In Maths, the Fibonacci numbers are the numbers ordered in a distinct Fibonacci sequence. These numbers were introduced to represent the positive numbers in a sequence, which follows a defined pattern. The list of the numbers in theFibonacci series is represented by the recurrence relation: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ……..,∞. In this article, let us discuss what is a Fibonacci number, properties, Fibonacci formulas and series in detail.
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What is Fibonacci Number?
A Fibonacci number is a series of numbers in which each Fibonacci number is obtained by adding the two preceding numbers. It means that the next number in the series is the addition of two previous numbers. Let the first two numbers in the series be taken as 0 and 1. By adding 0 and 1, we get the third number as 1. Then by adding the second and the third number (i.e) 1 and 1, we get the fourth number as 2, and similarly, the process goes on. Thus, we get the Fibonacci series as 0, 1, 1, 2, 3, 5, 8, ……. Hence, the obtained series is called the Fibonacci number series.
We can also obtain the Fibonacci numbers from the pascal’s triangle as shown in the below figure.
Here, the sum of diagonal elements represents the Fibonacci sequence, denoted by colour lines.
Fibonacci Series List
The list of numbers of Fibonacci Sequence is given below. This list is formed by using the formula, which is mentioned in the above definition.
Fibonacci Numbers Formula
The sequence of Fibonacci numbers can be defined as:
Fn = Fn-1 + Fn-2
Where Fn is the nth term or number
Fn-1 is the (n-1)th term
Fn-2 is the (n-2)th term
From the equation, we can summarize the definition as, the next number in the sequence, is the sum of the previous two numbers present in the sequence, starting from 0 and 1. Let us create a table to find the next term of the Fibonacci sequence, using the formula.
Fn-1
Fn-2
Fn
0
1
1
1
1
2
1
2
3
2
3
5
3
5
8
5
8
13
8
13
21
13
21
34
21
34
55
34
55
89
In the above table, you can see the numbers in each column are relational and diagonally the numbers are the same in all the three columns.
The following are the properties of the Fibonacci numbers.
In the Fibonacci series, take any three consecutive numbers and add those numbers. When you divide the result by 2, you will get the three numbers. For example, take 3 consecutive numbers such as 1, 2, 3. when you add these numbers, i.e. 1+ 2+ 3 = 6. When 6 is divided by 2, the result is 3, which is 3.
Take four consecutive numbers other than “0” in the Fibonacci series. Multiply the outer number and also multiply the inner number. When you subtract these numbers, you will get the difference “1”. For example, take 4 consecutive numbers such as 2, 3, 5, 8. Multiply the outer numbers, i.e. 2(8) and multiply the inner number, i.e. 3(5). Now subtract these two numbers, i.e. 16-15 =1. Thus, the difference is 1.
Write the first 6 Fibonacci numbers starting from 0 and 1.
Solution:
As we know, the formula for Fibonacci sequence is;
Fn = Fn-1 + Fn-2
Since the first term and second term are known to us, i.e. 0 and 1. Thus,
F0 = 0 and F1 = 1
Hence,
Third term, F2 = F0 + F1 = 0+1 = 1
Fourth term, F3 = F2+F1 = 1 + 1 = 2
Fifth term, F4 = F3+F2 = 1+2 = 3
Sixth term, F5 = F4+F3 = 3 + 2 = 5
So, the first six terms of Fibonacci sequence is 0,1,1,2,3,5.
Question 2:
Find the next term of Fibonacci series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
Solution:
Each next term of the Fibonacci series is the sum of the previous two terms.
Therefore, the required term is;
21 + 34 = 55
Watch the Below Video to understand the Fibonacci Numbers and Golden Ratio
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Frequently Asked Questions – FAQs
Q1
What are the first 10 Fibonacci numbers?
The first 10 Fibonacci numbers are given by: 1, 1, 2, 3, 5, 8, 13, 21, 34, and 55
Q2
What are the numbers in the Fibonacci sequence?
The Fibonacci sequence contains the numbers as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …,∞
Q3
What are Fibonacci numbers used for?
