Fibonacci Sequence
The Fibonacci sequence was first found by an Italian named Leonardo Pisano Bogollo (Fibonacci). Fibonacci numbers are a sequence of whole numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... This infinite sequence is called the Fibonacci sequence. Here each term is the sum of the two preceding ones, starting from 0 and 1. This 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 about the Fibonacci sequence and its interesting properties.
1.  What Is Fibonacci Sequence? 
2.  Fibonacci Sequence Formula 
3.  Fibonacci Sequence Properties 
4.  Applications of Fibonacci Sequence 
5.  FAQs on Fibonacci Sequence 
What is Fibonacci Sequence?
The Fibonacci sequence, in simple terms, says that every number in the Fibonacci sequence is the sum of two numbers preceding it in the sequence. The first 20 Fibonacci numbers are given as follows:
\(F_0\) = 0  \(F_{10}\) = 55 
\(F_1\) = 1  \(F_{11}\) = 89 
\(F_2\) = 1  \(F_{12}\) = 144 
\(F_3\) = 2  \(F_{13}\) = 233 
\(F_4\) = 3  \(F_{14}\) = 377 
\(F_5\) = 5  \(F_{15}\) = 610 
\(F_6\) = 8  \(F_{16}\) = 987 
\(F_7\) = 13  \(F_{17}\) = 1597 
\(F_8\) = 21  \(F_{18}\) = 2584 
\(F_9\) = 34  \(F_{19}\) = 4181 
The Fibonacci sequence is represented as the spiral shown below. The spiral represents the pattern of the Fibonacci numbers. This spiral starts with a rectangle whose length and width form the golden ratio(≈1.618). This rectangle is partitioned into two squares. Then the squares are further partitioned. Connecting the corners of the boxes, the spiral is drawn inside these squares. The larger the numbers in the Fibonacci sequence, the ratio becomes closer to the golden ratio.
The puzzle of rabbits explains the wonder behind this Fibonacci sequence.
 Two newborn rabbits are left in the field. They are still one pair at the end of the first month.
 They mate and produce a new pair, so there are 2 pairs in the field, at the end of the second month.
 The first pair produces the second pair, but the second pair is left without breeding, so 3 pairs in all at the end of the third month.
 The original pair produces another pair, the second pair produces their first pair and the third pair remains without breeding, making 5 pairs.
 The sequence continues in this pattern and at the end of the n^{th }month, the number of rabbits in the field is equal to the sum of the number of mature pairs (n2)^{th} month and the number of pairs alive last month(n1)^{th} month. This happens to be the n^{th} Fibonacci number.
Fibonacci Sequence Formula
The Fibonacci sequence formula for “\(F_n\)” is defined using the recursive formula by setting \(F_0\) = 0,\(F_1\)= 1 to give \(F_n\).
The Fibonacci formula is given as follows.
\(F_n= F_{n1} + F_{n2}\), 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, \(F_n =\dfrac{\phi^n  (1\phi)^n}{\sqrt{5}}\), Here φ is the golden ratio.
To find the 7^{th }term, we apply \(F_7 =\dfrac{(1.618034)^7  (11.618034)^7}{\sqrt{5}}\) = 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 abovesaid 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 × 1.618034... = 21.034442. This gives the next Fibonacci number 21 after 13.
2) Every n^{th} number is a multiple of \(F_n\). Observe the sequence to find another interesting pattern. Every 3rd number in the sequence is a multiple of 2. Every 4th number in the sequence is a multiple of 3 and every 5th number is a multiple of 5.
3) The Fibonacci sequence works below zero too. We write \(F_n = (1)^{n+1} F_n\). For example, the 4^{th} term = (1)^{5 }. 3 = 3
4) The sum of n terms of Fibonacci Sequence is given by \(\sum_{i=1}^{n} f_{i} = f_{n+1}  f_2\), 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 = \(f_{11}\)  1 = 89  1 = 88
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.
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.
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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.
Solution:
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
= 34 + 55
= 89
Answer: The 12^{th} term of the Fibonacci sequence is 89.

Example 2: The 14^{th} term in the sequence is 377. Find the next term.
Solution:
We know that 15^{th} term = 14^{th} term × the golden ratio.
\(F_{15}\) = 377 × 1.618034
≈ 609.99 = 610
Answer: The 15^{th} term in the Fibonacci sequence is 610.

Example 3: Calculate the value of the 12^{th }and the 13^{th }term of the Fibonacci sequence given that the 9^{th} and 10^{th} terms in the sequence are 21 and 34.
Solution
Using the Fibonacci sequence 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} term of the Fibonacci sequence are 89 and 144.
FAQs on Fibonacci Sequence
What Is 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, starting from 0 and 1. The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 , 144, .......
What Is Fibonacci Sequence Formula in Math?
The Fibonacci sequence formula deals with the Fibonacci sequence, finding its missing terms. The Fibonacci formula is given as,\(F_n= (F_{n1} + F_{n2})\), where n > 1
What Is The Fibonacci Sequence in Nature?
The Fibonacci sequence is We can spot the Fibonacci sequence as spirals in the petals of certain flowers, or the flower heads as in sunflower, broccoli, tree trunks, seashells, pineapples, and the pine cones. The spirals from the center to the outside edge create the Fibonacci sequence.
What Are the Applications of Fibonacci Sequence Formula?
The applications of the Fibonacci sequence include:
 the grouping of numbers and the brilliant proportion in music.
 the computer algorithms, interconnecting parallel, and distributed systems, or particularly coding.
 the fields of science including high energy physical science, quantum mechanics, Cryptography, etc.
 the setting of marketing and trade trends using Fibonacci retracements and Fibonacci ratios.
What Is the Recursive Formula for 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, \(F_n= F_{n1} + F_{n2}\), 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 \(F_n= (F_{n1} + F_{n2})\), where n >1.
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 \(\sum_{i=1}^{n} f_{i} = f_{n+1}  f_2\). The sum of 12 terms in a Fibonacci sequence is 14^{th} term 1 = 234 1 = 233
Why Is Fibonacci Sequence Important?
The Fibonacci sequence is important because of its relationship with the golden ratio. Except for the initial numbers, the numbers in the sequence have a pattern that each number ≈ 1.618 times its preceding number.