sequence definition, formula, list and examples in detail.

What is Fibonacci Sequence?

The Fibonacci sequence, also known as Fibonacci numbers, is defined as the sequence of numbers in which each number in the sequence is equal to the sum of two numbers before it. The Fibonacci Sequence is given as:

Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, ….

Here, the third term “1” is obtained by adding the first and second term. (i.e., 0+1 = 1)

“2” is obtained by adding the second and third term (1+1 = 2)

“3” is obtained by adding the third and fourth term (1+2) and so on.

For example, the next term after 21 can be found by adding 13 and 21. Therefore, the next term in the sequence is 34.

Fibonacci Sequence Formula

The Fibonacci sequence of numbers “F n ” is defined using the recursive relation with the seed values F 0 =0 and F 1 =1:

F n = F n-1 +F n-2

Here, the sequence is defined using two different parts, such as kick-off and recursive relation.

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 .

It is noted that the sequence starts with 0 rather than 1. So, F 5 should be the 6 th term of the sequence.

Fibonacci Sequence List

The list of first 20 terms in the Fibonacci Sequence is:

The list of Fibonacci numbers are calculated as follows:

Golden Ratio to Calculate Fibonacci Numbers

The Fibonacci Sequence is closely related to the value of the Golden Ratio. We know that the Golden Ratio value is approximately equal to 1.618034. It is denoted by the symbol “φ”. If we take the ratio of two successive Fibonacci numbers, the ratio is close to the Golden ratio. For example, 3 and 5 are the two successive Fibonacci numbers. The ratio of 5 and 3 is:

5/3 = 1.6666

Take another pair of numbers, say 21 and 34, the ratio of 34 and 21 is:

34/21 = 1.619

It means that if the pair of Fibonacci numbers are of bigger value, then the ratio is very close to the Golden Ratio.

So, with the help of Golden Ratio, we can find the Fibonacci numbers in the sequence.

The formula to calculate the Fibonacci numbers using the Golden Ratio is:

X n = [φ n – (1-φ) n ]/√5

φ is the Golden Ratio, which is approximately equal to the value of 1.618

n is the nth term of the Fibonacci sequence.

Fibonacci Sequence Solved Examples

Find the Fibonacci number when n=5, using recursive relation.

The formula to calculate the Fibonacci Sequence is: F n = F n-1 +F n-2

Take: F 0 =0 and F 1 =1

Using the formula, we get

F 2 = F 1 +F 0 = 1+0 = 1

F 3 = F 2 +F 1 = 1+1 = 2

F 4 = F 3 +F 2 = 2+1 = 3

F 5 = F 4 +F 3 = 3+2 = 5

Therefore, the fibonacci number is 5.

Find the Fibonacci number using the Golden ratio when n=6.

The formula to calculate the Fibonacci number using the Golden ratio is X n = [φ n – (1-φ) n ]/√5

We know that φ is approximately equal to 1.618.

Now, substitute the values in the formula, we get

X 6 = [1.618 6 – (1-1.618) 6 ]/√5

X 6 = [17.942 – (0.618) 6 ]/2.236

X 6 = [17.942 – 0.056]/2.236

X 6 = 17.886/2.236

X 6 = 7.999

X 6 = 8 (Rounded value)

The Fibonacci number in the sequence is 8 when n=6.

Practice Problems

• Find the Fibonacci number when n = 4, using the recursive formula.
• Find the next three terms of the sequence 15, 23, 38, 61, …
• Find the next three terms of the sequence 3x, 3x + y, 6x + y, 9x + 2y, …

Frequently Asked Questions on Fibonacci Sequence

The Fibonacci sequence is the sequence of numbers, in which every term in the sequence is the sum of terms before it.

Why is Fibonacci sequence significant?

The Fibonacci sequence is significant, because the ratio of two successive Fibonacci numbers is very close to the Golden ratio value.

What are two different ways to find the Fibonacci Sequence?

The two different ways to find the Fibonacci sequence are

• Recursive Relation Method
• Golden Ratio Method

Write down the list of the first 10 Fibonacci numbers.

The list of the first 10 Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.

What is the value of the Golden ratio?

The value of golden ratio is approximately equal to 1.618034…

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Table of Contents

Last modified on December 18th, 2023

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Fibonacci sequence.

The Fibonacci Sequence is a number series in which each number is obtained by adding its two preceding numbers. It starts with 0 and is followed by 1. The numbers in this sequence, known as the Fibonacci numbers, are denoted by F n .

The first few numbers of the Fibonacci Sequence are as follows.

The above sequence can be written as a ‘Rule’, which is expressed with the following equation.

Using this equation, we can conclude that the sequence continues to infinity.

The following table lists each term and term value in the Fibonacci Sequence till the 10 th .

The numbers in the sequence follow some interesting patterns: 

• Every third number in the series, starting at 2, is a multiple of 2. Similarly, every fourth number after 3 is a multiple of 3, every fifth number after 5 is a multiple of 5, and so on.
• Also, the sum of two odd numbers is always an even number, whereas the sum of an even and an odd number is an odd number. Thus, the Fibonacci sequence follows an even, odd, odd, even, odd, odd pattern.

Recursive Formula in Fibonacci Sequence

Since the Fibonacci sequence is formed by adding the previous two Fibonacci numbers, it is recursive in nature. 

For example,

To calculate the 50 th term, we need the sum of the 48 th and 49 th terms.

Geometrically, the sequence forms a spiral pattern. It starts with a small square, followed by a larger one adjacent to the first square. It is followed by the sum of the two previous squares, where each square fits into the next one, showing a spiral pattern expanding up to infinity.

The following graph represents it.

The limits of the squares of all the consecutive Fibonacci numbers create the Fibonacci Spiral.  

The Fibonacci Sequence has some important properties, which we will discuss below.

Fibonacci Sequence and Golden Ratio

Two successive Fibonacci numbers give the value ${\phi =\dfrac{1+\sqrt{5}}{2}}$ or, 1.618…, which is known as the Golden Ratio , also known as phi (an irrational number). 

For the given spiral, the Golden ratio follows the property:

Let the Fibonacci numbers be a, b, c, d, then ${\dfrac{a+b}{a}\approx \dfrac{b+c}{b}\approx \dfrac{c+d}{c}\approx 1.618}$

It follows a constant angle close to the Golden Ratio and is commonly known as the Golden Spiral. In geometry, this ratio forms a Golden rectangle, a rectangle whose ratio of its length and breadth gives the Golden Ratio. It appears in many works of art and architecture.

Fibonacci Sequence and Binet’s Formula

Using the Golden Ratio, we can approximately calculate any Fibonacci numbers as

${F\left( x_{n}\right) =\dfrac{\phi ^{n}-\left( 1-\phi \right) ^{n}}{\sqrt{5}}}$

Where, ${\phi =1.618034}$

This is known as Binet’s Formula.

The Sum of the Fibonacci Sequence

The sum of the Fibonacci Sequence is obtained by:

${\sum ^{n}_{i=0}F_{i} =F_{n+2} – F_{2}}$

= ${F_{n+2}-1}$

where Fn is the nth Fibonacci number, and the sequence starts from F 0 .

For example, 

The sum of the first 12 terms = (12+2) th term – 2 nd term 

= 14 th term – 2 nd term

= 233 – 1 = 232

Finding Lucas Numbers from the Fibonacci Sequence

We get another number sequence from the Fibonacci Sequence that follows the same rule mathematically.

Here, the number sequence starting from 2 is formed by adding two preceding numbers, known as Lucas numbers. 

The sequence is as follows:

2, 1, 3, 4, 7, ….

Thus, the Lucas numbers are found to get closer to the powers of the Golden Ratio.

Fibonacci Sequence in Pascal’s Triangle

We can also derive the sequence in Pascal’s triangle from the Fibonacci Sequence. It is a number triangle that starts with 1 at the top, and each row has 1 at its two ends. Here, the middle numbers of each row are the sum of the two numbers above it.

Fibonacci Sequence in Real-Life

We find the Fibonacci Sequence in various fields, from nature to the human body.  

• It appears in plants with many seed heads, pinecones, fruits, and vegetables. The pattern of seeds in the sunflower also follows this sequence.
• We can find this sequence in rabbits. In a pair of a male and a female rabbit, if no rabbits die or leave the place, it forms the Fibonacci sequence 1,1,2,3,5, and so on due to their reproduction. 
• The spiral can be seen in seashells and the shapes of snails.
• People claim this is ‘nature’s secret code’ for building the structures perfectly, just like the Great Pyramid of Giza.
• It is also found in the galaxy.
• It is used in coding (distributed systems, computer algorithms )
• It is used in stock prices and other financial data.
• It appears in various fields of study, including cryptography and quantum mechanics. 
• The golden ratio and the Fibonacci numbers guide design for websites, architecture, and user interfaces. 
• It is also found in paintings. One of the most famous examples was painted by Leonardo da Vinci, the Monalisa. The spiral begins from her left wrist and travels to the background of the painting that follows the sequence.

Solved Examples

Find the sum of the first 15 Fibonacci numbers.

As we know, The sum of the Fibonacci Sequence = ${\sum ^{n}_{i=0}F_{i} = F_{n+2} – F_{2}}$ = ${F_{n+2}-1}$, where F n is the nth Fibonacci number, and the sequence starts from F 0 . Thus, the sum of the first 15 Fibonacci numbers = (15+2) th term – 2 nd term = 17 th term – 1 = 987 – 1 = 986

Find the 5 th Fibonacci number.

As we know, The n th Fibonacci number is F(x n ) = F(x n-1 ) + F(x n-2 ), for n>2 Then the 5 th Fibonacci number is F(x 5 ) = F(x 5-1 ) + F(x 5-2 ), for n=5 = F(x 4 ) + F(x 3 ) = 2 + 1 = 3

Find the next number when F 14 = 377.

Here, F 15 = F 14 x Golden ratio = 377 x 1.618034 (up to 4 decimals) = 609.9988 (up to 4 decimals), which is approximately 610 Hence, F 15 = 610

Calculate the value of F -6.

As we know, F -n = (-1) n+1 F n Here, F -6 = (-1) 6+1 F 6 = (-1) x 5 = -5

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Fibonacci Sequence

The Fibonacci Sequence is a series of numbers that starts with 0 and 1, and each subsequent number is the sum of the two preceding numbers. So the sequence goes 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. This sequence is named after Leonardo Pica (who was also known as Fibonacci), an Italian mathematician who introduced it to the Western world in his book Liber Abaci in 1202. This sequence has been termed "nature's secret code".

We can spot the Fibonacci sequence in the spiral patterns of sunflowers, daisies, broccoli, cauliflowers, and seashells. Let us learn more about it and its interesting properties.

What is Fibonacci Sequence?

The Fibonacci sequence is the sequence formed by the infinite terms 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... In simple terms, it is a sequence in which every number in the Fibonacci sequence is the sum of two numbers preceding it in the sequence. Its first two terms are 0 and 1. The terms of this sequence are known as Fibonacci numbers . The first 20 terms of the Fibonacci sequence are given as follows:

Here, we can observe that F n = F n-1 + F n-2 for every n > 1. For example:

• F 2 = F 1 + F 0
• F 3 = F 2 + F 1
• F 4 = F 3 + F 2 , and so on.

The significance of the Fibonacci Sequence lies in its prevalence in nature and its applications in various fields, including mathematics, science, art, and finance. The sequence can be observed in the arrangement of leaves on a stem, the branching of trees, and the spiral patterns of shells and galaxies. It is also used to describe growth patterns in populations, stock market trends, and more.

