#### Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. In the 2nd century, the Indian poet and mathematician Pingala introduced a type of poetry in which each line had more syllables than the previous one. Later, the Italian mathematician Leonardo Fibonacci used this concept while researching a problem in the 12th century, leading to the discovery of the sequence. The sequence is used in various fields, from financial forecasts to algorithms in computer systems.

## What is the Fibonacci Sequence?

The Fibonacci sequence begins with any two numbers forming a series. The third number is calculated as the sum of the first two, the fourth number is the sum of the second and third, and the sequence continues in this manner. For example, the following is a Fibonacci sequence:

3 - 5 - 8 - 13 - 21 - 34

The series can theoretically continue indefinitely under the same rule. It is generally accepted that the first two numbers of the sequence are 0 and 1. The rabbit problem that Leonardo Fibonacci encountered also starts with 0 and 1. The problem is as follows: In a region, there is only one male and one female juvenile rabbit. Assuming that juvenile rabbits become adults in the second month and that the rabbits do not die, how many rabbits would there be in this area after one year if a rabbit is born each month?

The answer to the problem can be easily calculated using the Fibonacci sequence. You can see the number series below:

1 - 1 - 2 - 3 - 5 - 8 - 13 - 21 - 34 - 55 - 89 - 144

## What is the Logic of the Fibonacci Sequence?

The Fibonacci sequence has a mathematical logic. Knowing the first two numbers of the sequence is sufficient to determine any number in the sequence. To know any subsequent number, it is necessary to know the two preceding numbers. Without knowing the first two numbers or the preceding two numbers, it is impossible to determine the numbers in the sequence.

The logic of the Fibonacci sequence reveals interesting properties in the numbers that appear in the sequence. For example:

- Numbers in the sequence that are multiples of three are divisible by 2, those that are multiples of four are divisible by 3, and those that are multiples of five are divisible by 5.
- Every consecutive number in the sequence after the second number is prime with respect to each other.
- The sum of the squares of any two consecutive numbers is another number in the sequence.
- The sum of any ten consecutive numbers is a multiple of 11.
- What is the Relationship Between the Fibonacci Sequence and the Golden Ratio?
- There is a relationship between the Fibonacci sequence and the golden ratio. When Leonardo Fibonacci discovered the Fibonacci sequence, he also calculated the golden ratio. If we divide the larger number by the smaller number in any two consecutive numbers of the sequence, we find a value close to the golden ratio. As the sequence progresses, the ratio of two numbers approaches the golden ratio.
- After the 13th number, it becomes equal to the golden ratio. This value is accepted as 1.618.

You can see the logic of the Fibonacci sequence in all mathematical numbers arranged in a sequence in nature, such as the distances between consecutive rings in a snail's shell or the number of seeds in the spiraling rings from the center outward in a sunflower. The ratio of consecutive numbers also gives the golden ratio. The golden ratio observed in nature has also been used in art.

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