Fibonacci Sequence

The Fibonacci sequence was first found by an Italian named Leonardo Pisano Bogollo (Fibonacci). The Fibonacci sequence is a sequence of whole numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... This is an infinite sequence that starts with 0 and 1 and each term is the sum of the two preceding terms. This sequence 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 more about it and its interesting properties.

The Fibonacci sequence is the sequence formed by the infinite terms 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... In simple terms, it is a sequence in which every number in the Fibonacci sequence is the sum of two numbers preceding it in the sequence. Its first two terms are 0 and 1. The terms of this sequence are known as Fibonacci numbers. The first 20 terms of the Fibonacci sequence are given as follows:

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.

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.

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 form a golden rectangle.

The Fibonacci sequence formula for “F_n” is defined using the recursive formula by setting F₀ = 0, F₁ = 1, and using the formula below to find F_n. The Fibonacci formula is given as follows.

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

Note that F₀ is termed as the first term here (but NOT F₁). Thus, F_n represents the (n + 1)^th term of the Fibonacci sequence here.

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 =(Φⁿ - (1-Φ)ⁿ)/√5, Here φ is the golden ratio and Φ ≈ 1.618034.

To find the 7^th 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.

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 F_n. For example, F_-4 = (-1)⁵ . F₄ = (-1) 3 = -3.

4) The sum of n terms of Fibonacci Sequence is given by Σ_i=0ⁿ F_i = F_n+2 - F₂ (or) F_n+2 - 1, where F_n is the n^th Fibonacci number. (Note: the first term starts from F₀)

For example, the sum of first 10 terms of sequence = 12^th term - 1 = 89 - 1 = 88. It can be mathematically written as Σ_i=0⁹ F_i = F₁₁ - 1 = 89 - 1 = 88.

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.

☛ Related Articles:

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

What is the Definition of 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. It starts 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_n-1 + F_n-2, where n > 1.

What is Fibonacci Spiral?

First, take a small square of length 1 unit and attach it to an identical square vertically. Thus formed is a rectangle of vertical length 2 and width 1 unit. Adjacent to its length (2 units), attach a square of length 2 units. Thus formed is a rectangle of horizontal length 3 units and vertical width 2 units. If we continue the same process we get a big rectangle that is partitioned into squares where the length of each square is the sum of the lengths of two of its adjacent squares. The larger the rectangle, the more the chances for it to become a golden rectangle. If we join the centers of all squares, we get a spiral which is known as the Fibonacci spiral. For more information, click here.

What is The Fibonacci Sequence in Nature?

We can spot the Fibonacci sequence as spirals in the petals of certain flowers, or the flower heads as in sunflowers, broccoli, tree trunks, seashells, pineapples, and 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.

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 Σ_i=0ⁿ F_i = F_n+2 - 1. For example, the sum of the first 12 terms in a Fibonacci sequence is Σ_i=0¹¹ F_i = F₁₃ -1 = 233 -1 = 232. If we add the first 12 terms manually, we get 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 = 232, we got the same thing as formula.

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 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 Sequence?

The formula to find the n^th term of the sequence is denoted as F_n = F_n-1 + F_n-2, where n >1.

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.