Fibonacci Sequence Formula Recursive Rule and Solved Examples
Leonardo Pisano, also called Fibonacci (Fibonacci stands for filius Bonacii) was born in Pisa during 1170. His father, Guglielmo dei Bonacci, a rich Pisan merchant and a representative of the merchants of the Republic of Pisa present in the area of Bugia in Cabilia (in modern north-eastern Algeria), after 1192, he took his son with him, because he wanted Leonardo to become a merchant like him.
So the solution to this problem is the famous “Fibonacci sequence” which is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… a sequence of numbers where each of them is the sum of the previous two numbers.
What is Fibonacci Number?
Let’s know what a fibonacci number is. A Fibonacci number is known to be a series of numbers where each of the Fibonacci numbers is found by adding the two preceding numbers. It also means that the next number in the series is the addition of the two previous numbers. Let us take the first two numbers in the series as 0 and 1. So by adding 0 and 1, we will get the third number as 1, and by adding the second and the third number which is 1 and 1, we get the fourth number to be 2, and likely, the process goes on and on.
So, we get the Fibonacci series as 0, 1, 1, 2, 3, 5, 8, ……. Therefore, the obtained series is called to be the Fibonacci number series. Now we know what the fibonacci number is.
What is the Fibonacci Series?
We will discuss what is the Fibonacci series. The list of the numbers of Fibonacci Sequence is given below. This list is created by using the Fibonacci formula, which is also mentioned in the above definition.
The Fibonacci sequence is a set of the numbers that starts with a one or a zero, which are followed by a one, and then proceeds based on the rule that each of the numbers (called a Fibonacci number) equals to the sum of the preceding two numbers. If the Fibonacci sequence is put up as F (n), where n is the first term in the sequence, so the following equation obtains for n = 0, where the first two terms are put up as 0 and 1 by the convention:
Fibonacci Series Formula
F (0) equals 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 ...
In some of the texts, it is mandatory to use n = 1. So, the first two terms are defined as 1 and 1 by default, and we see:
F (1) equals 1, 1, 2, 3, 5, 8, 13, 21, 34 …
Fibonacci Numbers, Fibonacci Formula
The sequence of the Fibonacci numbers can be written as:
Fn = Fn-1 + Fn-2
Where Fn is the nth term or the number
Fn-1 is the (n-1)th term
Fn-2 is the (n-2)th term
From the equation above, we can also write up the definition as the next number in the sequence, and is the sum of the previous two numbers which are present in the sequence, starting from 0 and 1. So, let's make a table to find the next term of the Fibonacci sequence, using the above Fibonacci formula.
Fibonacci Series List
In the above table, we can see that the numbers in each of the columns are relational, and also diagonally the numbers are the same in all three columns.
Fibonacci Number Properties
The following points below are the properties for the Fibonacci numbers:
In the Fibonacci series, let us take any of the three consecutive numbers and add those numbers. When we divide the result by 2, we will get the three numbers. Example, let’s take 3 consecutive numbers like 1, 2, 3. when we add these numbers that are 1+ 2+ 3 = 6. When 6 is divided by 2, the result will be 3.
Take four of the consecutive numbers other than “0” in the Fibonacci series. Then multiply the outer number and also then multiply the inner number. When we subtract these numbers, we will get the difference “1”. For example, if we take 4 consecutive numbers like 2, 3, 5, 8. Multiply the outer numbers (i.e) 2(8) and then multiply the inner number which is 3(5). Now we subtract these two numbers 16-15 =1. So, the difference is 1.
Applications for the Fibonacci numbers will also include the computer algorithms like the Fibonacci search technique and then the Fibonacci heap data structure, and the graphs called as Fibonacci cubes which are used for interconnecting the parallel and distributed systems.
Limitations of Using the Fibonacci Numbers and Levels
The usage of the Fibonacci studies is very subjective since the trader has to use the highs and the lows of their choice. Whichever highs and lows that are chosen will also affect the results the trader gets.
Another argument against the Fibonacci number trading methods is that there are also so many of these levels that the market has to bounce or to change the direction near one of them, making the indicator look significantly in hindsight. The problem is that it is very difficult to know which of the numbers or the level will be important in real-time or in the future.
FAQs on Fibonacci Numbers Explained with Definition and Pattern
1. What are Fibonacci numbers?
The Fibonacci numbers are a sequence of numbers where each term is the sum of the two previous terms, starting with 0 and 1.
- The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ...
- Each term satisfies the rule: next term = sum of previous two terms.
- It is one of the most famous sequences in mathematics and appears in algebra, number theory, and nature.
2. What is the formula for the Fibonacci sequence?
The recursive formula for the Fibonacci sequence is F(n) = F(n−1) + F(n−2) with initial terms F(0) = 0 and F(1) = 1.
- This means each term is the sum of the two previous terms.
- Example: F(5) = F(4) + F(3) = 3 + 2 = 5.
- This is called a recurrence relation.
3. What is the nth term formula for Fibonacci numbers?
The nth term of the Fibonacci sequence is given by Binet’s Formula: F(n) = (φⁿ − ψⁿ)/√5, where φ = (1 + √5)/2 and ψ = (1 − √5)/2.
- φ is the golden ratio ≈ 1.618.
- This formula allows direct calculation without finding previous terms.
- Example: F(5) = 5.
4. How do you find the next Fibonacci number?
To find the next Fibonacci number, add the two previous numbers in the sequence.
- Example sequence: 3, 5
- Next term = 3 + 5 = 8
- This rule applies to every pair of consecutive Fibonacci numbers.
5. What is the golden ratio in Fibonacci numbers?
The golden ratio is the limit of the ratio of consecutive Fibonacci numbers and equals (1 + √5)/2 ≈ 1.618.
- As n increases, F(n+1)/F(n) approaches 1.618.
- Example: 21/13 ≈ 1.615, 34/21 ≈ 1.619.
- This connection links Fibonacci numbers to geometry, art, and nature.
6. What are the first 10 Fibonacci numbers?
The first 10 Fibonacci numbers starting from 0 are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
- Each term is obtained by adding the previous two terms.
- The sequence starts with 0 and 1.
- The 10th term (if counting from F(0)) is 34.
7. How do you find the sum of the first n Fibonacci numbers?
The sum of the first n Fibonacci numbers is given by F(n+2) − 1.
- This formula provides a quick way to calculate the total.
- Example: Sum of first 5 terms (0,1,1,2,3) = F(7) − 1 = 13 − 1 = 12.
- This identity is useful in algebra and sequence problems.
8. Where are Fibonacci numbers used in real life?
Fibonacci numbers appear in nature, computer science, mathematics, and financial markets.
- In nature: patterns of petals, pinecones, and shells follow Fibonacci spirals.
- In algorithms: used in dynamic programming examples.
- In finance: Fibonacci retracement levels are used in technical analysis.
9. Are Fibonacci numbers even or odd?
Fibonacci numbers follow a repeating pattern of odd, odd, even.
- Example: 1 (odd), 1 (odd), 2 (even), 3 (odd), 5 (odd), 8 (even).
- Every third Fibonacci number is even.
- This pattern repeats throughout the sequence.
10. What is the difference between Fibonacci sequence and geometric sequence?
The Fibonacci sequence is formed by adding the two previous terms, while a geometric sequence is formed by multiplying by a constant ratio.
- Fibonacci rule: F(n) = F(n−1) + F(n−2).
- Geometric rule: a, ar, ar², ar³, ...
- In Fibonacci, differences change; in geometric sequences, ratios stay constant.
