Fibonacci Numbers

Question 1

How to calculate the Fibonacci sequence?

Answer

Numbers from the Fibonacci sequence, noted $ F_n $ ou $ F(n) $, are equal to the addition of the 2 previous terms, they follow the recurrence formula: $$ F(n) = F(n-1) + F(n-2) $$ that can be also written $$ F(n+2) = F(n) + F(n+1) $$

To initiate the sequence, by default, the two first terms are $ F(0) = 0 $ and $ F(1) = 1 $

Example: $ F_2 = F_0+F_1 = 0+1 = 1 $
$ F_3 = F_1+F_2 = 1+1 = 2 $
$ F_{10} = F_8+F_9 $, etc.

Any other values of $ F_0 $ and $ F_1 $ will produce different Fibonacci sequences.

Question 2

What are the first terms of the Fibonacci sequence?

Answer

Fibonacci sequence first numbers are:

F(0)=	0
F(1)=	1
F(2)=	1
F(3)=	2
F(4)=	3
F(5)=	5
F(6)=	8
F(7)=	13
F(8)=	21
F(9)=	34
F(10)=	55

For the next Fibonacci terms, use the calculator above.

Question 3

How to compute the previous Fibonacci term?

Answer

Each term in the sequence is equal to the previous multiplied by approximately $ \varphi = 1.618 $ (golden number).

Example: $ F(10) = 55 $, $ 55/\varphi \approx 33.99 $ and in fact $ F(9) = 34 $

Question 4

What is the Fibonacci Rabbits' problem?

Answer

The rabbits' problem is a problem proposed by Leonardo Fibonacci in 1200.

There is a rabbit couple (male + female) and every month a couple breeds and gives birth to a new pair of rabbits which in turn can reproduce itself after 2 months. How many rabbits will be born after X months?

In the beginning there is 1 couple then

1 month	1 couple
two months	2 couples
three months	3 couples
4 months	5 couples
5 months	8 couples
6 months	13 couples
7 months	21 couples
8 months	34 couples

Each month, the total number of rabbits is equal to the sum of the numbers of the previous two months because it is the number of existing rabbits (the previous month) plus the number of babies born from rabbits couples who have at least two months (hence the number of rabbits 2 months ago). The numbers found are the numbers of the Fibonacci sequence.

Fibonacci Numbers