Fibonacci Sequence

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.

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 F₀=0 and F₁=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 F₀=0 and F₁=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, F₅ should be the 6^th term of the sequence.

The list of first 20 terms in the Fibonacci Sequence is:

The list of Fibonacci numbers are calculated as follows:

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 = [φⁿ – (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.

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: F₀=0 and F₁=1

Using the formula, we get

F₂ = F₁+F₀ = 1+0 = 1

F₃ = F₂+F₁ = 1+1 = 2

F₄ = F₃+F₂ = 2+1 = 3

F₅ = F₄+F₃ = 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 = [φⁿ – (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 = [φⁿ – (1-φ)ⁿ]/√5

X₆ = [1.618⁶ – (1-1.618)⁶]/√5

X₆ = [17.942 – (0.618)⁶]/2.236

X₆ = [17.942 – 0.056]/2.236

X₆ = 17.886/2.236

X₆ = 7.999

X₆ = 8 (Rounded value)

The Fibonacci number in the sequence is 8 when n=6.

Practice Problems

1. Find the Fibonacci number when n = 4, using the recursive formula.
2. Find the next three terms of the sequence 15, 23, 38, 61, …
3. Find the next three terms of the sequence 3x, 3x + y, 6x + y, 9x + 2y, …