**Introduction**

The Fibonacci sequence is a mathematical idea that can be represented as a series of numbers, sequences, or numbers where each number is equal to the sum of the two numbers that came before it, and the first two terms are 0 and 1. F_{n}, where n is a natural number, is the standard symbol for a Fibonacci number. The Fibonacci numbers are represented by the numbers 0, 1, 1, 2, 3, and so on.

Structures in plants and animals contain Fibonacci numbers. These figures are frequently referred to as the natural hidden code or the universal rule of nature.

Let us learn more about Fibonacci numbers in-depth in this article. We will discuss the golden ratio, its relationship to the Fibonacci numbers, and its rules and qualities. For a better understanding, we will solve several examples of Fibonacci numbers.

**What are Fibonacci numbers?**

**Definition**

The Fibonacci numbers consist of a collection of numbers, each of which is the sum of two numbers before it. The first two numbers start with 0 and 1. One of the most famous mathematical formulas is this sequence. A series of whole numbers ordered like 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34 are called the Fibonacci numbers.

The Fibonacci numbers can be visualized as a spiral if we create squares with specific widths, as illustrated below. We can see how the squares fit together perfectly in the given figure. For instance, the sum of 5 and 8 is 13, 8 and 13 is 21, and so on.

The series can theoretically go on forever if the formula is applied to each subsequent number. Although it’s not very frequent, some sources begin the Fibonacci sequence with a one rather than a zero.

**How to Calculate the Fibonacci Sequence?**

Mathematical calculations can determine the Fibonacci sequence. According to this method, every number in the sequence is regarded as a term denoted by the expression F_{n}. The n indicates where the given number falls in the sequence, which starts at 0. For instance, the fourth term is known as F_{3}, and the eighth term is known as F_{7}.

The following details can be used to define the Fibonacci sequence:

F_{0}=0 *the first term*

F_{1}=1 *the second term*

F_{n}=F_{n-1} + F_{n-2} *all other terms*

According to the first two equations, the terms in the first and second positions equal 0 and 1, respectively. The third equation is repetitive, meaning that each number in the sequence is defined using the numbers that came before it.

As an illustration, the terms F_{4} and F_{5} must previously be specified to define the sixth number (F_{6}). In effect, these two numbers demand that the numbers before them be defined. Throughout the sequence, the numbers keep adding to one another.

Let us examine how the first fifteen terms of the Fibonacci sequence came to be. When we tabulate the result, we find:

n | Term Position | F_{n-1} | F_{n-2} | F_{n}=F_{n-1}+F_{n-2} | Fibonacci Numbers |

