Fibonacci Series

The Fibonacci series, named after Italian mathematician named Leonardo Pisano Bogollo, later known as Fibonacci, is a series (sum) formed by Fibonacci numbers denoted as F_n. The fibonacci series numbers are 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 the first and second terms. In some old references, the term '0' might be omitted. The series has captured the interest of mathematicians and it continues to be studied and explored for its captivating properties.

We find applications of the Fibonacci series in nature. It is found in biological settings, like in the branching of trees, patterns of petals in flowers, etc. Let us understand the Fibonacci series formula, its properties, and its applications in the following sections.

The Fibonacci series is the sequence of numbers (also called Fibonacci numbers), where every number is the 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 thus be observed that every term can be calculated by adding the two terms before it.

Given the first term, F₀ and second term, F₁ as '0' and '1' respectively, the third term here can be given as, F₂ = 0 + 1 = 1

Similarly,

• F₃ = 1 + 1 = 2
• F₄ = 2 + 1 = 3
• F₅ = 2 + 3 = 5
• F₆ = 3 + 5 = 8
• F₇ = 5 + 8 = 13
• and so on

Therefore, to represent any (n+1)^th term in this 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 (n+1)^th term in the sequence is defined using the recursive formula, such that F₀ = 0, F₁ = 1 to give F_n.

The Fibonacci formula using recursion is given as follows.

F_n = F_n-1 + F_n-2, where n > 1

Fibonacci Series Spiral

The Fibonacci series spiral is a logarithmic spiral that is formed by joining the corners of squares that have side lengths the same as the Fibocacci numbers in the Fibonacci sequence. This spiral appears in nature, such as in the arrangement of leaves on a stem, the shell of a nautilus, the spiral arms of galaxies, etc. The Fibonacci series spiral has been studied extensively in mathematics and is known for its artistically pleasing and symmetrical appearance.

Here, the following rectangle with the Fibonacci series spiral is a golden rectangle. i.e., its dimensions are in the "Golden Ratio" (≈1.618).

Each term of a Fibonacci series is a sum of the two terms preceding it, given that the series starts from '0' and '1'. We can use this to find the terms in the 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 (in sigma notation) of all terms in this series is given as, Σ_j=0ⁿ F_j = F_n+2 - 1.
• The sum of all even index Fibonacci numbers in a this series is given as, Σ_j=1ⁿ F_2j = F₂ + F₄ + . . . + F_2n = F_2n+1 - 1.
• The sum of all odd index Fibonacci numbers in a this series is given as, Σ_j=1ⁿ F_2j-1 = F₁ + F₃ + . . . + F_2n-1 = F_2n.
• The numbers in a Fibonacci series are related to the golden ratio. Any Fibonacci number ((n + 1)^th term) can be calculated using the Golden Ratio using the formula, F_n = (Φⁿ - (1-Φ)ⁿ)/√5, Here φ is the golden ratio where φ ≈ 1.618034.
  For example: To find the 7^th term, we apply F₆ = (1.618034⁶ - (1-1.618034)⁶)/√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→∞ 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 the 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 = (Φⁿ - (1-Φ)ⁿ)/√5, where φ ≈ 1.618034 is the golden ratio.

The golden ratio is expressed 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→∞ 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. The 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 falling diagonal lines in Pascal’s triangle. We can observe this in the figure given below, considering the first element '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 series are given as,

• It is used in the grouping of numbers and used to study different other special mathematical sequences.
• It finds 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.

☛ Related Topics:

• Arithmetic Series Formula
• Geometric Series Formulas

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

What is the Meaning of the 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. Fibonacci series numbers are, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 , 144, .......

What is the 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, where F₀ = 0 and F₁ = 1.

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, and 4181. This series starts from 0 and 1, with every term being the sum of the preceding two terms.

What are the Examples of the Fibonacci Series in Nature?

The Fibonacci series can be spotted in the biological setting around us in different forms. It 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 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 Applications of the Fibonacci Series Formula?

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

• This is 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 Using Recursion?

Fibonacci series cannot be easily represented using an explicit formula. We, therefore, describe the Fibonacci series using a recursive formula, given as, F₀ = 0, F₁= 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 100^th Fibonacci Number in Fibonacci Series?

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

What is the Use of the Fibonacci Series?

The Fibonacci series is used in various fields, such as mathematics, finance, computer science, etc. Also, it is used as a basis for algorithms, models, and patterns.

How are Fibonacci Series and Golden Ratio Related?

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 can be expressed in terms of Golden ratio as, F_n = (Φⁿ - (1-Φ)ⁿ)/√5, where φ is the golden ratio. Similarly, the Golden ratio can be expressed as the ratio of successive terms of a Fibonacci series as, φ = lim _n→∞ F_n+1/F_n.