Fibonacci numbers play an essential role in financial analysis. Here, the Fibonacci number sequence can be used to generate ratios or percentages that are useful for business people.
Q4
What are the first 12 Fibonacci numbers?
The first 12 Fibonacci numbers are given by: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and 144
Q5
What is the 100th Fibonacci number?
The 100th Fibonacci number will be: 354,224,848,179,261,915,075 We can find this number using the Fibonacci sequence formula.
1170 – c. 1240–50), also known as Leonardo Bonacci, Leonardo of Pisa, or Leonardo Bigollo Pisano ('Leonardo the Traveller from Pisa'), was an Italian mathematician from the Republic of Pisa, considered to be "the most talented Western mathematician of the Middle Ages".
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, ….
The golden ratio, also known as the golden number, golden proportion, or the divine proportion, is a ratio between two numbers that equals approximately 1.618. Usually written as the Greek letter phi, it is strongly associated with the Fibonacci sequence, a series of numbers wherein each number is added to the last.
The number of petals in a flower consistently follows the Fibonacci sequence. Famous examples include the lily, which has three petals, buttercups, which have five (pictured at left), the chicory's 21, the daisy's 34, and so on.
The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. In mathematical terms, if F(n) describes the nth Fibonacci number, the quotient F(n)/ F(n-1) will approach the limit 1.618... for increasingly high values of n. This limit is better known as the golden ratio.
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, Fn = Fn-1 + Fn-2, where n > 1. It is used to generate a term of the sequence by adding its previous two terms.
The Fibonacci sequence, also known as the golden ratio, is utilized in architectural designs, creating aesthetically pleasing structures. In engineering and technology, Fibonacci numbers play a significant role, appearing in population growth models, software engineering, task management, and data structure analysis.
The Fibonacci sequence is important for many reasons. In nature, the numbers and ratios in the sequence can be found in the patterns of petals of flowers, the whorls of a pine cone, and the leaves on stems. As the sequence continues, the ratios of the terms approach a number known as the golden ratio.
Without getting too complicated, the golden ratio is 1.618 to 1. The golden spiral uses this ratio to create a series of squares. The size and placement of the squares are based on the Fibonacci sequence.
The further you go along the Fibonacci Sequence, the closer the answers get to Phi. But the answer will never equal Phi exactly. That's because Phi cannot be written as a fraction. It's irrational!
The spiritual meaning of the Fibonacci spiral is often associated with balance, harmony, and perfection. Some believe that this pattern represents the infinite and interconnected nature of all things. It symbolizes the natural order and balance found in the universe and signifies the beauty and efficiency of creation.
Animals. The Fibonacci sequence is common in the animal kingdom. The starfish has two manifestations of Fibonacci: It has five arms (a Fibonacci number), as well as a pentagon shape that reflects the golden ratio. Other examples are the horns of a ram, the tail of a seahorse, and the shells of snails and the nautilus.
The bones of your finger (including the bone from your knuckle to your wrist) follow the Fibonacci sequence. We have 8 fingers in total, 5 digits on each hand, 3 bones in each finger, 2 bones in 1 thumb, and 1 thumb on each hand. Many flowers also exhibit the Fibonacci sequence.
The golden ratio is 1.618, represented by the Greek letter 'phi', is said to be is a mathematical connection between two aspects of an object. It is also called the Fibonacci sequence and it can be found across all of nature: plants, animals, weather structures, star systems – it is ever-present in the universe.
In the Fibonacci sequence of numbers, each number in the sequence is the sum of the two numbers before it, with 0 and 1 as the first two numbers. The Fibonacci series of numbers begins as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on.
These levels are determined by calculating the vertical distances between the high and low points of an asset's price, and then dividing these distances by key Fibonacci ratios (23.6%, 38.2%, 50%, 61.8%, and 100%).
The notation that we will use to represent the Fibonacci sequence is as follows: f1=1,f2=1,f3=2,f4=3,f5=5,f6=8,f7=13,f8=21,f9=34,f10=55,f11=89,f12=144,…
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