Fibonacci Spiral

The Fibonacci spiral is a geometrical pattern that is derived from the Fibonacci sequence. It is created by drawing a series of connected quarter-circles inside a set of squares that are sized according to the Fibonacci sequence.

The spiral starts with a small square, followed by a larger square that is adjacent to the first square. The next square is sized according to the sum of the two previous squares, and so on. Each quarter-circle fits perfectly within the next square in the sequence, creating a spiral pattern that expands outward infinitely. The larger the numbers in the Fibonacci sequence, the ratio becomes closer to the golden ratio (≈1.618).

In this Fibonacci spiral, every two consecutive terms of the Fibonacci sequence represent the length and width of a rectangle. Let us calculate the ratio of every two successive terms of the Fibonacci sequence and see how they form the golden ratio.

• F 2 /F 1 = 1/1 = 1
• F 3 /F 2 = 2/1 = 2
• F 4 /F 3 = 3/2 = 1.5
• F 5 /F 4 = 5/3 = 1.667
• F 6 /F 5 = 8/5 = 1.6
• F 7 /F 6 = 13/8 = 1.625
• F 8 /F 7 = 21/13 = 1.615
• F 9 /F 8 = 34/21 = 1.619
• F 10 /F 9 = 55/34 = 1.617
• F 11 /F 10 = 89/55 = 1.618 = Golden Ratio

In this way, when the rectangle is very large, its dimensions are very close to form a golden rectangle.

Overall, the Fibonacci spiral and the golden ratio are fascinating concepts that are closely linked to the Fibonacci Sequence and are found throughout the natural world and in various human creations. Their applications in various fields make them a subject of continued study and exploration.

Fibonacci Sequence Formula

The Fibonacci sequence formula for “F n ” is defined using the recursive formula by setting F 0 = 0, F 1 = 1, and using the formula below to find F n . The Fibonacci formula is given as follows.

F n = F n-1 + F n-2 , where n > 1. Here

• F n represents the (n+1) th number in the sequence and
• F n-1 and F n-2 represent the two preceding numbers in the sequence.

The Fibonacci sequence formula is used to compute the terms of the sequence to obtain a new term. For example, since we know the first two terms of Fibonacci sequence are 0 and 1, the 3 r d term is obtained by the above formula as follows:

F 3 = F 1 + F 2 = 0 + 1 = 1.

In the same way, the other terms of the Fibonacci sequence using the above formula can be computed as shown in the figure below.

Note that F 0 is termed as the first term here (but NOT F 1 ). Thus, F n represents the (n + 1) th term of the Fibonacci sequence here.

Fibonacci Sequence Properties

The Fibonacci sequence has several interesting properties.

1) Fibonacci numbers are related to the golden ratio. Any Fibonacci number can be calculated (approximately) using the golden ratio, F n =(Φ n - (1-Φ) n )/√5 (which is commonly known as "Binet formula"), Here φ is the golden ratio and Φ ≈ 1.618034.

To find the F 7 , we apply F 7 = [(1.618034) 7 - (1-1.618034) 7 ] / √5 = 13

2) The ratio of successive terms in the Fibonacci sequence converges to the golden ratio as the terms get larger.

Just by multiplying the previous Fibonacci Number by the golden ratio (1.618034), we get the approximated Fibonacci number. For example, 13 is a number in the sequence, and 13 × 1.618034... = 21.034442. This gives the next Fibonacci number 21 after 13 in the sequence.

2) Observe the sequence to find another interesting pattern. Every 3rd number in the sequence (starting from 2) is a multiple of 2. Every 4th number in the sequence (starting from 3) is a multiple of 3 and every 5th number (starting from 5) is a multiple of 5; and so on.

3) The Fibonacci sequence works below zero too. We write F -n = (-1) n+1 F n . For example, F -4 = (-1) 5 . F 4 = (-1) 3 = -3.

4) The sum of n terms of the Fibonacci sequence is given by Σ i=0 n F i = F n+2 - F 2 (or) F n+2 - 1, where F n is the n th Fibonacci number. (Note: the first term starts from F 0 )

For example, the sum of first 10 terms of sequence = 12 th term - 1 = 89 - 1 = 88. It can be mathematically written as Σ i=0 9 F i = F 11 - 1 = 89 - 1 = 88.

5) The Fibonacci Sequence has connections to other mathematical concepts, such as the Lucas numbers and Pascal's triangle .

Applications of Fibonacci Sequence

The Fibonacci sequence can be found in a varied number of fields from nature, to music, and to the human body.

• used in the grouping of numbers and the brilliant proportion in music generally.
• used in Coding (computer algorithms, interconnecting parallel, and distributed systems)
• in numerous fields of science including high-energy physical science, quantum mechanics, Cryptography, etc.
• used to model various phenomena in biology, such as the growth patterns of plants and the arrangement of leaves on a stem.
• used in financial analysis to identify trends in stock prices and other financial data.

You can use the Fibonacci calculator that helps to calculate the Fibonacci Sequence. Look at a few solved examples to understand the Fibonacci formula better.

☛ Related Articles:

• Sequence and Series
• Arithmetic Sequence Formula
• Geometric Sequence Formulas

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Examples of Fibonacci Sequence

Example 1: Find the 12 th term of the Fibonacci sequence if the 10 th and 11 th terms are 34 and 55 respectively.

Using the Fibonacci sequence recursive formula, we can say that the 12 th term is the sum of 10 th term and 11 th term.

12 th term = 10 th term + 11 th term

Answer: The 12 th term is 89.

Example 2: The F 14 in the Fibonacci sequence is 377. Find the next term.

We know that F 15 = F 14 × the golden ratio.

F 15 = 377 × 1.618034

Answer: F 15 = 610.

Example 3: Calculate the value of the 12 th and the 13 th terms of the Fibonacci sequence given that the 9 th and 10 th terms in the sequence are 21 and 34.

Using the formula, we can say that the 11 th term is the sum of 9 th term and 10 th term.

11 th term = 9 th term + 10 th term = 21 + 34 = 55

Now, 12 th term = 10 th term + 11 th term = 34 + 55 = 89

Similarly,13 th term = 11 th term + 12 th term = 55 + 89 = 144

Answer: The 12 th and the 13 th terms are 89 and 144.

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Practice Questions on Fibonacci Sequence

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FAQs on Fibonacci Sequence

What is the definition of fibonacci sequence.

The Fibonacci sequence is an infinite sequence in which every number in the sequence is the sum of two numbers preceding it in the sequence and is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 , 144, ..... The ratio of consecutive numbers in the Fibonacci sequence approaches the golden ratio, a mathematical concept that has been used in art, architecture, and design for centuries. This sequence also has practical applications in computer algorithms, cryptography, and data compression.

What is the Formula for Generating the Fibonacci Sequence?

The Fibonacci sequence formula deals with the Fibonacci sequence, finding its missing terms. The Fibonacci formula is given as, F n = F n-1 + F n-2 , where n > 1. It is used to generate a term of the sequence by adding its previous two terms.

What is the Difference Between Fibonacci Sequence Formula and Fibonacci Series Formula?

What is fibonacci spiral.

Here are the steps of formation of Fibonacci spiral.

• First, take a small square of length 1 unit and attach it to an identical square vertically.
• Thus formed is a rectangle of vertical length 2 and width 1 unit.
• Adjacent to its length (2 units), attach a square of length 2 units.
• Thus formed is a rectangle of horizontal length 3 units and vertical width 2 units.
• If we continue the same process we get a big rectangle that is partitioned into squares where the length of each square is the sum of the lengths of two of its adjacent squares.
• The larger the rectangle, the more the chances for it to become a golden rectangle.
• If we join the centers of all squares, we get a spiral which is known as the Fibonacci spiral. For more information, click here .

What is The Fibonacci Sequence in Nature?

We can spot the Fibonacci sequence as spirals in the petals of certain flowers, or the flower heads as in sunflowers, broccoli, tree trunks, seashells, pineapples, and pine cones. The spirals from the center to the outside edge create the Fibonacci sequence.

How is the Fibonacci Sequence Related to the Golden Ratio?

The Fibonacci Sequence is closely related to the Golden Ratio, which is a mathematical ratio represented by the symbol phi (φ). The Golden Ratio is approximately equal to 1.61803398875. The ratio of each consecutive pair of Fibonacci numbers approximates the Golden Ratio as the numbers get higher. For example 21/13 = 1.615..., 34/21 = 1.619, ...

How Do You Find the Sum of The Fibonacci Sequence of n Terms?

The explicit formula to find the sum of the Fibonacci sequence of n terms is given by of the given generating function is the coefficient of Σ i=0 n F i = F n+2 - 1. For example, the sum of the first 12 terms in a Fibonacci sequence is Σ i=0 11 F i = F 13 -1 = 233 -1 = 232. If we add the first 12 terms manually, we get 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 = 232, we got the same thing as the formula.

What is the Recursive Formula to Find the nth Term of the Fibonacci Sequence?

We can't write a Fibonacci sequence easily using an explicit formula. Thus, we used to describe the sequence using a recursive formula, defining the terms of a sequence using previous terms. When F 0 = 0, F 1 = 1, F n = F n-1 + F n-2 , where n > 1.

What is the Formula for the n th Term of The Fibonacci Sequence?

The formula to find the n th term of the sequence is denoted as F n = F n-1 + F n-2 , where n >1.

Why is Fibonacci Sequence Important?

The Fibonacci sequence has many interesting mathematical properties, including the fact that the ratio of each consecutive pair of numbers approximates the Golden Ratio. It is also closely related to other mathematical concepts, such as the Lucas Sequence and the Pell Sequence. The Fibonacci sequence has many applications in science and engineering, including the analysis of population growth. The Fibonacci sequence appears in many forms in nature, including the branching of trees.

Plus.Maths.org

The Fibonacci sequence: A brief introduction

Leonardo Fibonacci c1175-1250.

The Fibonacci sequence

Where does it come from?

The Fibonacci is named after the mathematician Leonardo Fibonacci who stumbled across it in the 12th century while contemplating a curious problem. Fibonacci started with a pair of fictional and slightly unbelievable baby rabbits, a baby boy rabbit and a baby girl rabbit.

They were fully grown after one month

and did what rabbits do best, so that the next month two more baby rabbits (again a boy and a girl) were born.

The next month these babies were fully grown and the first pair had two more baby rabbits (again, handily a boy and a girl).

Ignoring problems of in-breeding, the next month the two adult pairs each have a pair of baby rabbits and the babies from last month mature.

Therefore, the total number of pairs of rabbits (adult+baby) in a particular month is the sum of the total pairs of rabbits in the previous two months:

Where does it go?

Real rabbits don't breed as Fibonacci hypothesised, but his sequence still appears frequently in nature, as it seems to capture some aspect of growth. You can find it, for example, in the turns of natural spirals, in plants, and in the family tree of bees. The sequence is also closely related to a famous number called the golden ratio . To find out more read The life and numbers of Fibonacci .

About the author

Rachel Thomas is Editor of Plus .

• Add new comment

5 occupies position number 5 in the Fibonacci sequence.

The number that occupies the 25th position, 75025, ends with 25 (which is 5^2 of course).

The125th position is occupied by a number ending in 125 (or 5^3): 59425114757512643212875125.

Ditto for 625 and 3125, which are 5^4 and 5^5 respectively

But this curious rule seems to start breaking down for powers of 5 greater than 5.

Hitting that square

Now let's pretend 5 is the first Fibonacci number instead of the usual 1, but still use the same addition algorithm: 5 5 10 15 25. That 25 is of course 5 squared.

Try this with 4 however: 4 4 8 12 20, and we don't land on 4 squared. 16 isn't in the sequence generated. The same goes for 6: 6 6 12 18 30 48 doesn't include 36. But then 4 and 6 aren't in the Fibonacci sequence either.