0 | First | – | – | – | 0 |

1 | Second | F_{0 }= 0 | F_{1 }= 1 | 1 | |

2 | Third | F_{1 }= 1 | F_{0 }= 0 | F_{2} = 0 + 1 | 1 |

3 | Fourth | F_{2 }= 1 | F_{1 }= 1 | F_{3} = 1 + 1 | 2 |

4 | Fifth | F_{3 }= 2 | F_{2 }= 1 | F_{4} = 2 + 1 | 3 |

5 | Sixth | F_{4 }= 3 | F_{3 }= 2 | F_{5} = 3 + 2 | 5 |

6 | Seventh | F_{5 }= 5 | F_{4 }= 3 | F_{6} = 5 + 3 | 8 |

7 | Eight | F_{6 }= 8 | F_{5 }= 5 | F_{7} = 8 + 5 | 13 |

8 | Ninth | F_{7 }= 13 | F_{6 }= 8 | F_{8} = 13 + 8 | 21 |

9 | Tenth | F_{8 }= 21 | F_{7 }= 13 | F_{9} = 21 + 13 | 34 |

10 | Eleventh | F_{9 }= 34 | F_{8 }= 21 | F_{10} = 34 + 21 | 55 |

11 | Twelfth | F_{10 }= 55 | F_{9 }= 34 | F_{11} = 55 + 34 | 89 |

12 | Thirteenth | F_{11}= 89 | F_{10 }= 55 | F_{12} = 89 + 55 | 144 |

13 | Fourteenth | F_{12 }= 144 | F_{11}= 89 | F_{13} = 144 + 89 | 233 |

14 | Fifteenth | F_{13 }= 233 | F_{12 }= 144 | F_{14} = 233+144 | 377 |

15 | Sixteenth | F_{14 }= 377 | F_{13 }= 233 | F_{15} = 377+233 | 610 |

16 | Seventeenth | F_{15 }= 610 | F_{14 }= 377 | F_{16} = 610+377 | 987 |

17 | Eighteenth | F_{16 }= 987 | F_{15 }= 610 | F_{17} = 987+610 | 1597 |

18 | Nineteenth | F_{17}=1597 | F_{16 }= 987 | F_{18} =1597+987 | 2584 |

19 | Twentieth | F_{18}=2584 | F_{17}=1597 | F_{19}=2584+1597 | 4181 |

**Fibonacci Numbers Formula**

The term F_{n} identifies the Fibonacci numbers, which are described as a recursive relationship with the initial values F_{0}=0 and F_{1}=1.

F_{n}=F_{n-1}+F_{n-2}

The following details can be used to define the Fibonacci sequence:

F_{0}=0 *the first term*

F_{1}=1 *the second term*

F_{n}=F_{n-1}+F_{n-2} *all other terms*

The F_{n} equation is repetitive, meaning that each number in the sequence is defined using the numbers that came before it.

For example, the terms F_{7} and F_{8} must previously be specified to define the ninth number (F_{9}). In effect, these two numbers demand that the numbers before them be defined. Throughout the sequence, the numbers keep adding to one another.

**Fibonacci Numbers List**

The Fibonacci sequence’s first 20 terms are listed below:

0, | 1, | 1, | 2, | 3, | 5, | 8, | 13, | 21, | 34, | 55, | 89, | 144, | 233, | 377, | 610, | 987, | 1597, | 2584, | 4181 |

The Fibonacci numbers consist of a collection of numbers, each of which is the sum of the two numbers before it. It begins with 0 and 1, which are the initial two numbers. This series is one of the most well-known mathematical formulas.

**The Golden Ratio and Fibonacci Sequence**

The golden ratio, which frequently occurs in nature and is used in many fields of human activity, is a ratio that is frequently linked to the Fibonacci sequence. The golden ratio and the Fibonacci sequence are used as principles in designing user interfaces, websites, nature, arts, architecture, and other things.

Any two consecutive Fibonacci numbers have a very close ratio of 1.618.

Let us take two consecutive random numbers from the list and get the ratio. Hence, we have,

$\frac{8}{5}$=1.6

$\frac{13}{8}$=1.625

$\frac{34}{21}$=1.619

$\frac{89}{55}$=1.618

The Greek letter ** phi** symbolizes the golden ratio. Usually, the lowercase form (ϕ or φ) is used. Using the Golden Ratio, we can calculate any Fibonacci number using the following formula:

F_{n} = [φ^{n} – (1 – φ)^{n}] ÷ √5

For example, let us calculate F_{5}. Let us use φ = 1.618 and n = √5.

F_{5} = [1.618^{5} -(1 – 1.618)^{5}] ÷ √5

F_{5} = 4.999

The result of rounding off 4.999 to the nearest whole number is 5. Thus, F_{5} = 5.

As another example, let us calculate F_{7}.

F_{7} = [1.618^{7 }– (1 – 1.618)^{7}] ÷ √5

F_{7} = 12.999

The result of rounding off 12.999 to the nearest whole number is 13. Thus, F_{7} = 13.

**Patterns in Nature (Fibonacci Numbers)**

In nature, Fibonacci numbers are visible. Here are a few of the Fibonacci number sequences and patterns that can be seen in nature:

**Flower petals**

The Fibonacci numbers can be seen on flower petals. The Fibonacci numbers appear on the number of petals of flowers like:

White calla lily: 1 petal

Euphorbia: 2 petals

Trillium: 3 petals

Columbine: 5 petals

Bloodroot: 8 petals

Black-eyed Susan: 13 petals

**Sunflower Seed**

The Fibonacci sequence can be visualized in the sunflower seed. Two spirals are arranged in an opposite pattern. It typically has 34 and 55 seeds, respectively, and the number of clockwise and counterclockwise spirals are consecutive Fibonacci numbers.

