The **Fibonacci sequence** is a type series where each number is the sum of the two that precede it. It starts from 0 and 1 usually. The Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The numbers in the Fibonacci sequence are also called Fibonacci numbers. In Maths, the sequence is defined as an ordered list of numbers that follow a specific pattern. The numbers present in the sequence are called the terms. The different types of sequences are arithmetic sequence, geometric sequence, harmonic sequence and Fibonacci sequence. In this article, we will discuss the Fibonacci sequence definition, formula, list and examples in detail.

**Table of Contents:**

- Definition
- Formula
- Sequence List
- Golden Ratio to Calculate Fibonacci Sequence
- Solved Examples
- Practice Problems
- FAQs

## What is Fibonacci Sequence?

The **Fibonacci sequence, **also known as Fibonacci numbers, is defined as the sequence of numbers in which each number in the sequence is equal to the sum of two numbers before it. The Fibonacci Sequence is given as:

**Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, ….**

Here, the third term “1” is obtained by adding the first and second term. (i.e., 0+1 = 1)

Similarly,

“2” is obtained by adding the second and third term (1+1 = 2)

“3” is obtained by adding the third and fourth term (1+2) and so on.

For example, the next term after 21 can be found by adding 13 and 21. Therefore, the next term in the sequence is 34.

## Fibonacci Sequence Formula

The Fibonacci sequence of numbers “F_{n}” is defined using the recursive relation with the seed values F_{0}=0 and F_{1}=1:

**F _{n} = F_{n-1}+F_{n-2}**

Here, the sequence is defined using two different parts, such as kick-off and recursive relation.

The kick-off part is F_{0}=0 and F_{1}=1.

The recursive relation part is F_{n} = F_{n-1}+F_{n-2}.

It is noted that the sequence starts with 0 rather than 1. So, F_{5 } should be the 6^{th} term of the sequence.

## Fibonacci Sequence List

The list of first 20 terms in the Fibonacci Sequence is:

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

The list of Fibonacci numbers are calculated as follows:

F_{n} | Fibonacci Number |

0 | 0 |

1 | 1 |

2 | 1 |

3 | 2 |

4 | 3 |

5 | 5 |

6 | 8 |

7 | 13 |

8 | 21 |

9 | 34 |

… and so on. | … and so on. |

## Golden Ratio to Calculate Fibonacci Numbers

The Fibonacci Sequence is closely related to the value of the Golden Ratio. We know that the Golden Ratio value is approximately equal to 1.618034. It is denoted by the symbol “φ”. If we take the ratio of two successive Fibonacci numbers, the ratio is close to the Golden ratio. For example, 3 and 5 are the two successive Fibonacci numbers. The ratio of 5 and 3 is:

5/3 = 1.6666

Take another pair of numbers, say 21 and 34, the ratio of 34 and 21 is:

34/21 = 1.619

It means that if the pair of Fibonacci numbers are of bigger value, then the ratio is very close to the Golden Ratio.

So, with the help of Golden Ratio, we can find the Fibonacci numbers in the sequence.

The formula to calculate the Fibonacci numbers using the Golden Ratio is:

**X _{n} = [φ^{n} – (1-φ)^{n}]/√5**

Where,

φ is the Golden Ratio, which is approximately equal to the value of 1.618

n is the nth term of the Fibonacci sequence.

## Fibonacci Sequence Solved Examples

**Example 1: **

Find the Fibonacci number when n=5, using recursive relation.

**Solution:**

The formula to calculate the Fibonacci Sequence is: **F _{n} = F_{n-1}+F_{n-2}**

Take: F_{0}=0 and F_{1}=1

Using the formula, we get

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

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

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

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

Therefore, the fibonacci number is 5.

**Example 2:**

Find the Fibonacci number using the Golden ratio when n=6.

**Solution:**

The formula to calculate the Fibonacci number using the Golden ratio is **X _{n} = [φ^{n} – (1-φ)^{n}]/√5**

We know that φ is approximately equal to 1.618.

n= 6

Now, substitute the values in the formula, we get

X_{n} = [φ^{n} – (1-φ)^{n}]/√5

X_{6} = [1.618^{6} – (1-1.618)^{6}]/√5

X_{6} = [17.942 – (0.618)^{6}]/2.236

X_{6} = [17.942 – 0.056]/2.236

X_{6} = 17.886/2.236

X_{6} = 7.999

X_{6} = 8 (Rounded value)

The Fibonacci number in the sequence is 8 when n=6.