In fact the only start numbers we can hit the square with seem to be the Fibonaccis and no others:

5 5 10 15 25

8 8 16 24 40 64

13 13 26 39 65 104 169

(My treatment of 1 seems a bit anomalous here since although it 's a perfect square, I haven't presented it as the result of any addition. This can be remedied perhaps with: 0 1 1)

Note also the number of numbers in each sequence, which is equal to the position of the start number in the standard Fibonacci sequence. For example 13 takes 7 numbers to get to its square, and 13 occupies position 7.

Lastly although I came across these results concerning Fibonacci powers on my own (see also my previous comment about 5), I daresay they aren't new discoveries. So please tell me of any similar work.

Nice conjecture!

Your conjecture that you only get to the square of the start number if it's one of the original Fibonacci sequence is lovely! I wonder if anyone can come up with a proof?

That conjecture

I liked it too at first, Rachel, but now I realise it's not so exciting after all, since it says nothing about Fibonacci numbers in particular unfortunately.

Take any random collection of numbers that includes 1. Beginning with 1, line them up in otherwise any way you like. Tell someone it's a "sequence" and that they are to multiply each successive number (including the 1) by any one number already there in the sequence. Obviously they're going to generate a new sequence which includes a number which is a perfect square of the first. For example 1 23 7 13 4 5 each multiplied by 4 is going to result in 4 92 28 52 16 20. That's effectively all I did in multiplying each item in the standard Fibonacci 1 1 2 3 5 8 by 8 for example to get 8 8 16 24 40 64 without spotting its triviality at the time.

Anyway I'm very pleased you had a look. I still hope my other results in this thread are interesting, especially the quantitative and structural differences in sequences resulting from varying the maturation delay. See "Fibonacci's fast and slow breeders". I'd be grateful for your opinion.

In defence of Fibonacci's breeding programme.

Yes, Fibonacci made certain assumptions, as we all do when figuring out what to expect. But is it fair of Rachel to describe them as a "highly unbelievable breeding process"?

True, he ignores potential in-breeding problems and mortality, but this isn't unduly optimistic given that he asked what quantity of rabbits could be expected at the end of only 12 months. And the birth of two babies of the same sex to one adult pair could be offset by two of the opposite sex being born to another.

What Fibonacci does refrain from assuming is a geometric population expansion such as 1 pair the first month, 2 the second, 4 the third, and so on. At 12 months this would lead to 2048 pairs of rabbits, compared to Fibonacci's much more modest 144. This is because he builds in a highly realistic maturation delay right from the start: 1 pair the first month, and still just one pair the second month. Only with the third do we have two pairs (two adults and two babies), 3 pairs the fourth month (the first pair of adults with another pair of babies, but the second generation still without issue). That second generation first produces its own pair in the fifth month along with yet another pair from the first two rabbits, making 5 pairs altogether, and so on.

Asking how many pairs after 12 months seems reasonable given our custom of yearly progress reviews. However the answer 144 must have intrigued Fibonacci the mathematician, as it does us today, since it is of course 12 squared.

I investigate this matter of powers in the Fibonacci sequence a bit more in my two previous comments, Fib 5 and Hitting that Square.

reply to your comment as a rabbit breeder

Male rabbits are not sexually mature until a minimum of 4 months and females a minimum of 6 months.

Fibonacci's fast and slow breeders

I've already observed how restrained Fibonacci rabbits are in their reproducton rate, due to the maturation delay (MD) of two months before producing their first offspring in the third. I've been constructing some of those well known family tree type diagrams for varying MD values: 1 month, 2 months (the Fibonacci case), 3 and 4. They seem to yield the following twelve-item sequences:

For MD = 1 month: 1 2 4 8 16 32 64 128 256 512 1024 2048 A simple doubling at each stage as every rabbit pair produces another in the second month of life. Fast.

2 months: 1 1 2 3 5 8 13 21 34 55 89 144

3 months: 1 1 1 2 3 4 6 9 13 19 24 31 The addition algorithm is a bit more complicated than for the familair Fibonacci one. It involves some leapfrogging (or should I say leapbunnying): 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, 2 + 4 = 6, 3 + 6 = 9, 4 + 9 = 13 and so on.

4 months: 1 1 1 1 2 3 4 5 7 10 14 19 Here we add each pair of numbers separated by two in between: 1 + 1 = 2, 1 + 2 = 3, 1 + 3 = 4, etc. Note that 10 is both the rabbit quantity and month number, just as 5 is for the Fibonacci case, so maybe the same sort of results as I observed before in "Fib 5".

OK I know my comments to Rachel's stimulating article have been proliferating like, well, you know what. That's just because they've been conceived and born with corresponding spontaneity.

Hitting those squares - with some help from Monsieur Lucas

Hi Rachel. This is to continue our interesting discussion about a result with squares I'd thought I'd found in the Fibonacci sequence (see below). This time compare some of the sequence to the left as well as the right of 0 with a corresponding segment of the famous Lucas sequence:

Fibonacci: . . . 34 -21 13 -8 5 -3 2 -1 1 0 1 1 2 3 5 8 15 21 34 . . .

Lucas: . . . 47 -29 18 -11 7 -4 3 -1 2 1 0 3 4 7 11 18 29 47 . . .

The Lucas can be seen as resulting from swapping round two consecutive Fibonacci terms, from 2, -1 to -1, 2 while retaining the same addition rule as Fibonacci, adding two consecutive numbers to get the third as you go right. To the right in the Lucas we now have not just one but two integer squares, those of -1 and 2, namely 1 and 4, in the Lucas. Hmm? Well, let's do the Lucas swap elsewhere in the Fibonacci to be a bit more convincing. What about from 13, -8 to -8, 13 (and get a different sequence because we're keeping to the same addition rule)?

. . . -8 13 5 18 23 41 64 85 169

Do you think this result escapes the charge of triviality that I felt bound to level at my first attempt to hit the square? Do you feel it's twice as lovely? I do for the moment, but am always ready to reconsider.

There shouldn't be a 0 in the Lucas sequence.

Hitting those squares - another correction

The last sequence should have 105, not 85, as the penultimate number in the last sequence. I do wish I could spot errors before hitting the save button. Anyway, at least it doesn't affect my point, which is that 64 and 169 are squares of -8 and 13 respectively.

Rabbit problem

can anybody answer this question?

Modify Fibonacci’s rabbit problem by introducing the additional rule that rabbit pairs stop reproducing after giving birth to three litters. How does this assumption change the recurrence relation? What changes do you need to make in the simple cases?

Tika's rabbit problem, second attempt.

It doesn’t look as if PlusMaths is going to post my first answer. Just as well, since it was wrong, so here’s my second attempt.

I printed off the biggest Fibonacci rabbit family tree diagram I could find on the net, showing 10 monthly generations resulting in a population of 55 pairs. I subtracted from each monthly population total those rabbit pairs which can't exist that month if each pair stops breeding after producing 3 litters. I entered the new sequence 1 1 2 3 5 7 11 16 24 35 into OEIS and got (amongst others) A023435 which followed on with 52 76 112 164 241 ...

The new recurrence relation, given by OEIS, is a(n)= a(n-1) + a(n-2) - a(n-5). Note that n=5 is the last index at which the Fibonacci and this new sequence continue to share terms. It marks the third and last time the first rabbit pair produces offspring.

The main problem I can see is OEIS also calls this sequence "Dying rabbits". That seems to negate the question, which assumes all rabbits once born continue to live and figure in the subsequent monthly population totals indefinitely. As I'm sure I only crossed off those which couldn't have been born, not those who simply stopped reproducing, this name appears to be wrong.

if you had 2 bunnies at the start you would have 18,454,930 rabbits after 3 years.

Rabbits Reproducing

Clever buggers, rabbits. They can have two pregnancies going at once - one on the back-burner. Allows them to get through lean times and still reproduce with a minimum of mating. Sort of damages their promiscuous reputation though! I suspect I haven't contributed value to the mathematical discussion. I respect the discussion, but it hurt my head trying to follow it.

You seem to be referring to the original Fibonacci sequence, term 35.

If you meant to reply to my post, then the number of rabbit pairs at this stage if they produce only 3 litters each is 504,355, or only 1,008,710 individuals.

It's still a lot though.

Your rabbits are really well pictured :)

other versions

are there other versions of the rabbit problem

model of pairs of rabbits

okay so what if i had a model of the number of pairs of rabbits up to 30 time periods but then i wanted to modify the model so that in the fifth period of life the rabbits cease to produce offspring and in the next period they die

Found in nature?

I didn't know that about bee family trees. That's very interesting because it seems legitimate, and actual instances of the Fibonacci sequence in nature and art are more few and far between than people think. For instance, Fibonacci spirals are not especially common. Your source never claims that they are found in nature; it just makes the extremely weak claim that spirals are found in nature. This has nothing to do with the Fibonacci sequence unless they are Fibonacci spirals, so that section should be omitted if you ask me.

Some bunnies aren't so nice.

In fact they've been known to cannibalise each other. Well, that famous variant on the Fibonacci sequence, known as the Lucas sequence, can be used to model this. It goes 2 1 3 4 7 11 18 29 47 76 and so on, but like Fibonacci adding each successive two numbers to get the next.

For our rabbits this means start with 2 pairs and one eats the other, so now only 1. However that 1 then gives birth to 3. Of those 3, 1 gives birth to 2, but the other 2 don't give birth yet, so now we have 4. Next the 2 that didn't give birth last time now give birth to 3 each, while of the remaining 2, one eats the other, leaving us with a total of 7. And so on.

Probability

Each die is a standard 6-sided cube, but the six faces bear the first six numbers of the Fibonacci sequence, 1, 1, 2, 3, 5, and 8. If this pair of dice is rolled once, what is the probability that the sum of the numbers showing on the top faces of the two dice is a Fibonacci number?

Fibonacci Sequence

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...

The next number is found by adding up the two numbers before it:

• the 2 is found by adding the two numbers before it (1+1),
• the 3 is found by adding the two numbers before it (1+2),
• the 5 is (2+3),

It is that simple!

Here is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,233,377,610,987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, ...

Can you figure out the next few numbers?

Makes A Spiral

When we make squares with those widths, we get a nice spiral:

Do you see how the squares fit neatly together? For example 5 and 8 make 13, 8 and 13 make 21, and so on.

The Fibonacci Sequence can be written as a "Rule" (see Sequences and Series ).

First, the terms are numbered from 0 onwards like this:

So term number 6 is called x 6 (which equals 8).

So we can write the rule:

The Rule is x n = x n−1 + x n−2

• x n is term number "n"
• x n−1 is the previous term (n−1)
• x n−2 is the term before that (n−2)

Example: term 9 is calculated like this:

Golden ratio.

And here is a surprise. When we take any two successive (one after the other) Fibonacci Numbers, their ratio is very close to the Golden Ratio " φ " which is approximately 1.618034...

In fact, the bigger the pair of Fibonacci Numbers, the closer the approximation. Let us try a few:

We don't have to start with 2 and 3 , here I randomly chose 192 and 16 (and got the sequence 192, 16,208,224,432,656, 1088, 1744, 2832, 4576, 7408, 11984, 19392, 31376, ... ):

It takes longer to get good values, but it shows that not just the Fibonacci Sequence can do this!

Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

x n = φ n − (1−φ) n √5

The answer comes out as a whole number , exactly equal to the addition of the previous two terms.

Example: x 6

x 6 = (1.618034...) 6 − (1−1.618034...) 6 √5

When I used a calculator on this (only entering the Golden Ratio to 6 decimal places) I got the answer 8.00000033 , a more accurate calculation would be closer to 8.