**Pineapples**

Pineapples feature hexagonal nubs that form spirals. Depending on the pineapple’s size, the nubs produce five spirals, eight spirals, or 13 spirals that revolve diagonally upward and to the right. The Fibonacci numbers are 5, 8, and 13.

**Pinecones**

Pinecones are made up of Fibonacci numbers. Fibonacci numbers are used to describe spirals in pinecones that have 5 and 8 or 8 and 13 arms from the center.

**Animals**

Numerous animals have essential aspects that are positioned and proportioned according to phi or φ.

Examples include the body parts of ants and other insects, animal body parts, and seashell spirals.

**Human Body**

Human body shape and structure are seen to follow the golden ratio. The Fibonacci numbers appear in various parts of the human body, including the two hands with five digits and each finger with three parts. The proportion of the forearm to the hand, as well as other body parts, is phi.

**Examples**

**Example 1**

Suppose the 7^{th} and 8^{th} terms in the Fibonacci sequence are 8 and 13. Find the value of the 11^{th} and 12^{th} terms in the Fibonacci numbers.

**Solution:**

Since the 7^{th} and 8^{th} terms are 8 and 13, we can find the 9^{th} term by adding these numbers. Hence, we have,

9^{th} term = 8 + 13 = 21

Let us continue the sequence by adding the 8^{th} and 9^{th} terms, 9^{th} and 10^{th} terms, and 10^{th} and 11^{th} terms.

10^{th} term = 8^{th} and 9^{th} term = 13 + 21 = 34

11^{th} term = 9^{th} and 10^{th} term = 21 + 34 = 55

12^{th} term = 10^{th} and 11^{th} term = 34 + 55 = 89.

Therefore, in the Fibonacci numbers, the 11^{th} term is 55, and the 12^{th} is 89.

**Example 2**

Find the sum of the following:

( a ) the sum of the first 5 Fibonacci numbers

( b ) the sum of the first 10 Fibonacci numbers

( c ) the sum of the first 13 Fibonacci numbers

**Solution:**

( a ) The first 5 Fibonacci numbers list is 0, 1, 1, 2, 3. Hence, we have,

Sum of the first 5 Fibonacci numbers = 0 + 1 + 1 + 2 + 3 = **7**.

( b ) The list of the first 10 Fibonacci numbers is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. So, we get,

Sum of the first 10 Fibonacci numbers = 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 = **88**.

( c ) The list of the first 13 Fibonacci numbers is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. Thus, we have, Sum of the first 15 Fibonacci numbers = 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 + 89 + 144 = **376**.

**Example 3**

Using the formula F_{n}=F_{n-1}+F_{n-2}, find the Fibonacci number when n = 7.

**Solution:**

Since F_{0}=0 and F_{1}=1, using the formula, we will have,

F_{2} = F_{2-1} + F_{2-2} = F_{1 }+ F_{0} = 1 + 0 = **1**

F_{3} = F_{3-1} + F_{3-2} = F_{2 }+ F_{1} = 1 + 1 = **2**

F_{4} = F_{4-1} + F_{4-2} = F_{3 }+ F_{2} = 2 + 1 = **3**

F_{5} = F_{5-1} + F_{5-2} = F_{4 }+ F_{3} = 3 + 2 = **5**

F_{6} = F_{6-1} + F_{6-2} = F_{5 }+ F_{4} = 5 + 3 = **8**

F_{7} = F_{7-1} + F_{7-2} = F_{6 }+ F_{5} = 8 + 5 = **13**

Hence, when n = 7, the Fibonacci number is 13.

**Example 4**

Using the golden ratio, find the Fibonacci number when:

( a ) n = 4

( b ) n = 8

( c ) n = 12

**Solution**

Using the Golden Ratio, we can calculate any Fibonacci number using the following formula:

F_{n} = [φ^{n} – (1 – φ)^{n}] ÷ √5

Let us now substitute the given n to the formula.

( a ) n = 4

F_{4} = [φ^{4} – (1 – φ)^{4}] ÷ √5

F_{4} = [1.618^{4} – (1 – 1.618)^{4}] ÷ √5

F_{4} = 2.999

The result of rounding off 2.999 to the nearest whole number is 3. Thus, F_{4}=3.

