A Fibonacci number is defined as the sequence of positive numbers in math that follows a defined pattern. These Fibonacci numbers were introduced to represent the sequence of positive numbers in mathematics. Using the recurrence relation, we can generate the Fibonacci series number following sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ………, infinity. Fibonacci numbers are a series of infinitely repeating numbers, where each Fibonacci number is calculated by adding the 2 preceding ones. That is, the next number in the series is the result of adding the 2 previous ones.
We can take the first two numbers in the series as 0 and 1. By adding 0 and 1, we can obtain the third number, which is 1; then by adding the second and third numbers (i.e. 1 and 1) we can obtain the fourth number, 2, and so on. Therefore we get the following Fibonacci number series starting from 0 and continuing till infinity: 0, 1, 1, 2, 3, 5, 8, ……….The series is called the Fibonacci number series. It can also be obtained from the Pascal triangle.
Properties of Fibonacci numbers:
- If you add three consecutive numbers to each other in the Fibonacci series, when you divide the result by 2, you’ll get the three numbers. When you add 3 consecutive numbers, such as 1, 2, 3, you get 6, and dividing 6 by 2 yields 3, so 3 is the result.
- Add four consecutive Fibonacci numbers, except “0”. You can multiply the outer and inner numbers simultaneously, and when you subtract these numbers, you get the difference “1”. Take 4 numbers such as 2, 3, 5, 8, then multiply the outer number (2 & 8 in this example) and the inner number (3 & 5 in this example). Now subtract the two numbers, 16-15 = 1, to get the difference of 1.
Aside from applications for Fibonacci numbers, there are also computer algorithms, including the Fibonacci search algorithm and the Fibonacci heap data structure, and the graph called a Fibonacci cube, used to connect systems in parallel and distributed mode. It is important to know that Fibonacci numbers play an important role in financial analysis. They can calculate useful ratios and percentages which are used by business houses.
The formula for Fibonacci numbers:
As an example, the Fibonacci sequence can be represented as:
F n = F n-1 + F n-2
Where Fn is the nth term or the number
F n-1 is the (n-1)th term
F n-2 is the (n-2)th term
Based upon the equation, we can say that the number to come is the sum of the two numbers that have already appeared in the sequence, beginning with 0 & 1.
Examples:
- Write the first 8 Fibonacci numbers starting from 0 and 1.
Ans: As we know, the formula for the Fibonacci sequence is;
F n = F n-1 + F n-2
Since the first term and second term are known to us, i.e. 0 and 1. Thus,
F 0 = 0 and F 1 = 1
Hence,
Third term, F 2 = F 0 + F 1 = 0+1 = 1
Fourth term, F 3 = F 2 + F 1 = 1 + 1 = 2
Fifth term, F 4 = F 3 + F 2 = 1+2 = 3
Sixth term, F 5 = F 4 + F 3 = 3 + 2 = 5
Seventh term, F 6 = F 5 + F 4 = 5 +3 = 8
Eighth term, F 7 = F 6 + F 5 = 8 + 5 = 13
So, the first six terms of the Fibonacci sequence are 0,1,1,2,3,5,8,13.
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