Try n=12 and see what you get.

You can also calculate a Fibonacci Number by multiplying the previous Fibonacci Number by the Golden Ratio and then rounding (works for numbers above 1):

Example: What is the next in the sequence after 8 ?

It will be 8 times φ:

8φ = 8 × 1.618034... = 12.94427... = 13 (rounded)

Some Interesting Things

An odd fact:.

The sequence goes even, odd , odd , even, odd , odd , even, odd , odd , ... :

Because adding two odd numbers produces an even number, but adding even and odd (in any order) produces an odd number.

Lucas Numbers

Starting the sequence with 2 and 1 we get the "Lucas Numbers". They get closer and closer to the powers ( exponents ) of the Golden Ratio:

For example, the 15th Lucas Number is approximately φ 15 = 1364.0007..., so is exactly 1364 . Try it yourself!

Here is the Fibonacci sequence again:

There is an interesting pattern:

• Look at the number x 3 = 2 . Every 3 rd number is a multiple of 2 (2, 8, 34,144,610, ...)
• Look at the number x 4 = 3 . Every 4 th number is a multiple of 3 (3, 21,144, ...)
• Look at the number x 5 = 5 . Every 5 th number is a multiple of 5 (5, 55,610, ...)

And so on (every n th number is a multiple of x n ).

1/89 = 0.011235955056179775...

Notice the first few digits (0, 1, 1, 2, 3, 5) are the Fibonacci sequence?

In a way they all are, except multiple digit numbers (13, 21, etc) overlap , like this:

Fibonacci Words

by Rod Pierce

Terms Below Zero

The sequence works below zero also, like this:

(Prove to yourself that each number is found by adding up the two numbers before it!)

In fact the sequence below zero has the same numbers as the sequence above zero, except they follow a +-+- ... pattern. It can be written like this:

x −n = (−1) n+1 x n

Which says that term "−n" is equal to (−1) n+1 times term "n", and the value (−1) n+1 neatly makes the correct +1, −1, +1, −1, ... pattern.

Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

About Fibonacci The Man

His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.

"Fibonacci" was his nickname, which roughly means "Son of Bonacci".

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.

Fibonacci Day

Fibonacci Day is November 23rd, as it has the digits "1, 1, 2, 3" which is part of the sequence. So next Nov 23 let everyone know!

What Is the Fibonacci Sequence?

The Fibonacci sequence is a series of numbers made famous by Leonardo Fibonacci in the 12th century. It has been described in texts for over two millennia, with the earliest description found in Indian texts in 200 BC , and further development throughout the first millennium . It appears commonly in mathematics and in nature, and for that reason has become a popular pedagogical tool. 

Why Is the Fibonacci Sequence Important?

The Fibonacci sequence has many interesting properties and appears in various areas of mathematics, science and nature. Given how often it can be found in nature, some have suggested that the sequence has some underlying mathematical principles at work in nature. The Fibonacci Sequence also has connections to other areas of mathematics such as number theory, algebra and geometry.

More From the Built In Tech Dictionary What Is Schrödinger’s Cat?

How Does the Fibonacci Sequence Work?

The formula that defines the Fibonacci sequence is:

F n =F n-1 +F n-2

We can also describe this by stating that any number in the Fibonacci sequence is the sum of the previous two numbers.

For the most common representation of the Fibonacci sequence, the first two terms are defined as F 0 =0, F 1 =1 . This leads to the sequence 0, 1, 1, 2, 3, 5, 8, 13, . . .

It’s possible to calculate other Fibonacci sequences by starting with different base numbers, for example:

F 0 =0, F 1 =2 ; Fibonacci sequence: 0, 2, 2, 4, 6, 10, 16, 26, . . .

F 0 =2, F 1 =1 ; Fibonacci sequence: 2, 1, 3, 4, 7, 11, 18, 29, 47, . . .

The Fibonacci sequence also has a closed form representation, known as Binet’s formula. With the closed formula it’s possible to calculate the n th  value in the Fibonacci sequence directly, without calculating each of the previous numbers.

Is the Fibonacci Sequence the Same as the Golden Ratio?

No, the Fibonacci sequence and the golden ratio are not the same. That said, the Fibonacci sequence is intimately related to the golden ratio, a value with significant cultural importance. The golden ratio has fascinated people across numerous fields , from art to architecture to music.

The golden ratio can be approximately derived by dividing any Fibonacci number by the previous one. This ratio becomes more accurate the further you proceed down the sequence. You can see the results below.

1/1 = 1 2/1 = 2 3/2 = 1.5 5/3 = 1.3 . . .  89/55 = 1.62 . . .  F n /F n-1 ≈ 1.618 . . .

More From Jye Sawtell-Rickson Rage Against the Machine Learning: My War With Recommendation Engines

What Are Other Sequences Similar to the Fibonacci Sequence?

The Fibonacci sequence is an example of a number sequence, of which there are many others in mathematics. Some other common sequences include:  

Arithmetic Sequence

You can reach each number by adding a fixed number to the previous one. Each pair of numbers has a common difference.

• A n = A n-1 +a 1  = a 0 + n x a 1
• 1, 3, 5, 7, 9, 11, . . .

Triangular Numbers

The n th triangle number is the number of dots in a triangle with n dots on a side. You can also state it as the sum of all the numbers from 1 to n .

• T n  = n(n+1) / 2 
• 1, 3, 6, 10, 15, 21, . . .

Magic Squares Constant

In magic squares, a set of numbers is arranged in a square to such that the rows, columns and diagonals all sum up to the same value. A magic square of size n is typically filled with the numbers from 1 to n 2 . The common sum is known as the magic constant.

• M n =n(n 2 +1) / 2
• 1, 5, 15, 34, 65, 111, . . .

Fibonacci Sequence In Science 

Fibonacci search is a key application of the Fibonacci sequence in the space of computer science . In Fibonacci search, the search space is divided up into segments according to the Fibonacci numbers, differing from common search algorithms such as binary search . This algorithm isn’t commonly used today, but it has niche applications. 

For example, when the array you’re searching is very large and cannot fit in memory, Fibonacci search can be more efficient. You can also use Fibonacci search when only the addition and subtraction operations are available, as opposed to binary search which requires division or multiplication. However, on average, Fibonacci search requires four percent more comparisons compared to binary search.

The Fibonacci Sequence in Nature and Art

Outside of human applications, we find the Fibonacci sequence in nature. For example, the arrangement of seeds on pinecones follow the Fibonacci sequence, as do the seeds in a sunflower and the sections of a Roman cauliflower.

These various applications are interesting discoveries, however there has been no strong justification for why these various phenomena occur in nature. Similarly, in various artworks and architectural findings, there is limited evidence that the creators specifically built the Fibonnaci sequence into their works.

Humans are good at finding patterns , even when no patterns exist. In the case of Fibonacci, we have to be careful not to over-analyze unrelated patterns.  

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• Munem Shahriar
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The Fibonacci sequence is an integer sequence defined by a simple linear recurrence relation . The sequence appears in many settings in mathematics and in other sciences. In particular, the shape of many naturally occurring biological organisms is governed by the Fibonacci sequence and its close relative, the golden ratio .

The Fibonacci numbers appear as numbers of spirals in leaves and seedheads as well.

The Fibonacci numbers are the terms of a sequence of integers in which each term is the sum of the two previous terms with \[\begin{array} &F_1 = F_2 = 1, &F_n = F_{n-1} + F_{n-2}.\end{array}\]

The first few terms are

\[1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, \ldots.\]

Closed-form Expression

Continued fraction, enumerative problems, identities involving fibonacci series, generating function, divisibility properties, zeckendorf's theorem.

Let \( \phi = \frac{1+\sqrt{5}}2 \) be the golden ratio. Let \( {\overline \phi} = \frac{-1}{\phi} = \frac{1-\sqrt{5}}2. \) Then \[{ F }_{ n }= \frac { \phi^n-{\overline \phi}^n }{ \sqrt { 5 } } .\]

The formula (often called Binet's formula ) comes from a general result for linear recurrence relations , but it can be proved directly by induction. Let \( G_n = \frac { \phi^n-{\overline \phi}^n }{ \sqrt { 5 } } \). The goal is to prove that \( F_n=G_n\) by induction on \(n \). The base cases are \( G_1 = G_2 = 1 \), which is clear. Now suppose \( G_k=F_k \) for all \( k<n\), where \( n \) is at least \( 3 \). Then \[ \begin{align} F_n &= F_{n-1}+F_{n-2} \\&= G_{n-1}+G_{n-2} & \text{(inductive hypothesis)} \\ &= \frac1{\sqrt{5}} \Big(\phi^{n-1}-{\overline{\phi}}^{n-1}\Big)+\frac1{\sqrt{5}}\Big(\phi^{n-2}-{\overline{\phi}}^{n-2}\Big) \\ &= \frac1{\sqrt{5}} \Big(\phi^{n-1}+\phi^{n-2}-{\overline\phi}^{n-1}-{\overline\phi}^{n-2}\Big) \\ &= \frac1{\sqrt{5}} \Big(\phi^n-{\overline\phi}^n\Big) = G_n, \end{align} \] where the last line comes from the fact that \( \phi\) and \(\overline\phi\) are the two roots of the equation \( x^2=x+1\). \(_\square\)

Note that for \( n \ge 1 \), the term \( \frac{{\overline\phi}^n}{\sqrt{5}} \) is small, certainly between \( -0.3 \) and \( 0.3\). So \( F_n \) is the nearest integer to \( \frac{\phi^n}{\sqrt{5}} \).

We have \[\frac{\phi^{10}}{\sqrt{5}} = 55.0036\ldots, \quad \frac{\phi^{11}}{\sqrt{5}} = 88.9977\ldots,\] so \( F_{10} = 55 \) and \( F_{11} = 89\).

The ratios of successive Fibonacci numbers should therefore be roughly

\[\displaystyle \phi =\lim _{ n\rightarrow \infty }{ \frac { { F }_{ n+1 } }{ { F }_{ n } } } = \frac{1+\sqrt5}{2}.\]

This can be proved easily using Binet's formula.

We have \[ \begin{align} \frac{F_{n+1}}{F_n} &= \frac{\phi^{n+1}-{\overline{\phi}}^{n+1}}{\phi^n-{\overline{\phi}}^n} \\ &= \frac{\phi-\frac{{\overline{\phi}}^{n+1}}{\phi^n}}{1-\frac{{\overline{\phi}}^n}{\phi^n}}, \end{align} \] and the terms \( \frac{{\overline{\phi}}^{n+1}}{\phi^n} \) and \( \frac{{\overline{\phi}}^n}{\phi^n} \) approach \( 0 \) as \( n \to \infty \), because the numerator approaches \( 0 \) and the denominator approaches \( \infty \). Therefore, the limit is \( \frac{\phi}1 = \phi \). \(_\square\)

The Fibonacci sequence appears as the numerators and denominators of the convergents to the simple continued fraction

\[ [1,1,1,\ldots] = 1+\frac1{1+\frac1{1+\frac1{\ddots}}}. \]

This continued fraction equals \( \phi,\) since it satisfies \( x = 1+\frac1{x} \) (and it is greater than 1).

The \( n^\text{th}\) convergent to this continued fraction is \( \frac{F_{n+1}}{F_n} \), so this gives another proof that \( \lim\limits_{n\to\infty} \frac{F_{n+1}}{F_n} = \phi \).

Like the Catalan numbers , the Fibonacci numbers count many types of combinatorial objects. Here are four examples.