( b ) n = 8

F_{8} = [φ^{8} – (1 – φ)^{8}] ÷ √5

F_{8} = [1.618^{8} – (1 – 1.618)^{8}] ÷ √5

F_{8}= 20.996

The result of rounding off 20.996 to the nearest whole number is 3. Hence, F_{8}=21.

( a ) n = 12

F_{12} = [φ^{12} – (1 – φ)^{12}] ÷ √5

F_{12}= [1.618^{12} – (1 – 1.618)^{12}] ÷ √5

F_{12}= 143.96

The result of rounding off 143.96 to the nearest whole number is 144. Thus, F_{12}=144.

**Example 5**

Calculate the Fibonacci number when n = 13. (Use the formula F_{n}=F_{n-1}+F_{n-2})

**Solution**

Since F_{0}=0 and F_{1}=1, using the formula, we will have,

F_{2} = F_{2-1} + F_{2-2} = F_{1 }+ F_{0} = 1 + 0 = **1**

F_{3} = F_{3-1} + F_{3-2} = F_{2 }+ F_{1} = 1 + 1 = **2**

F_{4} = F_{4-1} + F_{4-2} = F_{3 }+ F_{2} = 2 + 1 = **3**

F_{5} = F_{5-1} + F_{5-2} = F_{4 }+ F_{3} = 3 + 2 = **5**

F_{6} = F_{6-1} + F_{6-2} = F_{5 }+ F_{4} = 5 + 3 = **8**

F_{7} = F_{7-1} + F_{7-2} = F_{6 }+ F_{5} = 8 + 5 = **13**

F_{8} = F_{8-1} + F_{8-2} = F_{7 }+ F_{6} = 13 + 8 = **21**

F_{10} = F_{10-1} + F_{10-2} = F_{9 }+ F_{8} = 34 + 21 = **55**

F_{11} = F_{11-1} + F_{11-2} = F_{10 }+ F_{9} = 55 + 34 = **89**

F_{12} = F_{12-1} + F_{12-2} = F_{11 }+ F_{10} = 89 + 55 = **144**

F_{13} = F_{13-1} + F_{13-2} = F_{12 }+ F_{11} = 144 + 89 = **233**

Therefore, the Fibonacci number is 233 when n = 13.

**Summary**

The ** Fibonacci numbers** consist of a collection of numbers, each of which is the sum of two numbers before it. The first two numbers start with 0 and 1. One of the most famous mathematical formulas is this sequence. A series of whole numbers ordered like 0, 1, 1, 2, 3, 5, 8, 13, 21, and 34 are called the Fibonacci numbers.

*Formula*

The term F_{n} identifies the Fibonacci numbers, which are described as a recursive relationship with the initial values F_{0}=0 and F_{1}=1.

F_{n}=F_{n-1}+F_{n-2}

The following details can be used to define the Fibonacci sequence:

F_{0}=0 *the first term*

F_{1}=1 *the second term*

F_{n}=F_{n-1}+F_{n-2} *all other terms*

*List of the first 20 Fibonacci numbers:*

0, | 1, | 1, | 2, | 3, | 5, | 8, | 13, | 21, | 34, | 55, | 89, | 144, | 233, | 377, | 610, | 987, | 1597, | 2584, | 4181 |

**Frequently Asked Questions on Fibonacci Numbers (FAQs)**

**What are Fibonacci numbers?**

The Fibonacci numbers consist of a collection of numbers, each of which is the sum of the two numbers before it. It begins with 0 and 1, which are the initial two numbers. This series is one of the most well-known mathematical formulas. The sequence of whole numbers known as the Fibonacci numbers goes like this:

0, | 1, | 1, | 2, | 3, | 5, | 8, | 13, | 21, | 34, | 55, | 89, | 144, | 233, | 377, | 610, | 987, | 1597, | 2584, | 4181 |

**What is the formula for calculating the Fibonacci numbers?**

The term F_{n} identifies the Fibonacci numbers, which are described as a recursive relationship with the initial values F_{0}=0 and F_{1}=1.

