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.
Table of Contents:- Definition
- Formula
- Sequence List
- Golden Ratio to Calculate Fibonacci Sequence
- Solved Examples
- Practice Problems
- FAQs
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)
Similarly,
“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 “Fn” is defined using the recursive relation with the seed values F0=0 and F1=1:
Fn = Fn-1+Fn-2
Here, the sequence is defined using two different parts, such as kick-off and recursive relation.
The kick-off part is F0=0 and F1=1.
The recursive relation part is Fn = Fn-1+Fn-2.
It is noted that the sequence starts with 0 rather than 1. So, F5 should be the 6th term of the sequence.
Fibonacci Sequence List
The list of first 20 terms in the Fibonacci Sequence is:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181.
The list of Fibonacci numbers are calculated as follows:
| Fn | Fibonacci Number |
| 0 | 0 |
| 1 | 1 |
| 2 | 1 |
| 3 | 2 |
| 4 | 3 |
| 5 | 5 |
| 6 | 8 |
| 7 | 13 |
| 8 | 21 |
| 9 | 34 |
| … and so on. | … and so on. |
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:
Xn = [φn – (1-φ)n]/√5
Where,
φ is the Golden Ratio, which is approximately equal to the value of 1.618
n is the nth term of the Fibonacci sequence.
Related Articles
- Sequence And Series
- Arithmetic Progression
- Geometric Progression
- Harmonic Progression
Fibonacci Sequence Solved Examples
Example 1:
Find the Fibonacci number when n=5, using recursive relation.
Solution:
The formula to calculate the Fibonacci Sequence is: Fn = Fn-1+Fn-2
Take: F0=0 and F1=1
Using the formula, we get
F2 = F1+F0 = 1+0 = 1
F3 = F2+F1 = 1+1 = 2
F4 = F3+F2 = 2+1 = 3
F5 = F4+F3 = 3+2 = 5
Therefore, the fibonacci number is 5.
Example 2:
Find the Fibonacci number using the Golden ratio when n=6.
Solution:
The formula to calculate the Fibonacci number using the Golden ratio is Xn = [φn – (1-φ)n]/√5
We know that φ is approximately equal to 1.618.
n= 6
Now, substitute the values in the formula, we get
Xn = [φn – (1-φ)n]/√5
X6 = [1.6186 – (1-1.618)6]/√5
X6 = [17.942 – (0.618)6]/2.236
X6 = [17.942 – 0.056]/2.236
X6 = 17.886/2.236
X6 = 7.999
X6 = 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
Q1
What is Fibonacci Sequence?
The Fibonacci sequence is the sequence of numbers, in which every term in the sequence is the sum of terms before it.
Q2
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.
Q3
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
Q4
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.
Q5
What is the value of the Golden ratio?
The value of golden ratio is approximately equal to 1.618034…
FAQs
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.
What is the formula and example of Fibonacci sequence? ›
Fibonacci sequence, the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …, each of which, after the second, is the sum of the two previous numbers; that is, the nth Fibonacci number Fn = Fn − 1 + Fn − 2.
What are the 100 Fibonacci numbers? ›
First 100 terms of Fibonacci series are :- 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 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 ...
What is the formula for the sum of the Fibonacci sequence? ›
The formula is: F(1) + F(2) + F(3) +…. + F(n) = F(n+2) - 1.
What is the Fibonacci sequence defined by? ›
The Fibonacci sequence is defined by 1=a1=a2 and an= an 1+an 2, n > 2.
What is the sequence formula? ›
The formula for the nth term in an arithmetic sequence is an=a1+(n−1)d. This formula can be used to determine the value of any term in an arithmetic sequence. An arithmetic sequence has a common difference between every term.
What is the full Fibonacci sequence? ›
Fibonacci sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946,….
What are the numbers of God Fibonacci? ›
In this sequence the numbers are: 1,1,2,3,5,8,13,21,34,55,89,144,233, ad infinitum. Each succeeding number is the sum of the two preceding numbers. Leonardo Fibonacci, alias Leonardo of Pisa, who was born around A.D. 1180, discovered this sequence. It is described in his famous book, Liber Abaci.
How to calculate Fibonacci sequence? ›
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.
What is the golden rule of the Fibonacci numbers? ›
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.
This Fibonacci calculator is a tool for calculating the arbitrary terms of the Fibonacci sequence. Never again will you have to add the terms manually – our calculator finds the first 250 terms for you! You can also set your own starting values of the sequence and let this calculator do all the work for you.
What is the golden ratio to calculate Fibonacci sequence? ›
Golden ratio is represented using the symbol “ϕ”. Golden ratio formula is ϕ = 1 + (1/ϕ). ϕ is also equal to 2 × sin (54°) If we take any two successive Fibonacci Numbers, their ratio is very close to the value 1.618 (Golden ratio).
How is the Fibonacci series used in real life? ›
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.
What is the Fibonacci sequence answer? ›
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.
What are examples of 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.
What is an example of the Fibonacci method? ›
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.
What is the formula for the golden ratio of the Fibonacci numbers? ›
Golden ratio is represented using the symbol “ϕ”. Golden ratio formula is ϕ = 1 + (1/ϕ). ϕ is also equal to 2 × sin (54°) If we take any two successive Fibonacci Numbers, their ratio is very close to the value 1.618 (Golden ratio).
What are 3 examples of where Fibonacci numbers are used or seen? ›
On many plants, the number of petals is a Fibonacci number: buttercups have 5 petals; lilies and iris have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 whereas daisies can be found with 34, 55 or even 89 petals.
What is the closed formula for the Fibonacci sequence? ›
The closed-form expression of the nth n t h Fibonacci number is thus given by: Fn=1√5[(1+√52)n−(1−√52)n]. F n = 1 5 [ ( 1 + 5 2 ) n − ( 1 − 5 2 ) n ] .
