# Fibonacci Numbers

The Fibonacci sequence was invented by an Italian named Leonardo Pisano Bigollo.

It was an outcome of a mathematical problem in order to keep a check on the pairs of rabbits born every year.

Once you understand the basic concept, can you help us find the first 60 Fibonacci numbres? If you solve this successfully without a Fibonacci number calculator, then you will have grasped the Fibonacci sequence facts, as well as understood the nature of Fibonacci numbers.

Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

**Lesson Plan**

**What Are Fibonacci Numbers?**

- Fibonacci numbers are a sequence of whole numbers arranged as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34,...

- This sequence is called the Fibonacci sequence.
- Each number in the Fibonacci sequence is given as F
_{n}. - It is an infinite sequence.
- It is represented as the spiral below.

**Did you know: **

**Fibonacci numbers in nature can be shown by an example of petals of a sunflower. They follow the Fibonacci sequence.**

Some important Fibonacci sequence facts are as follows:

- Fibonacci sequence is only applicable for whole numbers and decimal numbers from a financial perspective.
- It doesn't apply for fractions.
- The first number in a Fibonacci sequence is always 0.
- The second number in a Fibonacci sequence is always 1.

**What is the Rule for Fibonacci Numbers?**

The rule for the Fibonacci numbers is given as :

- The first number in the Fibonacci sequence is given as F
_{0}= 0 - The second number in the Fibonacci sequence is given as F
_{1}= 1 - Fibonacci numbers follow a rule according to which,

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

- The third number in the Fibonacci sequence is given.

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

As we know, F_{0} = 0 and F_{1} = 1, the value of F_{2} = 0 + 1 = 1

- The Fibonacci sequence goes like 0, 1, 1, and so on.
- The rule for Fibonacci numbers, if explained in simple terms, says that every number in the sequence is the sum of two numbers preceding it in the sequence.

**How Can We Calculate Fibonacci Numbers?**

Let's learn calculating the Fibonacci numbers as per the rule learned in the above section.

The sequence is given as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144....

Let's see how the first thirteen terms come about in the sequence.

If we tabulate the calculation, we get

n | Term | F_{(n-1)} |
F_{(n-2)} |
F_{n} = F_{(n-1)} + F_{(n-2) ,} ( for n >1) |
---|---|---|---|---|

0 | First | - | - | F_{0} = 0 |

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

Inferences obtained from the above table are:

- The first term is always 0, and the second term is always 1.
- The output obtained in the 4
^{th}column is the summation of the values in the 2^{nd}column and 3^{rd}column which represent the two preceding numbers.

Now we understand how to calculate Fibonacci numbers.

Let's see how a chart of the first 60 Fibonacci Numbers in the Fibonacci sequence looks like:

**Here is a Fibonacci numbers calculator.**

**Enter the term of your choice, and the calculator will find the value of the term.**

Did you know:

- When any two consecutive Fibonacci numbers are taken, their ratio is very close to 1.618034
- Let's take a random example of two consecutive numbers,

Let A = 13, B = 21

- Let's divide B by A. We get

\[ 21 \div 13 = 1.625 \]

- True, isn't it?
- This ratio of successive Fibonacci numbers is known as the Golden Ratio.
- We can calculate any Fibonacci number using this Golden Ratio as per the below given formula:

\( x_n = ((ɸ)^n - (1 - ɸ)^n) \div \sqrt{5} \) |

** **

- Find the seventieth, eightieth, and ninetieth term in the Fibonacci sequence using the Golden Ratio formula.

**Solved Examples**

Example 1 |

Leslie's teacher asked her to give two real-life examples of the Fibonacci sequence. Can you help her do so?

**Solution**

Leslie told her teacher that the Fibonacci sequence exists in daisy flowers and cherry plants.

Daisy and Cherry |

Example 2 |

Howard had rabbits on his farm and wanted to know how many rabbits he would have at the end of 5 months? How do you think he'll find that out?

**Solution**

Howard knew that rabbits give birth in the order of the Fibonacci sequence.

He knew a few facts about rabbits too:

- Newborns can't reproduce.
- One-month olds can't reproduce.
- Mature pairs can only reproduce.

Month | Newborns | One-month olds | Mature pairs | Total pairs |
---|---|---|---|---|

1 | 1 | 0 | 0 | 1 |

2 | 0 | 1 | 0 | 1 |

3 | 1 | 0 | 1 | 2 |

4 | 1 | 1 | 1 | 3 |

5 | 2 | 1 | 2 | 5 |

Howard will have 5 pairs of rabbits at the end of 5 months. |

Example 3 |

Help Linda calculate the value of the 12^{th } and the 13^{th }term of the Fibonacci sequence given that the 9^{th} and 10^{th} terms in the sequence are 21 and 34.

**Solution**

Linda would have calculated the 12^{th} and the 13^{th} term of the Fibonacci sequence in the following way:

11^{th} term will be obtained by summation of 9^{th} and 10^{th} term which is given by \( 21 + 34 = 55 \)

12^{th} term will be given as \( 34 + 55 = 89 \)

13^{th} term will be given as \( 55 + 89 = 144 \)

Hence, Linda found the answer.

Linda found that the 12^{th} term is 89 and the 13^{th} term is 144. |

Example 4 |

Can Elijah find the sum of the first ten numbers in the Fibonacci sequence? How will he do it?

**Solution**

The Fibonacci sequence is given as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34

On summation of numbers in the sequence, we get

Sum = 0+1+1+2+3+5+8+13+21+34

The value of sum = 88

Elijah found that the sum of the first 10 numbers is 88. |

Example 5 |

Can Max prove, using the Golden ratio, that the 4th term in the sequence is 2? How do you think he'll prove it?

**Solution**

Max will use the following steps to prove that the 4th term in sequence is 2:

x_{n} = \( ((ɸ)^n - (1 - ɸ)^n) \div \sqrt{5} \)

Here, n = 3, which represents the 4th term of the sequence.

Hence,

x_{3 }= \( ((1.618)^3 - (1 - 1.618)^3) \div \sqrt{5} \)

x_{3} = 1.99

1.99 is approximately equal to 2.

Hence proved. |

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

We hope you enjoyed learning about Fibonacci numbers with the simulations and practice questions. Now you will be able to easily solve problems on Fibonacci numbers in nature, Fibonacci numbers calculator, Fibonacci sequence facts, and the first 60 Fibonacci numbers.

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**Frequently Asked Questions (FAQs)**

## 1. Why is the Fibonacci sequence so important?

The Fibonacci sequence has practical applications in the movement of financial markets. It also exists in nature.

## 2. Is 0 a Fibonacci number?

Yes, 0 is a Fibonacci number.

## 3. Is the Fibonacci sequence infinite?

Yes, the Fibonacci sequence is infinite.

## 4. Is there a formula for Fibonacci?

Yes, there is a formula for Fibonacci. The formula is given as xn = \( ((ɸ)^n - (1 - ɸ)^n) \div \sqrt{5} \)