The Fibonacci sequence is a type series where each number is the sum of the two that precede it. It starts from 0 and 1 usually. The Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The numbers in the Fibonacci sequence are also called Fibonacci numbers. In Maths, the sequence is defined as an ordered list of numbers that follow a specific pattern. The numbers present in the sequence are called the terms. The different types of sequences are arithmetic sequence, geometric sequence, harmonic sequence and Fibonacci sequence. In this article, we will discuss the Fibonacci sequence definition, formula, list and examples in detail.
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What is Fibonacci Sequence?
The Fibonacci sequence, also known as Fibonacci numbers, is defined as the sequence of numbers in which each number in the sequence is equal to the sum of two numbers before it. The Fibonacci Sequence is given as:
Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, ….
Here, the third term “1” is obtained by adding the first and second term. (i.e., 0+1 = 1)
Similarly,
“2” is obtained by adding the second and third term (1+1 = 2)
“3” is obtained by adding the third and fourth term (1+2) and so on.
For example, the next term after 21 can be found by adding 13 and 21. Therefore, the next term in the sequence is 34.
Fibonacci Sequence Formula
The Fibonacci sequence of numbers “F_{n}” is defined using the recursive relation with the seed values F_{0}=0 and F_{1}=1:
F_{n} = F_{n-1}+F_{n-2}
Here, the sequence is defined using two different parts, such as kick-off and recursive relation.
The kick-off part is F_{0}=0 and F_{1}=1.
The recursive relation part is F_{n} = F_{n-1}+F_{n-2}.
It is noted that the sequence starts with 0 rather than 1. So, F_{5 } should be the 6^{th} term of the sequence.
The list of first 20 terms in the Fibonacci Sequence is:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181. |
The list of Fibonacci numbers are calculated as follows:
F_{n} | Fibonacci Number |
0 | 0 |
1 | 1 |
2 | 1 |
3 | 2 |
4 | 3 |
5 | 5 |
6 | 8 |
7 | 13 |
8 | 21 |
9 | 34 |
… and so on. | … and so on. |
Golden Ratio to Calculate Fibonacci Numbers
The Fibonacci Sequence is closely related to the value of the Golden Ratio. We know that the Golden Ratio value is approximately equal to 1.618034. It is denoted by the symbol “φ”. If we take the ratio of two successive Fibonacci numbers, the ratio is close to the Golden ratio. For example, 3 and 5 are the two successive Fibonacci numbers. The ratio of 5 and 3 is:
5/3 = 1.6666
Take another pair of numbers, say 21 and 34, the ratio of 34 and 21 is:
34/21 = 1.619
It means that if the pair of Fibonacci numbers are of bigger value, then the ratio is very close to the Golden Ratio.
So, with the help of Golden Ratio, we can find the Fibonacci numbers in the sequence.
The formula to calculate the Fibonacci numbers using the Golden Ratio is:
X_{n} = [φ^{n} – (1-φ)^{n}]/√5
Where,
φ is the Golden Ratio, which is approximately equal to the value of 1.618
n is the nth term of the Fibonacci sequence.
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Fibonacci Sequence Solved Examples
Example 1:
Find the Fibonacci number when n=5, using recursive relation.
Solution:
The formula to calculate the Fibonacci Sequence is: F_{n} = F_{n-1}+F_{n-2}
Take: F_{0}=0 and F_{1}=1
Using the formula, we get
F_{2} = F_{1}+F_{0} = 1+0 = 1
F_{3} = F_{2}+F_{1} = 1+1 = 2
F_{4} = F_{3}+F_{2} = 2+1 = 3
F_{5} = F_{4}+F_{3} = 3+2 = 5
Therefore, the fibonacci number is 5.
Example 2:
Find the Fibonacci number using the Golden ratio when n=6.
Solution:
The formula to calculate the Fibonacci number using the Golden ratio is X_{n} = [φ^{n} – (1-φ)^{n}]/√5
We know that φ is approximately equal to 1.618.
n= 6
Now, substitute the values in the formula, we get
X_{n} = [φ^{n} – (1-φ)^{n}]/√5
X_{6} = [1.618^{6} – (1-1.618)^{6}]/√5
X_{6} = [17.942 – (0.618)^{6}]/2.236
X_{6} = [17.942 – 0.056]/2.236
X_{6} = 17.886/2.236
X_{6} = 7.999
X_{6} = 8 (Rounded value)
The Fibonacci number in the sequence is 8 when n=6.
Practice Problems
- Find the Fibonacci number when n = 4, using the recursive formula.
- Find the next three terms of the sequence 15, 23, 38, 61, …
- Find the next three terms of the sequence 3x, 3x + y, 6x + y, 9x + 2y, …
Frequently Asked Questions on Fibonacci Sequence
What is Fibonacci Sequence?
The Fibonacci sequence is the sequence of numbers, in which every term in the sequence is the sum of terms before it.
Why is Fibonacci sequence significant?
The Fibonacci sequence is significant, because the ratio of two successive Fibonacci numbers is very close to the Golden ratio value.
What are two different ways to find the Fibonacci Sequence?
The two different ways to find the Fibonacci sequence are
- Recursive Relation Method
- Golden Ratio Method
Write down the list of the first 10 Fibonacci numbers.
The list of the first 10 Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34.
What is the value of the Golden ratio?
The value of golden ratio is approximately equal to 1.618034…