(1) \( F_n \) is the number of compositions of \( n-1 \) consisting of \( 1\)s and \( 2\)s. (A composition of \(n-1\) is an expression of \( n-1\) as a sum of parts, where the order of the parts matters.) For instance, \[ \begin{align} 5 &= 1+1+1+1+1\\&=1+1+1+2\\&=1+1+2+1\\&=1+2+1+1\\&=2+1+1+1 \\&=1+2+2\\&=2+1+2\\&=2+2+1, \end{align} \] so \( F_6 = 8. \) (2) \( F_n \) is the number of ways to tile a \( 2\times (n-1)\) board with \( 1\times 2\) dominoes. (3) \( F_n \) is the number of binary sequences of length \( n-2\) with no consecutive \( 0\)s. (4) \( F_n \) is the number of subsets of \( \{ 1,2,\ldots,n-2\} \) that do not contain any pair of consecutive numbers.

To see that the Fibonacci numbers count these objects, let the number of objects equal \( G_n \) and show that \( G_n=G_{n-1}+G_{n-2}\). Then verify that \( F_1=G_1\) and \( F_2 = G_2 \), and the proof is complete.

For instance, to prove (4), start with \( G_1=1, G_2 =1, G_3= 2, G_4 = 3 \), and then suppose \( S \) is a subset of \( \{1,2,\ldots,n-2\}\) without any consecutive numbers in it. There are \( G_{n-1} \) such sets \( S\) that don't contain \( n-2 \). If \( S \) contains \( n-2\), then it doesn't contain \( n-3\), so \( S \) is obtained by taking a subset of \( \{1,2,\ldots,n-4\} \) and throwing in \( n-2\). So there are \( G_{n-2}\) such sets \( S\). This proves that \( G_n = G_{n-1}+G_{n-2},\) as desired.

A composition of \( n \) is an expression of \( n \) as a sum of not necessarily distinct positive integers, where the order matters. Note that \( n = n \) counts as a composition of \( n \).

Let \( C_n \) be the number of compositions of \( n \) with no part equal to 1.

For instance, \( C_6 = 5 \) because \( 6 = 6 =4+2=3+3=2+4=2+2+2.\)

Find \( C_{15} \).

A clown can climb a staircase either by one step or by two steps. For example, he can climb from the floor to the first step and then to the third, or he can climb from the floor to the first step, then to the second, and then finally to the third.

If a staircase has 10 steps, in how many ways can the clown climb it?

Clarification:

• When he climbs from the \({ 9 }^\text{th}\) step to the \({ 10 }^\text{th},\) he has climbed the whole stair; that is, the final step is the second floor.
• The order in which he climbs the staircase matters!

Bonus: Generalize it.

The Fibonacci numbers are given by

• \(F(0) = 0\)
• \(F(1) = 1\)
• \(F(n) = F(n-1) + F(n-2).\)

So, the first few are \(0, 1, 1, 2, 3, 5, 8, 13.\)

If we generalize this to negative numbers, what is \(F(-8)?\)

There are quite a few identities relating different Fibonacci numbers. For instance,

\[ F_n^2-F_{n-1}F_{n+1} = (-1)^{n+1}, \]

which has the useful corollary that consecutive Fibonacci numbers are coprime .

As is typical, the most down-to-earth proof of this identity is via induction. It is clear for \( n = 2,3 \), and now suppose that it is true for \( n \). Then \[ \begin{align} F_{n+1}^2-F_nF_{n+2} &= F_{n+1}^2-F_n(F_n+F_{n+1})\\ &= F_{n+1}(F_{n+1}-F_n) - F_n^2 \\ &= F_{n+1}F_{n-1}-F_n^2 \\ &= -(-1)^{n+1} = (-1)^{n+2}, \end{align} \] so it is true for \( n+1\). \(_\square\)

The proofs of the rest of these identities are similar.

(1) \( F_mF_n + F_{m-1}F_{n-1} = F_{m+n-1}\) (1a) \( F_{2n-1}=F_{n-1}^2+F_n^2 \) (1b) \( F_{2n} = F_n(F_{n-1}+F_{n+1}) \) (2) \( F_1+F_2+F_3+\cdots+F_n = F_{n+2}-1 \) (3) \( F_1+F_3+F_5+\cdots+F_{2n-1} = F_{2n} \) and \( F_2+F_4+F_6+\cdots+F_{2n}=F_{2n+1}-1 \) (4) \( F_{n+1} = \sum\limits_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k} \)

Once upon a time, a rabbit and a turtle were competing in a race.

The fast rabbit could hop in an increasing distance similar to the Fibonacci sequence (omitting the first 1-term) as shown above: \(1, 2, 3, 5, 8, 13,\ldots.\)

The slow turtle could roll in an increasing distance of an arithmetic sequence of 1-interval as shown: \(1, 2, 3, 4, 5,\ldots\)

Though seemingly even at the first three steps, soon afterwards, the rabbit rapidly went ahead of his opponent. However, at one point, the rabbit, confident of his victory, stopped for a nap. Later on, the turtle continued his track in the same pattern and met the rabbit at that same distance. The turtle then carried on his effort before eventually winning the race.

According to this tale, what is the least possible distance from the start to the rabbit's sleeping point?

I was cleaning up my attic recently and found a set of at least 14 sticks which a curious Italian gentleman sold me some years ago. Trying hard to figure out why I bought it from him, I realized that the set has the incredible property that there are no \(3\) sticks that can form a triangle. If the set has two sticks of length \(1\), which are the smallest, what is the least possible length of the \({ 14 }^\text{th}\) stick?

\[y^2 - xy - x^2 = 1\]

Let \((x,y)\) be the non-negative integer solutions to the hyperbolic graph above.

If \(x+y = n\) for some perfect square \(n\), what is the sum of all possible \(n?\)

Hint: The only Fibonacci numbers that are perfect squares are 0, 1, and 144.

The generating function of the Fibonacci numbers is

\[ \sum_{n=1}^\infty F_n x^n = \frac{x}{1-x-x^2}. \]

This follows from the expansion: \[\begin{align} \big(1-x-x^2\big)\big(F_1x + F_2x^2 + F_3x^3 + \cdots\big) &= F_1x + (F_2-F_1)x^2+(F_3-F_2-F_1)x^3+(F_4-F_3-F_2)x^4+ \cdots \\ &= x+0x^2+\sum_{k=3}^\infty (F_k-F_{k-1}-F_{k-2})x^k \\ &= x.\ _\square \end{align}\]

\[ \frac{1}{10}+ \frac{1}{10^2}+ \frac{2}{10^3}+ \frac{3}{10^4}+ \frac{5}{10^5}+ \frac{8}{10^6}+ \cdots \]

Find the sum of the above fractions, where the denominators follow a geometric progression and the numerators follow the Fibonacci sequence.

If your answer is \(\frac{A}{B}\), where \(A\) and \(B\) are coprime positive integers, submit your answer as \(A+B\).

\[ \sum_{n=1}^ \infty \dfrac{ n F_{n}}{ 2^n } = k{\sum_{n=1}^ \infty \dfrac{ F_{n}}{ 2^n }}\]

Let \(F_n\) denote the \(n^\text{th} \) Fibonacci number, where \(F_0 = 0\), \(F_1 = 1\) and \(F_n = F_{n-1} + F_{n-2} \) for \(n=2,3,4, ....\)

Find the value of \(k\) satisfying the equation above.

\[ \begin{eqnarray} 0. && 00000 \quad 00001 \quad 00001 \quad 00002 \quad 00003 \quad 00005 \quad 00008 \\ && 00013 \quad 00021 \quad 00034 \quad 00055 \quad 00089\quad 00144\quad \ldots \\ \end{eqnarray} \]

The above shows the first few digits (actually 65) of the decimal representation of the fraction \( \large \frac1{9,999,899,999}. \) If we split the digits into partitions of 5, we can see that the numbers form a Fibonacci sequence: \(0,1,1,2,3,5,8,13,\ldots \). How many positive Fibonacci numbers can we find before the pattern breaks off?

Note: For example, suppose that the fraction equals \[0.00000 \quad 00001 \quad 00001 \quad 00002 \quad 00003 \quad 00009 \ldots \] instead of the one given at the top. Then you could only find the first five Fibonacci numbers, namely \(0,1,1,2,3\). So your answer would then be that there are 4 positive Fibonacci numbers before the pattern breaks off.

• Bonus : Generalize this.
• Try Daniel Liu's problem that was inspired by this problem.

Identity (1) above can be used to show that if \( a|b \), then \( F_a|F_b\). In fact, more is true:

\[ \text{gcd}(F_a,F_b) = F_{\text{gcd}(a,b)}. \]

This follows from (1) and a process similar to the Euclidean algorithm ; write \( a = bq+r, \, 0 \le r < b, \) and then

\[ \begin{align} \text{gcd}(F_a,F_b) &= \text{gcd}(F_{bq+r},F_b) \\ &= \text{gcd}(F_{bq}F_{r+1}+F_{bq-1}F_r,F_b) \\ &= \text{gcd}(F_{bq-1}F_r,F_b) &&\qquad (\text{because } F_b|F_{bq}) \\ &= \text{gcd}(F_r,F_b), &&\qquad \big(\text{because gcd} (F_{bq-1},F_{bq})=1\big) \end{align} \]

We have \( \text{gcd}(F_{10},F_{15}) = \text{gcd}(55,610)=5=F_5. \)

Note that this discussion implies that if \( F_p \) is prime, then \( p \) is prime or \( p =4 \). The converse is not true \((F_2 = 1, F_{19} = 37 \cdot 113), \) and in fact it is not known whether there are infinitely many primes \( p \) such that \( F_p \) is prime.

In the Fibonacci sequence, \(F_{0}=1\), \({F_1}=1\), and for all \(N>1\), \(F_N=F_{N-1}+F_{N-2}\).

How many of the first 2014 Fibonacci terms end in 0?

A Zeckendorf representation of a positive integer is an expression of the integer as a sum of (at least one) distinct non-consecutive Fibonacci numbers. For instance, \( 41 = 34+5+2 \) is a Zeckendorf representation of \( 41 \).

Every positive integer has exactly one Zeckendorf representation.

To show first that there cannot be more than one representation, use the identities in item (3) above to see that the sum of any non-consecutive Fibonacci numbers of which the largest is \( F_n \) cannot be larger than \( F_{n+1} \) \((\)note that \( F_1 \) and \( F_3 \) are consecutive Fibonacci numbers since \( F_1 = F_2). \) Then suppose that there are two Zeckendorf representations of an integer, and subtract out all the common Fibonacci numbers in the two sums. Then the resulting two sums are still equal, and consist of two disjointed sets of Fibonacci numbers. Suppose the largest Fibonacci number in the first sum is \( F_a \) and the largest Fibonacci number in the second sum is \(F_b\); suppose without loss of generality that \( a < b \). Then the first sum is less than \( F_{a+1} \) but the second sum is clearly \( \ge F_b \), so they cannot be equal. To see that a Zeckendorf representation always exists, proceed by induction. The base case is clear \((1=1,2=2),\) and now suppose the result holds for all \( k < n \). Let \( F_a \) be the largest Fibonacci number less than or equal to \( n \). If \( F_a = n,\) then that is a Zeckendorf representation, so suppose \( F_a < n \). Then \( n-F_a \) has a Zeckendorf representation \( F_{b_1}+F_{b_2} + \cdots + F_{b_k} \) by the inductive hypothesis, so \( n =F_a + F_{b_1}+F_{b_2}+\cdots+F_{b_k} \). The \( b_i \) are non-consecutive, and furthermore all of the \( b_i \) are less than \( a-1, \) because if \( n-F_a \ge F_{a-1}, \) then \( n \ge F_a + F_{a-1} = F_{a+1}, \) which contradicts the minimality of \( a \). So this is a Zeckendorf representation. \(_\square\)

The proof implies that the Zeckendorf representation can be found by always taking the largest Fibonacci number less than \( n,\) subtracting it out, and repeating the process.

