Fibonacci Sequence
Each number in the Fibonacci sequence is the sum of the two numbers that precede it. Typically, it begins at 0 and 1. The Fibonacci sequence consists of the numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. Fibonacci numbers are also known as the numbers in the Fibonacci sequence. In mathematics, a sequence is an ordered list of integers that follows a particular pattern. The numbers in the sequence are referred to as words. Arithmetic sequences, geometric sequences, harmonic sequences, and Fibonacci sequences are the various types of sequences. In this article, we will examine the definition, formula, list, and instances of the Fibonacci sequence in depth.
What is Fibonacci Sequence?
The Fibonacci sequence consists of the infinite terms 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and… In simple words, the Fibonacci sequence is a sequence in which each number is the sum of the two numbers preceding it. Its initial two terms are zero and one. These numbers are known as the Fibonacci sequence.
The phrase “1” is created by adding the terms “1” and “2” together. (i.e., 0+1 = 1)
In a similar manner, “2” is obtained by adding the second and third terms (1+1 = 2).
By adding the third and fourth terms (1+2), “3” is obtained.
For example, the term after 21 can be calculated by adding 13 and 21. Therefore, the following term in the sequence is 34. Following are the first 20 terms of the
Fibonacci Sequence List :
| F0 = 0 | F10 = 55 |
| F1 = 1 | F11 = 89 |
| F2 = 1 | F12 = 144 |
| F3 = 2 | F13 = 233 |
| F4 = 3 | F14 = 377 |
| F5 = 5 | F15 = 610 |
| F6 = 8 | F16 = 987 |
| F7 = 13 | F17 = 1597 |
| F8 = 21 | F18 = 2584 |
| F9 = 34 | F19 = 4181 |
Here, we can observe that \( F_n=F_{n-1}+F_{n-2}\) for every n > 1. For example:
F2 = F1 + F0
F3 = F2 + F1
F4 = F3 + F2, and so on.
Fibonacci Spiral
The Fibonacci sequence is illustrated below as a spiral. The spiral depicts the Fibonacci number sequence. This spiral commences with a rectangle whose length and width constitute the golden ratio (1.618). This rectangular area is divided into two squares. The squares are then subdivided further. A spiral is drawn within these squares by connecting the corners of the boxes. With increasing Fibonacci sequence numbers, the ratio approaches the golden ratio.
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.
F2/F1 = 1/1 = 1
F3/F2 = 2/1 = 2
F4/F3 = 3/2 = 1.5
F5/F4 = 5/3 = 1.667
F6/F5 = 8/5 = 1.6
F7/F6 = 13/8 = 1.625
F8/F7 = 21/13 = 1.615
F9/F8 = 34/21 = 1.619
F10/F9 = 55/34 = 1.617
F11/F10 = 89/55 = 1.618 = Golden Ratio
In this way, when the rectangle is very large, its dimensions are very close to forming a golden rectangle.
Fibonacci Sequence Formula
The Fibonacci sequence formula for “Fn” is defined using the recursive formula by setting F0 = 0, F1 = 1, and using the formula below to find Fn. The Fibonacci formula is given as follows.
\( F_n=F_{n-1}+F_{n-2}\), where n > 1
Fibonacci Sequence Properties
The interesting properties of the Fibonacci sequence are as follows:
1) Fibonacci numbers are related to the golden ratio. Any Fibonacci number can be calculated using the golden ratio, Fn =(Φn – (1-Φ)n)/√5, Here φ is the golden ratio and Φ ≈ 1.618034.
To find the 7th term, we apply F7 = [(1.618034)7 – (1-1.618034)7] / √5 = 13
2) The ratio of successive Fibonacci numbers is called the “golden ratio”. Let A and B be the two consecutive numbers in the Fibonacci sequence. Then B/A converges to the Golden ratio. to find any term in the Fibonacci sequence, we could apply the above-said formula.
| A | B | A/B |
| 2 | 3 | 1.5 |
| 3 | 5 | 1.6 |
| 5 | 8 | 1.6 |
| 8 | 13 | 1.625 |
| 144 | 233 | 1.618055555555556 |
| 233 | 377 | 1.618025751072961 |
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 Fn. For example, F-4 = (-1)5 . F4 = (-1) 3 = -3.
4) The sum of n terms of the Fibonacci Sequence is given by\( \sum_{i=0}^{n}F_i=F_{n+2}-F_2 or F_{n+2}-1\) , where Fn is the nth Fibonacci number. (Note: the first term starts from F0)
For example, the sum of the first 10 terms of sequence = 12th term – 1 = 89 – 1 = 88. It can be mathematically written as Σi=09 Fi = F11 – 1 = 89 – 1 = 88.
Applications of the Fibonacci Sequence
The Fibonacci sequence can be observed in numerous domains, including nature, music, and the human body.
- Utilized in grouping numbers and the dazzling proportion in music in general.
- Applied to Coding (computer algorithms, interconnecting parallel, and distributed systems)
- Numerous scientific domains, such as high-energy physics, quantum mechanics, cryptography, etc.
You can figure out the Fibonacci Sequence with the help of a Fibonacci calculator. Look at some examples of how the Fibonacci formula has been used to help you understand it better.
Fibonacci Sequence Solved Examples
Example 1: Find the 12th term of the Fibonacci sequence if the 10th and 11th terms are 34 and 55 respectively.
Solution:
Using the Fibonacci Sequence recursive formula, we can say that the 12th term is the sum of the 10th term and 11th term.
12th term = 10th term + 11th term
= 34 + 55
= 89
Answer: The 12th term is 89.
Example 2: 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 3: Calculate the value of the 12th and the 13th term of the Fibonacci sequence given that the 9th and 10th terms in the sequence are 21 and 34.
Solution :
Using the formula, we can say that the 11th term is the sum of 9th term and 10th term.
11th term = 9th term + 10th term = 21 + 34 = 55
Now, 12th term = 10th term + 11th term = 34 + 55 = 89
Similarly,13th term = 11th term + 12th term = 55 + 89 = 144
Answer: The 12th and the 13th term are 89 and 144.
Frequently Asked Questions on Fibonacci Sequence
Q.1 What is Fibonacci sequence explain?
The Fibonacci sequence is a set of whole numbers called the Fibonacci numbers. It starts with a zero, then a one, then another one, and then a series of numbers that keep getting bigger. The order of the numbers is based on the rule that each number is equal to the sum of the two numbers before it.
Q.2 What are the first 10 terms of the Fibonacci sequence?
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.
Q.3 What is the formula for Fibonacci sequence?
The Fibonacci formula is given as, Fn = Fn-1 + Fn-2, where n > 1.
Q.4 What Are the Applications of Fibonacci Sequence Formula?
The Fibonacci sequence has many uses, such as:
how numbers are put together and how music has great proportions.
the computer algorithms, linking parallel and distributed systems, or coding in general.
high-energy physics, quantum mechanics, cryptography, and other areas of science.
the use of Fibonacci retracements and ratios to set marketing and trade trends.
