Fibonacci Series

Fibonacci Series

A Fibonacci series, also called a Fibonacci number, is a sequence of numbers in which each Fibonacci number is formed by adding the two preceding numbers. It denotes that the next number in the series is the result of adding two preceding ones. These numbers were created to represent positive numbers in a sequence that followed a predetermined pattern. Let 0 and 1 be the first two integers in the series. By adding 0 and 1, we get 1 as the third number. The fourth number is obtained by adding the 2nd and 3rd numbers that are 1 and 1, and the process continues in such a manner. The recurrence relation describes the Fibonacci series’ list of numbers. The Fibonacci series :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ……..,∞.

Fibonacci Series Properties

The following are the properties of the Fibonacci series:

  • Take three consecutive numbers in the Fibonacci series and add them together. The third number is acquired by dividing the result by 2. Let’s take three consecutive numbers, such as 1, 2, and 3. When these numbers are added together, you will get (1+ 2+ 3) When you divide 6 and 2, you will get 3.
  • Take four consecutive numbers that aren’t “0.” Multiply the outer number and the inner number together. The difference “1” is acquired by subtracting these values. Take four numbers in a row, such as 2, 3, 5, and 8. Multiply the surrounding numbers, such as 2(8), and the interior number, such as 3 (5). Subtract these two integers, the result will be 16-15 = 1.

Golden Ratio to Calculate Fibonacci Numbers

The Golden Ratio can be used to approximate the Fibonacci sequence. If the value of consecutive Fibonacci numbers increases, the ratio is very close to the Golden Ratio. We can find the Fibonacci numbers in the series this way. Approximately 1.618034 is the Golden Ratio. The φ symbol is frequently used to represent it.

For instance, the Fibonacci numbers 3 and 5 are two consecutive Fibonacci numbers.

The 5:3 ratio is as follows:

5/3 = 1.6666

Fibonacci Numbers Examples

Let’s solve an example to understand the Fibonacci series.

Question 1:Write the first 6 Fibonacci numbers starting from 0 and 1.

Solution: As we know, the formula for Fibonacci sequence is;

Since the first term and second term are known to us, i.e. 0 and 1. Thus,

= 0 and  = 1

Hence,

Third term,  =  +  = 0 + 1 = 1

Fourth term,  = +  = 1 + 1 = 2

Fifth term,  = +  = 1+2 = 3

Sixth term,  =  +  = 3 + 2 = 5

So, the first six terms of the Fibonacci sequence are 0,1,1,2,3,5.

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