This theorem has applications in coding theory .

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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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7.2: The Golden Ratio and Fibonacci Sequence

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In this section, we will discuss a very special number called the Golden Ratio. It is an irrational number, slightly bigger than 1.6, and it has (somewhat surprisingly) had huge significance in the world of science, art and music. It was also discovered that this number has an amazing connection with what is called the Fibonacci Sequence, originally studied in the context of biology centuries ago. This unexpected link among algebra, biology, and the arts suggests the mathematical unity of the world and is sometimes discussed in philosophy as well.

Golden Ratio

With one number \(a\) and another smaller number \(b\), the ratio of the two numbers is found by dividing them. Their ratio is \(a/b\). Another ratio is found by adding the two numbers together \(a+b\) and dividing this by the larger number \(a\). The new ratio is \((a+b)/a\). If these two ratios are equal to the same number, then that number is called the Golden Ratio. The Greek letter \(\varphi\) (phi) is usually used to denote the Golden Ratio.

For example, if \(b = 1\) and \(a / b=\varphi\), then \(a=\varphi\). The second ratio \((a+b)/a\) is then \((\varphi+1) / \varphi\). Because these two ratios are equal, this is true:

\[\varphi=\dfrac{\varphi+1}{\varphi}\nonumber \]

(This equation has two solutions, but only the positive solution is referred to as the Golden Ratio \(\varphi\)).

One way to write this number is

\[\varphi=\dfrac{1+\sqrt{5}}{2} \nonumber \]

\(\sqrt{5}\) is the positive number which, when multiplied by itself, makes \(5: \sqrt{5} \times \sqrt{5}=5\).

The Golden Ratio is an irrational number. If a person tries to write the decimal representation of it, it will never stop and never make a pattern, but it will start this way: 1.6180339887... An interesting thing about this number is that you can subtract 1 from it or divide 1 by it, and the result will be the same.

\[\varphi-1=1.6180339887 \ldots-1=0.6180339887 \nonumber \]

\[1 / \varphi=\frac{1}{1.6180339887}=0.6180339887 \nonumber \]

Golden rectangle

If the length of a rectangle divided by its width is equal to the Golden Ratio, then the rectangle is called a "golden rectangle.” If a square is cut off from one end of a golden rectangle, then the other end is a new golden rectangle. In the picture, the big rectangle (blue and pink together) is a golden rectangle because \(a / b=\varphi\). The blue part (B) is a square. The pink part by itself (A) is another golden rectangle because \(b /(a - b)=\varphi\).

Assume that \(\varphi=\dfrac{a}{b}\) , and \(\varphi\) is the positive solution to \(\varphi^{2}-\varphi-1=0\). Then , \(\dfrac{a^{2}}{b^{2}}-\dfrac{a}{b}-\dfrac{b}{b}=0\). Multiply by \(b^{2}, a^{2}-a b-b^{2}=0\). So, \(a^{2}-a b=b^{2}\). Thus, \(a(a-b)=b^{2}\). We then get \(\dfrac{a}{b}=\dfrac{b}{a-b}\). Both sides are \(\varphi\) .

Fibonacci Sequence

The Fibonacci sequence is a list of numbers. Start with 1, 1, and then you can find the next number in the list by adding the last two numbers together. The resulting (infinite) sequence is called the Fibonacci Sequence. Since we start with 1, 1, the next number is 1+1=2. We now have 1, 1, 2. The next number is 1+2=3. We now have 1, 1, 2, 3. The next number is 2+3=5. The next one is 3+5=8, and so on. Each of these numbers is called a Fibonacci number. Originally, Fibonacci (Leonardo of Pisa, who lived some 800 years ago) came up with this sequence to study rabbit populations! He probably had no idea what would happen when you divide each Fibonacci number by the previous one, as seen below.

Here is a very surprising fact:

The ratio of two consecutive Fibonacci numbers approaches the Golden Ratio.

It turns out that Fibonacci numbers show up quite often in nature. Some examples are the pattern of leaves on a stem, the parts of a pineapple, the flowering of artichoke, the uncurling of a fern and the arrangement of a pine cone. The Fibonacci numbers are also found in the family tree of honeybees.

Meanwhile, many artists and music researchers have studied artistic works in which the Golden Ratio plays an integral role. These include the works of Michelangelo, Da Vinci, and Mozart. Interested readers can find many resources and videos online. Perhaps it is not surprising that numbers like 3, 5, 8, and 13 are rather important in music theory; just take a quick look at the piano keys!

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Fibonacci Sequence | Formula, Spiral, Properties, Examples

Fibonacci sequence is a series of numbers where each number is the sum of the two numbers that come before it. The numbers in the Fibonacci sequence are known as Fibonacci numbers and are usually represented by the symbol Fₙ. Fibonacci sequence numbers begins with the following 14 integers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.

Fibonacci sequence in nature can be seen many places, such as in the growth of trees. As the tree grows, the trunk grows and spirals outward. The branches also follow the Fibonacci sequence, starting with one trunk that splits into two, then one of those branches splits into two, and so on.

Let’s learn about Fibonacci Sequence in detail, including Fibonacci sequence formula, properties, and examples.

Table of Content

Fibonacci Sequence

Fibonacci sequence formula, fibonacci spiral, golden ratio, fibonacci series in pascal’s triangle, fibonacci sequence in real life, important facts about fibonacci numbers, fibonacci sequence properties, fibonacci sequence examples, practice problems on fibonacci sequence.

Fibonacci Sequence is a series of numbers in which each number, starting with 0 and 1, is generated by adding the two preceding numbers . It forms the sequence of 0, 1, 1, 2, 3, 5, 8, 13, 21,… Each number in the Fibonacci series is the sum of the two numbers before it.

Fibonacci sequence is a special sequence of numbers that starts from 0 and 1 and then the next terms are the sum of the previous terms and they go up to infinite terms. This sequence is represented as, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

Fibonacci formula is used to find the nth term of the sequence when its first and second terms are given.

The nth term of the Fibonacci Sequence is represented as F n. It is given by the following recursive formula,

F n = F n-1 + F n-2

• First term is 0 i.e., F 0 = 0
• Second term is 1 i.e., F 1 = 1

Using this formula, we can easily find the various terms of the Fibonacci Sequence. Suppose we have to find the 3rd term of this Sequence then we would require the 2nd and the 1st term according to the given formula, then the 3rd term is calculated as,

F 3 = F 2 + F 1 = 1 + 0 = 1

Thus, the third term in the Fibonacci Sequence is 1, and similarly, the next terms of the sequence can also be found as,

F 4 = F 3 + F 2 = 1 + 1 = 2 F 5 = F 4 + F 3 = 2 + 1 = 3

Check: Nth Fibonacci Number

List of first 20 numbers of Fibonacci sequence are represented in the table below.

• Fibonacci Sequences have infinite terms.
• By closely observing the table we can say that F n = F n-1 + F n-2 for every n > 1.

Recursive Formula

A Fibonacci spiral is a geometric pattern derived from the Fibonacci sequence.

This pattern is created by drawing a series of connected quarter-circles inside a set of squares that have their side according to the Fibonacci sequence. We start the construction of the spiral with a small square, followed by a larger square that is adjacent to the first square. The side of the next square is the sum of the two previous squares, and so on.

Each quarter-circle fits perfectly within the next square in the sequence, creating a spiral pattern that expands outward infinitely.

After studying the Fibonacci spiral we can say that every two consecutive terms of the Fibonacci sequence represent the length and breadth of a rectangle.

Let us now calculate the ratio of every two successive terms of Fibonacci sequence and see the result.

F 2 /F 1 = 1/1 = 1 F 3 /F 2 = 2/1 = 2 F 4 /F 3 = 3/2 = 1.5 F 5 /F 4 = 5/3 = 1.667 F 6 /F 5 = 8/5 = 1.6 F 7 /F 6 = 13/8 = 1.625 F 8 /F 7 = 21/13 = 1.615 F 9 /F 8 = 34/21 = 1.619 F 10 /F 9 = 55/34 = 1.617 F 11 /F 10 = 89/55 = 1.618 (Golden Ratio)

Thus, we see that for the larger term of the Fibonacci sequence, the ratio of two consecutive terms forms the Golden Ratio .

Check: A Fibonacci spiral is a geometric pattern derived from the Fibonacci sequence.

The golden ratio is a ratio between two numbers that is approximately 1.618 . It is represented by the Greek letter phi “Φ” , and is also known as the golden number, golden proportion, or the divine proportion. We have observed that by taking the ratio of two consecutive terms of the Fibonacci Sequence we get the ratio called the “ Golden Ratio “.

Φ = F n /F n-1

Golden Ratio Formula

The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. The formula for the golden ratio is ϕ = 1 + (1/ϕ).

We can calculate the golden ratio of Fibonacci sequence using the formula

F 11 = 89 F 10 = 55

The ratio of these two terms are,

F 11 /F 10 = 89/55 = 1.618 (Golden Ratio)

Here the ratio so obtained is called the golden ratio. {Φ = 1.618 (Golden Ratio)}

We can also calculate the Fibonacci number using the golden ratio by the formula:

F n = (Φ n – (1-Φ) n )/√5

where, Φ is the Golden ratio.

Check: Fibonacci Series

Pascal’s triangle is a triangular array of numbers that begins with 1 at the top and 1s running down the two sides of a triangle. Each new number is the sum of the two numbers above it.

Pascal’s triangle contains the Fibonacci sequence, which is an infinite sequence of numbers that are generated by adding the two previous terms in the sequence. The Fibonacci sequence in Pascal’s triangle is 1, 1, 2, 3, 5, 8, 13, 21, and so on.

Fibonacci Numbers in Pascal’s Triangle

To find the Fibonacci series in Pascal’s triangle, you can draw “shallow diagonals” from the top to the bottom of the triangle . The sum of the diagonals of Pascal’s triangle is equal to the corresponding Fibonacci sequence term.

• The Fibonacci sequence is evident in nature’s patterns, including plant growth, shell spirals, and leaf arrangements.
• It influences architectural design, with structures like the Parthenon and spiral staircases adhering to Fibonacci ratios.
• Artists use Fibonacci proportions to create aesthetically pleasing compositions.
• In financial markets, Fibonacci retracement levels are utilized for technical analysis to identify potential areas of support and resistance.
• Fibonacci sequence is a series of numbers where each number is the sum of the two numbers that come before it. For example, 0 + 1 = 1, 1 + 1 = 2, and 1 + 2 = 3.
• Fibonacci sequence is a never-ending series of numbers.
• Fibonacci sequence is described by the mathematical equation: Fn+2 = Fn+1 + Fn.
• Fibonacci sequence was first described in Indian mathematics around 200 BC.
• The sequence is named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci.
• Fibonacci sequence is used in many applications, including computer algorithms and graphs.
• Golden ratio of 1.618 is derived from the Fibonacci sequence.
• Many things in nature have dimensional properties that adhere to the golden ratio of 1.618.
• November 23 is Fibonacci Day as it forms the first 4 digits of Fibonacci numbers 11/23.

Important properties of Fibonacci Sequence are :

• We can easily calculate the Fibonacci Numbers using the Binet Formula:

where Φ is called Golden Ratio and its value is, Φ ≈ 1.618034.