F_{n}=F_{n-1}+F_{n-2}

**What are examples of Fibonacci numbers in Arts?**

The golden ratio, which frequently occurs in nature and is used in many fields of human activity, is a ratio that is frequently linked to the Fibonacci sequence. The golden ratio and the Fibonacci sequence are principles to achieve beauty, balance, and harmony in art, architecture, and design. The following are some examples of Fibonacci numbers and the golden ratio in arts:

- The Parthenon in Athens embodies the golden ratio.
- Leonardo da Vinci’s creations were believed to have incorporated the golden ratio in his paintings, such as the Vitruvian Man, The Last Supper, Monalisa, and St. Jerome in the Wilderness.
- Some of the creations of Michelangelo, like “The Creation of Adam” and the “Holy Family.”

**What are examples of Fibonacci numbers in Architecture?**

Many architectural structures, like the Great Pyramid, Notre Dame, Taj Mahal, Chartres Cathedral, United Nations Building, Eiffel Tower, and others, are examples that the Fibonacci numbers and the Golden ratio are applied.

The Great Pyramid of Giza follows the golden ratio in its proportions. The base-to-height ratio is approximately 1.5717, which is near the Golden ratio.

Golden proportions are also visible in the Eiffel Tower in Paris. In perfect accordance with the golden ratio, the bottom is wider, and the top is narrower.

**What are the first 30 Fibonacci Numbers?**

These are the first 30 Fibonacci numbers:

0, | 1, | 1, | 2, | 3, | 5, | 8, | 13, | 21, | 34, | 55, | 89, | 144, | 233, | 377, | 610, | 987, | 1597, | 2584, | 4181, |

6765, | 10946, | 17711, | 28657, | 46368, | 75025, | 121393, | 196418, | 317811, | 514229. |

**What are examples of Fibonacci Numbers in Nature?**

In nature, Fibonacci numbers are visible. Here are a few of the Fibonacci number sequences and patterns that can be seen in nature:

Flower petals ( the count of petals is Fibonacci numbers )

Sunflower Seed ( the spirals are successive Fibonacci numbers )

Pineapples ( the nubs produce spirals that are Fibonacci numbers )

Pinecones: ( Fibonacci numbers are used to describe spirals )

Animals ( proportion of body parts )

Human Body ( shape and structure follow the golden ratio )

**What does the Golden ratio mean?**

The golden ratio, which frequently occurs in nature and is used in many fields of human activity, is a ratio that is frequently linked to the Fibonacci sequence.

Any two consecutive Fibonacci numbers have a very close ratio of 1.618.

The Greek letter ** phi** symbolizes the golden ratio. Usually, the lowercase form (ϕ or φ) is used. Using the Golden Ratio, we can calculate any Fibonacci number using the following formula:

F_{n} = [φ^{n} – (1 – φ)^{n}] ÷ √5

**What are the first two terms in the Fibonacci Numbers?**

In the Fibonacci Sequence, 0 and 1 are the first two terms. F_{0} = 0 and F_{1} = 1.

**How to use the Golden ratio to calculate Fibonacci numbers?**

The golden ratio, which frequently occurs in nature and is used in many fields of human activity, is a ratio that is frequently linked to the Fibonacci sequence. The ratio between any two consecutive Fibonacci numbers is very close to 1.618. Let us take two consecutive random numbers from the list and get the ratio. Hence, we have,

$\frac{8}{5}$=1.6

$\frac{13}{8}$=1.625

$\frac{34}{21}$=1.619

$\frac{89}{55}$=1.618

The Greek letter ** phi** symbolizes the golden ratio. Usually, the lowercase form (ϕ or φ) is used. Using the Golden Ratio, we can calculate any Fibonacci number using the following formula:

F_{n} = [φ^{n} – (1 – φ)^{n}] ÷ √5

For example, let us calculate F_{5}. Let us use φ=1.618 and n=4.

F_{4} = [1.618^{4} – (1 – 1.618)^{4}] ÷ √5

F_{4} = 2.999

The result of rounding off 2.999 to the nearest whole number is 3. Thus, F_{4}=3.

As another example, let us calculate F_{8}.

F_{8} = [1.618^{8} – (1 – 1.618)^{8}] ÷ √5

F_{8}=20.996

The result of rounding off 20.996 to the nearest whole number is 21. Thus, F_{8}=21.

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