Fibonacci Series

The Fibonacci series, named after Italian mathematician named Leonardo Pisano Bogollo, later known as Fibonacci, is a series formed by Fibonacci numbers denoted as F\(_n\). Fibonacci series is given as: 0, 1, 1, 2, 3, 5, 8, 13, 21, 38, . . . In a Fibonacci series, every term is the sum of the preceding two terms, starting from 0 and 1 as first and second terms. In some old references, the term '0' might be omitted.

We find applications of the Fibonacci series around us in our day-to-day lives. It is also found in biological settings, like in the branching of trees, patterns of petals in flowers, etc. Let us understand the Fibonacci series formula, properties, and its applications in the following sections.

The Fibonacci series is the sequence of numbers(also called Fibonacci numbers), where every number is sum of the preceding two numbers, such that the first two terms are '0' and '1'. In some older versions of the series, the term '0' might be omitted. A Fibonacci series can thus be given as, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, . . . It can be thus be observed that every term can be calculated by adding the two terms before it.

Given the first, F\(_0\) and second term, F\(_1\) as '0' and '1', the third term here can be given as, F\(_2\) = 0 + 1 = 1
Similarly,
F\(_3\) = 1 + 1 = 2
F\(_4\) = 2 + 1 = 3
Therefore, to represent any (n+1)th term in Fibonacci series, we can give the expression as, F\(_n\) = F\(_{n-1}\) + F\(_{n-2}\). We can thus represent a Fibonacci series as shown in the image below,

The Fibonacci series formula in maths can be used to find the missing terms in a Fibonacci series. The formula to find the Fibonacci series can be given to express the (n+1)th term in the sequence is defined using the recursive formula, such that \(F_0\) = 0,\(F_1\) = 1 to give \(F_n\).

The Fibonacci formula is given as follows.

\(F_n = F_{n-1} + F_{n-2}\), where n > 1

Fibonacci Series Spiral

The Fibonacci spiral is the representation of the pattern formed by the Fibonacci numbers in a grid. Fibonacci series spiral starts in a plane in the shape of a rectangle whose dimensions(length × breadth) follow the principle of a "Golden Ratio"(≈1.618), and is therefore referred to as "Golden Rectangle". The following image depicts the Fibonacci spiral starting with a rectangle partitioned into 2 squares. Fibonacci spiral is an approximation of the golden spiral.

Each term of a Fibonaccis series is a sum of the two terms preceding it, given that the series starts from '0' and '1'. We can use this concept to find the terms in a Fibonacci series.The first 20 numbers in a Fibonacci series are given below in the Fibonacci series list.

☛Also Check: You can use the Fibonacci calculator that helps to calculate the terms in a Fibonacci Series.

There are some very interesting properties associated with Fibonacci Series. They are given below,

• The sum of all terms in Fibonacci series is given as, \(\sum_{j=0}^n\) F\(_j\) = F\(_{n+1}\) - 1.
• The sum of all even index Fibonacci numbers in a Fibonacci series is given as, \(\sum_{j=1}^n\) \(F_{2j}\) = F\(_2\) + F\(_4\) + . . . + F\(_{2n}\) = F\(_{2n+1}\) - 1.
• The sum of all odd index Fibonacci numbers in a Fibonacci series is given as, \(\sum_{j=1}^n\) \(F_{2j-1}\) = F\(_1\) + F\(_3\) + . . . + F\(_{2n-1}\) = F\(_{2n}\).
• The numbers in a Fibonacci series are related to the golden ratio. Any Fibonacci number in a Fibonacci series can be calculated using the Golden Ratio using the formula, \(F_n =\dfrac{\phi^n - (1-\phi)^n}{\sqrt{5}}\), Here φ is the golden ratio.
  For example: To find the 7^th term, we apply \(F_6 =\dfrac{(1.618034)^6 - (1-1.618034)^6}{\sqrt{5}}\) = 8.
• As we discussed in the previous property, we can also calculate the golden ratio using the ratio of consecutive Fibonacci numbers. For 2 consecutive Fibonacci numbers, given as, F\(_{n+1}\) and F\(_{n}\), the value of φ can be calculated as,\( \lim_{n \to \infty}\) F\(_{n+1}\)/F\(_{n}\)

We will understand this relationship between the Fibonacci series and the Golden ratio in detail in the next section.

In mathematics, the Fibonacci series and Golden ratio are closely connected. The Fibonacci series is given as, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, ... The following expression explains the interrelationship between both attributes.
\(F_n =\dfrac{\phi^n - (1-\phi)^n}{\sqrt{5}}\), where φ is the golden ratio.