Using this formula we can easily calculate the nth term of the Fibonacci sequence as, for

F3 4 = (Φ 4 – (1-Φ) 4 )/√5 = ({1.618034} 4 – (1-1.618034) 4 )/√5 = 3

• For larger terms the ratio of two consecutive terms of the Fibonacci Sequence converges to the Golden Ratio.

This can be understood by the table added below,

Thus, it is evident that as the number becomes larger their ratio converges close to the Golden Ratio (1.618034).

• Multiplying a term of Fibonacci Sequence with Golden Ratio gives the next term of the Fibonacci sequence as,

F 7 in Fibonacci Sequence is 13 then F 8 is calculated as,

F 8 = F 7 (1.618034) = 13(1.618034) = 21.0344 = 21 (approx.) Thus, the F 8 in the Fibonacci Sequence is 21.

• We can also calculate the Fibonacci Sequence for below zero numbers as,

F -n = (-1) n+1 F n For example, F -2 = (-1) 2+1 F 2 = -1

• Fibonacci Numbers are used to define other mathematical concepts such as Pascal Triangle and Lucas Number.

Check: Fibonacci Sequence Formula

We have solved some questions on Fibonacci Sequence to help you consolidate your concepts.

Example 1: Find the 7th term of the Fibonacci sequence if the 5th and 6th terms are 3 and 5 respectively.

Using the Fibonacci sequence recursive formula, 7th term = 6th term + 5th term F 6 = 3 + 5 = 8 Thus, the 7th term of the Fibonacci Sequence is F 6 = 8

Example 2: If F 9 in the Fibonacci sequence is 34. Find the next term(F 10 )

We know that, F n = F n-1 × Φ where, Φ is golden ration and its value is 1.618034 F 9 = 34 × Φ = 34 × (1.618034) = 55.0131 = 55 Thus, the F 9 term in the Fibonacci Sequence is 55.

Example 3: Find the 10th term of the Fibonacci sequence if the 8th and 9th terms are 13 and 21 respectively.

Using the Fibonacci sequence recursive formula, 10th term = 9th term + 8th term F 9 = 13 + 21 = 34 Thus, the 10th term of the Fibonacci Sequence is F 9 = 34

Example 4: If F 12 in the Fibonacci sequence is 144. Find the next term(F 13 )

We know that, F n = F n-1 × Φ where, Φ is golden ration and its value is 1.618034 F 13 = 144 × Φ = 144 × (1.618034) = 232.996 = 233 Thus, the F 13 term in the Fibonacci Sequence is 233.

Related Articles:

Arithmetic Progression Geometric Progression Sequences and Series

1. What is the next number in the Fibonacci sequence: 0, 1, 1, 2, 3, 5, …?

2. what is the sum of the first five fibonacci numbers: 0, 1, 1, 2, 3, 3. in the fibonacci sequence, if f(6) represents the 6th term, what is the value of f(6), 4. what is the common ratio between consecutive fibonacci numbers as you move further along the sequence, 5. which fibonacci number is known as the “golden ratio,” often denoted by the greek letter phi (φ).

6. What is the only even number in the first ten Fibonacci numbers?

7. If F(0) = 0 and F(1) = 1, what is the value of F(2)?

8. Which Fibonacci property leads to the appearance of Fibonacci sequence in nature, such as in the arrangement of leaves or seeds?

• Prime property
• Golden ratio property
• Exponential growth property
• Palindrome property

9. If the Fibonacci sequence starts with F(0) = 1 and F(1) = 2, what is the third term, F(2)?

10. What is the relationship between consecutive Fibonacci numbers as you move further along the sequence?

• Subtraction
• Multiplication

Conclusion of Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1 . The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Named after Leonardo of Pisa, commonly known as Fibonacci, the sequence has fascinated mathematicians, scientists, and artists for centuries due to its intriguing properties and widespread occurrence in nature.

Fibonacci Sequence – FAQs

What is fibonacci sequence.

Fibonacci Sequence is the sequence of the number that is generated by adding the last two numbers of the term when the first term and the second term of the sequence are, 0 and 1.

What is Fibonacci Sequence Formula?

Formula for generating the Fibonacci Sequence is F n = F n-1 + F n-2 where n > 1.

What is the sum of Fibonacci Sequence?

In Fibonacci Sequence after the first two terms each new term is the sum of the previous two terms. The following first 14 integers of the Fibonacci Sequence are, 0, 1, 1, 2, 3,5, 8, 13, 21, 34, 55, 89, 144, 233,…

What is Fibonacci Spiral?

A geometric pattern observed in the nature derived from the Fibonacci sequence is called the Fibonacci Spiral. This pattern is observed in the nature in various aspects.

How is Fibonacci Sequence Related to the Golden Ratio?

By closely observing the Fibonacci Sequence we see that the ratio of two consecutive terms of the Fibonacci Terms converges to the Golden Ratio.

What is formula of Fibonacci Sequence for nth term?

Formula to find the nth term of the Fibonacci Sequence is, F n = F n-1 + F n-2 where n >1

Who discovered Fibonacci Sequence?

Fibonacci sequence was first discovered by the famous Italian mathematician “Leonardo Fibonacci” in the early 13th century. But in Indian literature, the Fibonacci sequence was mentioned in early 200 BC literature.

What is the application of Fibonacci Sequence?

Fibonacci sequence is used in fields like art, architecture, and nature due to its occurrence in patterns such as the Golden Ratio. It is also used in finance for predicting market trends and in computer science for algorithm design.

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The Fibonacci Sequence: its Significance and how it is used.

The Fibonacci sequence, named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1.

Page Contents

The sequence goes as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. This sequence has many unique properties and can be used in various fields such as mathematics, nature, and art.

In mathematics, the Fibonacci sequence has many interesting properties. For example, the ratio of any two successive numbers in the sequence approximates the golden ratio, which is approximately 1.6180339887498948482045868343656 .

This ratio is often denoted by the Greek letter phi (φ) and appears in many natural phenomena such as the arrangement of leaves on a stem, the branching of trees, and the spiral patterns in seashells and pinecones.

How is the Fibonacci sequence used? 

The Fibonacci sequence is a series of numbers in which each number is equal to the sum of the two preceding numbers.  The golden ratio of 1.618 is derived from the Fibonacci sequence.

Many items in nature have dimensional features that adhere to the golden ratio of 1.618. Four strategies, including retracements, arcs, fans, and time zones, may be used to apply the Fibonacci sequence to banking.

The Fibonacci sequence is also used in the field of computer science. It is used in algorithms that generate fractals, which are shapes that are self-similar across different scales.

The Fibonacci sequence is also used to model population growth. It can be used to model how a population of rabbits will grow over time if each pair of rabbits produces another pair of rabbits each month, with the first pair of rabbits being born mature.

The Fibonacci Sequence used in Art & Architects 

In art, the Fibonacci sequence can be used to create a sense of balance and harmony in composition.

Artists and architects have used the golden ratio to create pleasing proportions in their work. The most famous example is the Parthenon in Athens, which is said to be based on the golden ratio.

Golden Ratio 

By dividing each number in the Fibonacci sequence by its immediate predecessor, the golden ratio is obtained. Where F(n) is the nth Fibonacci number, the quotient F(n)/ F(n-1) approaches the golden ratio limit of 1.618.

The golden ratio also appears in the arts, and rectangles with golden ratio-based dimensions can be found at the Parthenon in Athens and the Great Pyramid of Giza.

Many natural objects, such as the honeybee, have dimensions that comply to the ratio of 1.618 . In every given hive, dividing the number of female bees by the number of male bees yields a figure close to 1.618.

The Fibonacci spiral In Nature?

The Fibonacci spiral utilizes (phi) or the golden ratio as its foundation, and this spiral can be observed in nature and art. Fibonacci Patterns in Nature Observation is one of the earliest scientific approaches employed by humans to problems they did not fully comprehend.

A number of generations of scientists, philosophers, and mystics have attempted to explain a mystery using symmetry. Why is this pattern so prevalent in nature?

We may understand what symmetry is, but nobody knows why it occurs. It is probable that the Indian mathematicians, along with Fibonacci, simply watched nature and identified an occurrence for which a mathematical explanation could be provided.

Observation is one of the earliest scientific tools used by humans to approach challenges they did not fully comprehend. On pine cones, seashells, sunflowers, and flower petals, as well as in numerous other forms of life, the Fibonacci sequence can be plainly observed.

Nobody knows exactly how or why these patterns emerge. The most plausible explanation would be that a flower’s leaves follow the sun and grow on the portion of the flower that receives the most growth hormone and sunshine.

The Fibonacci Sequence used in Music

The Fibonacci Sequence has a significant role in Western musical harmony and scales. Here are the specifics:

On the piano, an octave consists of thirteen notes. There are eight white keys and five black ones. A scale consists of eight notes, the third and fifth of which form the basis of a basic chord. The dominant note in a scale is the fifth note, which is also the eighth note of the 13 notes that comprise an octave. Eight divided by thirteen is around 0.61538… the Golden Ratio )

In conclusion, the Fibonacci sequence is a fascinating mathematical concept that has many interesting properties and can be used in various fields. It is used in mathematics to approximate the golden ratio, in computer science to generate fractals and model population growth, and in art to create a sense of balance and harmony. It is a simple sequence of numbers that has deep mathematical significance.

Julianne has a bachelor’s in communication and journalism working with Psychic Spirituality & Relationships. She has also practiced numerology, tarot, and other psychic arts.

Fibonacci sequence

What is the fibonacci sequence and how does it impact design.

Designers have found inspiration in the Fibonacci sequence and its related mathematical concept known as the golden ratio for centuries. This ratio is often represented by the Greek letter phi (Φ) and is derived from the Fibonacci sequence. The golden ratio is believed to represent aesthetically pleasing proportions found in nature, art, and design.

Fibonacci Sequence

What is the fibonacci sequence.

The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones, usually starting with 0 and 1. In mathematical terms, the sequence is defined recursively as follows:

Fibonacci sequence formula

F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2) for n > 1.

The Fibonacci sequence begins with 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Each subsequent number is the sum of the two preceding numbers.

The Fibonacci sequence has numerous fascinating properties and appears in various natural phenomena, such as the growth patterns of plants, the arrangement of leaves on stems, and the structure of pinecones. It also has connections to mathematics, art, and computer science.

Who was Fibonacci?

Fibonacci, also known as Leonardo of Pisa or Leonardo Fibonacci, was an Italian mathematician born around 1170 and is best known for introducing the Hindu-Arabic numeral system and popularizing the Fibonacci sequence in the Western world.

Fibonacci’s most famous work is his book “Liber Abaci” (Book of Calculation), published in 1202. In this book, he introduced the decimal number system we use today (0-9) and the concept of place value, including the use of zero. Prior to this, Roman numerals were predominantly used in Europe. Fibonacci’s book played a crucial role in promoting and advancing the adoption of the decimal numeral system in Europe.

In “Liber Abaci,” Fibonacci also presented various mathematical problems and solutions, including the Fibonacci sequence. However, the sequence itself had been previously described in Indian mathematics, particularly by the mathematician Pingala, several centuries before Fibonacci. Fibonacci, however, brought the sequence to the attention of the Western world and demonstrated its various properties and applications.

Fibonacci’s sequence gained significant recognition and interest due to its intriguing mathematical properties and its presence in various natural phenomena. While Fibonacci made important contributions to mathematics, his work extended beyond that and encompassed topics such as arithmetic, algebra, geometry, and commercial mathematics.

Fibonacci’s work laid the foundation for many subsequent developments in mathematics and had a lasting impact on the field. He is considered one of the most influential mathematicians of the Middle Ages and is recognized for his role in popularizing and advancing arithmetic and number theory in Europe.