## Practice Problems

- Find the Fibonacci number when n = 4, using the recursive formula.
- Find the next three terms of the sequence 15, 23, 38, 61, …
- Find the next three terms of the sequence 3x, 3x + y, 6x + y, 9x + 2y, …

## Frequently Asked Questions on Fibonacci Sequence

Q1

### What is Fibonacci Sequence?

The Fibonacci sequence is the sequence of numbers, in which every term in the sequence is the sum of terms before it.

Q2

### Why is Fibonacci sequence significant?

The Fibonacci sequence is significant, because the ratio of two successive Fibonacci numbers is very close to the Golden ratio value.

Q3

### What are two different ways to find the Fibonacci Sequence?

The two different ways to find the Fibonacci sequence are

- Recursive Relation Method
- Golden Ratio Method

Q4

### Write down the list of the first 10 Fibonacci numbers.

The list of the first 10 Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.

Q5

### What is the value of the Golden ratio?

The value of golden ratio is approximately equal to 1.618034…

## FAQs

### Fibonacci Sequence - Definition, List, Formulas and Examples? ›

The Fibonacci sequence is **a type series where each number is the sum of the two that precede it**. It starts from 0 and 1 usually. The Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The numbers in the Fibonacci sequence are also called Fibonacci numbers.

**What are the first 100 terms of the Fibonacci sequence? ›**

First 100 terms of Fibonacci series are :- 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 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 ...

**What is the pattern of 1 1 2 3 5 8? ›**

**The Fibonacci sequence** is a famous group of numbers beginning with 0 and 1 in which each number is the sum of the two before it. It begins 0, 1, 1, 2, 3, 5, 8, 13, 21 and continues infinitely.

**What are some practical examples of the Fibonacci sequence? ›**

For example, **many flowers have a Fibonacci number as their number of leaves or petals, such as 3, 5, 8, or 13**. The lily has 5 petals, some daisies have 13 petals, and a chicory has 21 petals. The number of petals on a lily is the Fibonacci number 5.

**Is there a formula for the Fibonacci sequence? ›**

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 the next 3 terms of the Fibonacci sequence 0 1 1 2 3 5? ›**

The Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55… This guide provides you with a framework for how to transition your team to agile.

**What is the formula for the nth term of the Fibonacci sequence? ›**

First, calculate the first 20 numbers in the Fibonacci sequence. Remember that the formula to find the nth term of the sequence (denoted by F[n]) is **F[n-1] + F[n-2]**.

**What is simple Fibonacci example? ›**

The Fibonacci sequence begins with the following 14 integers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 ... Each number, starting with the third, adheres to the prescribed formula. For example, **the seventh number, 8, is preceded by 3 and 5, which add up to 8**.

**What is the rule of 2 5 8 11 14? ›**

Answer and Explanation:

The next number in the list of numbers 2, 5, 8, 11, 14, . . . is 17. Notice that the difference between each consecutive term in this sequence is 3. Therefore, this is an arithmetic sequence with a common difference of 3. Thus, to find the next number in the sequence, we simply **add 3 to 14**.

**What is the rule for 5 8 11 14 17? ›**

Sequence A: 5 , 8 , 11 , 14 , 17 , ... For sequence A, **if we add 3 to the first number we will get the second number**. This works for any pair of consecutive numbers. The second number plus 3 is the third number: 8 + 3 = 11, and so on.

### What is the rule for the pattern 5 8 11 14? ›

This is an arithmetic sequence since there is a common difference between each term. In this case, **adding 3 to the previous term in the sequence gives the next term**.

**What are the 12 terms of the Fibonacci sequence? ›**

The first 12 terms of the Fibonacci sequence are **1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144**. The 12th term (144) gives the number of rabbits after one year, which answers Fibonacci's original question to his readers.

**Is egg a Fibonacci? ›**

**Egg is an example for Fibonacci spiral**.

**Is banana a Fibonacci sequence? ›**

Fruit: Bananas and apples **when cut in half, not lengthwise, show ridges that appear in the fibonacci sequence**, that is, 3 or 5, respectively.

**What are the first 200 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 is the 100th term? ›**

To find the 100th term of an arithmetic sequence, we add the first term of the sequence to the product of 99 and the common difference of the sequence. That is the 100th term, a100, of an arithmetic sequence is found using the formula **a1 + 99d**.

**What are the first six terms of the Fibonacci sequence 1 1 2 3 5 8? ›**

Fibonacci Numbers (Sequence):

**1,1,2,3,5,8,13,21,34,55,89,144,233,377**,... Fn=Fn−2+Fn−1 where n≥2 . Each term of the sequence , after the first two, is the sum of the two previous terms.

**What is the 99th term of the Fibonacci sequence? ›**

n | f(n) |
---|---|

97 | 83621143489848422977 |

98 | 135301852344706746049 |

99 | 218922995834555169026 |

100 | 354224848179261915075 |