The golden ratio is the expressed in terms of Fibonacci series as the limit of the ratios of successive terms of the Fibonacci series (or any Fibonacci-like sequence), as depicted by Kepler in the expression given below,
φ = \( \lim_{n \to \infty}\) F\(_{n+1}\)/F\(_{n}\).

In other words, if a Fibonacci number is divided by its immediate predecessor in the given Fibonacci series, the quotient approximates φ. This accuracy of this value increases with the increase in the value of 'n', i.e., as n approaches infinity. We have also discussed in the previous section, that how a Fibonacci spiral approximates a Golden spiral.

Another interesting method used to find the numbers in a Fibonacci series is Pascal's triangle. Pascal's triangle, in mathematics, is a triangular array comprising the binomial coefficients. In a Fibonacci series, Fibonacci numbers can be derived by calculating the sum of elements on the rising diagonal lines in Pascal’s triangle. We can observe this in the figure given below, considering the first element in a Fibonacci series as '0', the following terms can be calculated by summing the diagonal elements as given below,

The Fibonacci series finds application in different fields in our day-to-day lives. The different patterns found in a varied number of fields from nature, to music, and to the human body follow the Fibonacci series. Some of the applications of the Fibonacci series are given as,

• It used in the grouping of numbers and used to study different other special mathematical sequneces.
• It find application in Coding (computer algorithms, distributed systems, etc). For example, Fibonacci series are important in the computational run-time analysis of Euclid's algorithm, used for determining the GCF of two integers.
• It is applied in numerous fields of science like quantum mechanics, Cryptography, etc.
• In finance market trading, Fibonacci retracement levels are widely used in technical analysis.

☛ Topics Related to Fibonacci Series:

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

Let us understand the concept of Fibonacci series better using the following solved examples.

What is Fibonacci Series?

The Fibonacci series is an infinite series, starting from '0' and '1', in which every number in the series is the sum of two numbers preceding it in the series. The Fibonacci series is given as, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 , 144, .......

What is Fibonacci Series Formula in Math?

The Fibonacci series formula is the formula used to find the terms in a Fibonacci series in math. The Fibonacci formula is given as, \(F_n= (F_{n-1} + F_{n-2})\), where n > 1.

What are the Examples of Fibonacci Series in Nature?

The Fibonacci series is can be spotted in the biological setting around us in different forms. The Fibonacci series can be found in spirals in the petals of certain flowers such as in the flower heads of sunflowers. It can also be found in the branching of trees.

What are the Applications of Fibonacci Series Formula?

The applications of the Fibonacci series include different fields like finance, music, etc. These applications are given as,

• Fibonacci series are important in the computational run-time analysis of Euclid's algorithm, used for determining the GCF of two integers.
• It can be applied in numerous fields of science like quantum mechanics, physics, Cryptography, etc.
• In finance market trading, Fibonacci retracement levels are used in the technical analysis of data.

What is the Fibonacci Series Recursive Formula?

Fibonacci series cannot be easily represented using an explicit formula. We therefore describe the Fibonacci series using a recursive formula, given as, \(F_0\) = 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 Series?

The formula to find the (n)^th term of the series is given as \(F_{n-1}= (F_{n-2} + F_{n-3})\), where n >1.

What is the Importance of the Fibonacci Series?

The Fibonacci series is important because of its relationship with the golden ratio and Pascal's triangle. Except for the initial numbers, the numbers in the series have a pattern that each number ≈ 1.618 times its preceding number. This value becomes more accurate as the number of terms in the Fibonacci series increases.

What are the First 10 Fibonacci Numbers in Fibonacci Series?

The first 10 terms in a Fibonacci series are given as, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181. Fibonacci series starts from 0 and 1, with every term being the sum of preceding two terms.

What is the 100th Fibonacci Number in Fibonacci Series?

The 100th term in a Fibonacci series is 354,224,848,179,261,915,075. Using the Fibonacci series formula, 100th term in Fibonacci series can be given as the sum of 98th and 99th term.

What is Use of Fibonacci Series?

Fibonacci series find application in different fields in our day-to-day lives. They are used to study patterns of numbers and data in mathematical analysis in various fields.

How is a Fibonacci Series Related to Golden Ratio?

The terms in a Fibonacci series share a relationship with the Golden ratio. The following expressions can be used to depict this inter-relationship:

(n+1)th term of a Fibonacci series an be expressed in term of Golden ratio as, \(F_n =\dfrac{\phi^n - (1-\phi)^n}{\sqrt{5}}\), where φ is the golden ratio.

Similarly, Golen ratio can be expressed as ratio of successive terms of a Fibonacci series as, φ = \( \lim_{n \to \infty}\) F\(_{n+1}\)/F\(_{n}\).