The Fibonacci sequence itself was not “discovered” in the sense of being an invention or creation by a specific individual. The sequence and its related patterns have been observed and studied for centuries. However, the sequence was popularized and introduced to the Western world by Leonardo Fibonacci.

The sequence had been previously described in Indian mathematics, particularly in connection with the poetic meter by the mathematician Pingala, in texts dating back to around 200 BC. Pingala’s work referred to the sequence as the “mātrāmeru” pattern.

Fibonacci encountered the Fibonacci sequence during his travels to various countries in the Islamic world, where he learned about the decimal system and other mathematical concepts. He recognized the significance of the sequence and included it in his book “Liber Abaci,” along with its various properties and applications.

While Fibonacci did not discover the sequence itself, he played a crucial role in bringing it to the attention of the Western world and showcasing its fascinating mathematical properties. As a result, the sequence became widely known as the Fibonacci sequence, named after Leonardo Fibonacci.

Fibonacci sequence in nature

The Fibonacci sequence and its related concepts, such as the golden ratio, can be observed in numerous natural phenomena. Here are a few examples:

Flower petals: Many flowers exhibit a specific number of petals that follows a Fibonacci sequence or a number that is a Fibonacci number. For example, lilies often have three petals, buttercups have five, some delphiniums have eight, and daisies usually have 34, 55, or 89 petals. Rose petals are arranged in a Fibonacci spiral. This means that petal number one and six will be on the same vertical imaginary line.

Seed arrangements: The arrangement of seeds in various plants often follows Fibonacci patterns. For instance, in sunflowers, the seeds are typically arranged in two sets of spirals—one clockwise and the other counterclockwise—and the number of spirals often follows Fibonacci numbers.

Pinecones: The scales on a pinecone are arranged in a spiral pattern, with each new scale positioned at a particular angle relative to the previous one. These spirals often follow the Fibonacci sequence.

Tree branches: The branching patterns of trees, especially in certain species like the oak tree, can exhibit Fibonacci-like patterns. The main trunk typically splits into two branches, then each branch splits into two more, and so on, following a pattern akin to the Fibonacci sequence.

Shell structures: The shells of many mollusks, such as nautilus shells, often exhibit spiral growth patterns that approximate the golden ratio or follow Fibonacci spirals.

Body proportions: The proportions of various body parts, such as the segments of an insect’s body, lengths of bones in the human hand, or the distribution of feathers on a bird can approximate the golden ratio or follow Fibonacci ratios.

While these patterns can often be found in nature, they do not always strictly adhere to the Fibonacci sequence or the golden ratio. Nature is complex, and other factors can influence the specific arrangements and proportions observed.

why is the Fibonacci sequence important

The Fibonacci sequence is significant for several reasons:

Mathematical Properties: The sequence has interesting mathematical properties and relationships. For instance, the ratio of consecutive Fibonacci numbers tends to approximate the golden ratio (approximately 1.618), which is a fundamental ratio found in nature, art, and design. The Fibonacci sequence is also closely connected to the Lucas sequence and has numerous other intriguing mathematical properties.

Nature and Biology: The Fibonacci sequence appears in various natural phenomena. It can be observed in the growth patterns of plants, such as the spirals in sunflowers, pinecones, and the arrangement of leaves on stems. This pattern allows plants to efficiently utilize space and resources for optimal growth.

Art and Design: Artists, architects, and designers have drawn inspiration from the Fibonacci sequence and the golden ratio. It is believed that using these proportions in artwork and design creates aesthetically pleasing and visually harmonious compositions.

Algorithms and Problem-Solving: The Fibonacci sequence is often used as an example in algorithmic and problem-solving exercises. It provides a straightforward way to introduce concepts like recursion and dynamic programming. Understanding the Fibonacci sequence helps develop problem-solving skills and lays the foundation for more advanced mathematical concepts.

Number Theory and Combinatorics: The Fibonacci sequence has connections to various areas of number theory and combinatorics. It can be explored in depth to understand the properties of numbers, divisibility, modular arithmetic, and more.

Overall, the Fibonacci sequence is important because it not only showcases fascinating mathematical properties but also offers insights into the patterns and structures found in the natural world and serves as a foundation for various mathematical and scientific disciplines.

Ways the Fibonacci sequence has influenced design:

Typography: The golden ratio is often used to determine ideal proportions for typography, such as the width of text columns, line spacing, and font sizes. Applying these ratios can improve readability and create a harmonious typographic system.

Product design: Fibonacci-based proportions and the golden ratio are utilized in product design to create aesthetically pleasing and ergonomic forms. From architecture to industrial design, designers apply these principles to achieve visually balanced and functional products.

Nature-inspired design: The Fibonacci sequence is often associated with natural forms and patterns. Designers draw inspiration from these patterns to create organic and biomimetic designs that mimic the aesthetic and efficiency found in nature.

Fibonacci sequence influence on visual hierarchy or visual weight

The Fibonacci sequence can guide designers in establishing visual hierarchies within a design. By applying the sequence, designers can determine the relative size and positioning of elements to draw attention and guide the viewer’s eye through the composition.

The Fibonacci sequence and its related concept, the golden ratio, have been utilized in visual design to create a sense of balance, harmony, and aesthetic appeal. They can influence visual hierarchy in the following ways:

Proportional Relationships: The Fibonacci sequence and golden ratio provide a set of proportions that are visually pleasing and are believed to create a sense of balance. By applying these proportions to elements in a design, such as the size of objects or the placement of visual elements, a sense of hierarchy can be established. Larger elements can be placed in relation to smaller elements using Fibonacci ratios to create a pleasing and visually appealing composition.

Grid Systems: Grid systems based on the Fibonacci sequence or golden ratio can be used to establish a hierarchy in layout design. These grids create a structure that helps organize visual elements and guides the viewer’s attention. Elements can be placed along the grid lines or at key intersections to create a hierarchy based on their size, importance, or relevance.

Composition and Placement: The Fibonacci spiral, derived from the Fibonacci sequence, is often used as a guide for placing visual elements in a composition. The spiral starts from a central point and expands outward in a logarithmic spiral, with each quarter-circle arc corresponding to a Fibonacci number. Placing important elements along this spiral can create a natural flow and guide the viewer’s gaze through the composition, establishing a visual hierarchy.

Scaling and Resizing: Applying Fibonacci ratios when scaling or resizing elements can help maintain visual harmony. For example, reducing the size of an element by a Fibonacci ratio can create a visually pleasing progression of sizes and help establish a hierarchy within a set of related elements.

While the Fibonacci sequence and the golden ratio have been influential in design, they are not universally applicable or considered a rigid rule. Designers use these principles as guidelines to achieve aesthetically pleasing results, but creativity and individual judgment still play a significant role in the design process.

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mathsisfun.com // wikipedia.org // techtarget.com // livescience.com // mathworld.wolfram.com // youtube.com // britannica.com // clevelanddesign.com // howstuffworks.com // thoughtco.com //

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What Is the Fibonacci Sequence?

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The Bottom Line

• Technical Analysis
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Fibonacci Sequence: Definition, How it Works, and How to Use It

Investopedia / Yurle Villegas

The Fibonacci sequence was developed by the Italian mathematician, Leonardo Fibonacci, in the 13th century. The sequence of numbers, starting with zero and one, is a steadily increasing series where each number is equal to the sum of the preceding two numbers.

Some traders believe that the Fibonacci numbers and ratios created by the sequence play an important role in finance that traders can apply using technical analysis .

Key Takeaways

• The Fibonacci sequence is a set of steadily increasing numbers where each number is equal to the sum of the preceding two numbers.
• The golden ratio of 1.618 is derived from the Fibonacci sequence.
• Many things in nature have dimensional properties that adhere to the golden ratio of 1.618.

The Fibonacci sequence can be applied to finance by using four techniques including retracements, arcs, fans, and time zones.

Understanding the Fibonacci Sequence

The numbers in the Fibonacci Sequence don't equate to a specific formula, however, the numbers tend to have certain relationships with each other. Each number is equal to the sum of the preceding two numbers. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377.

Fibonacci Sequence Rule

xn = xn−1 + xn−2

xn is term number "n"

xn−1 is the previous term (n−1)

xn−2 is the term before that (n−2)

The  golden ratio  of 1.618, important to mathematicians, scientists, and naturalists for centuries is derived from the Fibonacci sequence. The quotient between each successive pair of Fibonacci numbers in the sequence approximates 1.618, or its inverse 0.618.

Golden Ratio

The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. Where F(n) is the nth Fibonacci number, the quotient F(n)/ F(n-1) will approach the limit 1.618, known as the golden ratio.

Many things in nature have dimensional properties that adhere to the ratio of 1.618, like the honeybee. If you divide the female bees by the male bees in any given hive, you will get a number near 1.618. The golden ratio also appears in the arts and rectangles whose dimensions are based on the golden ratio appear at the Parthenon in Athens and the Great Pyramid in Giza.

How to Use the Fibonacci Sequence

Fibonacci retracements require two price points chosen on a chart, usually a swing high and a swing low . Once two points are chosen, the Fibonacci numbers and lines are drawn at percentages of that move. If a stock rises from $15 to $20, then the 23.6% level is $18.82, or $20 - ($5 x 0.236) = $18.82. The 50% level is $17.50, or $15 - ($5 x 0.5) = $17.50.

Image by Sabrina Jiang © Investopedia 2021

Fibonacci retracements are the most common form of technical analysis based on the Fibonacci sequence. During a trend, Fibonacci retracements can be used to determine how deep a pullback may be. Traders tend to watch the Fibonacci ratios between 23.6% and 78.6% during these times. If the price stalls near one of the Fibonacci levels and then start to move back in the trending direction, an investor may trade in the trending direction.

Arcs, fans, and time zones are similar concepts but are applied to charts in different ways. Each one shows potential areas of support or resistance, based on Fibonacci numbers applied to prior price moves. These supportive or resistance levels can be used to forecast where prices may fall or rise in the future.

What Is the Fibonacci Spiral?

The limits of the squares of successive Fibonacci numbers create a spiral known as the Fibonacci spiral. It follows turns by a constant angle close to the golden ratio and is commonly called the golden spiral. The numbers of spirals in pinecones are Fibonacci numbers, as is the number of petals in each layer of certain flowers. In spiral-shaped plants, each leaf grows at an angle compared to its predecessor, and sunflower seeds are packed in a spiral formation in the center of their flower in a geometry governed by the golden ratio.

Where Is the Fibonacci Sequence Evident?

In almost all flowering plants, the number of petals on the flower is a Fibonacci number. It is extremely rare for the number of petals not to be so and examples of this phenomenon include corn marigold, cineraria, and daisies with 13 petals and asters and chicory with 21 petals.

How Can the Fibonacci Sequence Affect Trading Behavior?

Humans tend to identify patterns and traders easily equate patterns in charts through the Fibonacci sequence. It's unproven that Fibonacci numbers relate to fundamental market forces, however, markets by design react to the beliefs of their players. Consequently, if investors buy or sell because of Fibonacci analysis, they tend to create a self-fulfilling prophecy that affects the market trends.

The Fibonacci sequence is a set of steadily increasing numbers where each number is equal to the sum of the preceding two numbers. Many things in nature have dimensional properties that adhere to the golden ratio of 1.618, a quotient derived from the Fibonacci sequence. When applied to finance and trading, investors apply the Fibonacci sequence through four techniques including retracements, arcs, fans, and time zones.

Wild Maths. " Fibonacci and Bees ."

Smithsonian Magazine. " The Fibonacci Sequence Is Everywhere - Even the Troubled Stock Market ."

Science Struck. " 13 Real-Life Examples of the Golden Ratio ."

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