Fibonacci Sequence

The Fibonacci sequence is one of the most well-known sequences in mathematics. Named in honor of the Italian mathematician Leonardo of Pisa, who was known as Fibonacci, the sequence was first introduced in his 1202 book "Liber Abaci" (The Book of Calculation). The sequence arises in the context of a problem about rabbit population growth and has since found applications in various fields such as mathematics, computer science, biology, art, and finance.

1.0Fibonacci Sequence

The Fibonacci sequence is a sequence of numbers in which each number (called a Fibonacci number) is the sum of the two preceding ones. Typically, the sequence starts with 0 and 1. That is,

${\mathrm{F}}_0=0,{\mathrm{F}}_1=1$

and for $\mathrm{n}\ge 2,F_n=F_{n-1}+F_{n-2}$

The Fibonacci sequence starts as follows: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and so on.

2.0What is Fibonacci Sequence?

The Fibonacci sequence is a numerical series of numbers in which each number is the addition of the two preceding numbers. It progresses as 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, and extends infinitely. Each number in this sequence is called a Fibonacci Number.

Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, …

Here, the first term is 0, second term is 1 so third term is obtained by adding first term and second term.

0 + 1 = 1

So, now the fourth term will be added to the second and third term.

1 + 1 = 2 

So, now the fifth term will be added to the third term and fourth term.

1 + 2 = 3 and so on.

3.0Fibonacci Sequence Formula

The Fibonacci sequence formula is a recursive relation where each term is obtained by adding the two preceding ones. It is defined as:

F₀ = 0

F₁ = 1

$F_n=F_{n-1}+F_{n-2}\text{ for }n\ge 2$

Additionally, the nth Fibonacci number can be calculated using Binet's formula:

${\mathrm{F}}_{\mathrm{n}}=\frac{{\varphi }^n-(1-\varphi {)}^n}{\sqrt{5}}$where $\varphi$ (the golden ratio) is $\frac{1+\sqrt{5}}{2}$ .

4.0Fibonacci Sequence List

Here is the list of the first 20 Fibonacci Sequences.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181

In tabular form

5.0Golden Ratio to Calculate Fibonacci Sequence

Golden Ratio: The ratio of successive Fibonacci numbers converges to the golden ratio:

${\lim }_{n\to \infty }\frac{F_{n+1}}{F}=\varphi =\frac{1+\sqrt{5}}{2}\approx 1.618033988749895\dots$

It is deeply connected to the Fibonacci sequence. The relationship can be seen in Binet's formula, which provides a closed-form expression for the n-th Fibonacci number. Binet's formula is:

$F_n=\frac{{\varphi }^n-(1-\varphi {)}^n}{\sqrt{5}}$

where $\varphi =\frac{1+\sqrt{5}}{2}\approx 1.618033988749895$ and $1-\varphi =\frac{1-\sqrt{5}}{2}\approx -0.618033988749895$ .

This formula allows for the direct computation of any Fibonacci number without needing to calculate all the preceding terms. As n increases, $(1-\varphi {)}^{\mathrm{n}}$ becomes very small, making F_n approximately equal to $\frac{{\varphi }^n}{\sqrt{5}}$ , illustrating the close relationship between the Fibonacci sequence and the golden ratio.

6.0Fibonacci Sequence in Real Life

The Fibonacci sequence appears in various natural and human-made phenomena. Some examples include:

1. Nature: The pattern of leaf arrangement on a stem and the pattern of branching of trees, the flowering of artichokes, and the arrangement of seeds in a sunflower follow the Fibonacci sequence. Pinecones, pineapples, and shells also exhibit Fibonacci spirals.
2. Art and Architecture: the golden ratio, derived from the Fibonacci sequence, is utilized to achieve visually appealing designs. Renowned artworks and architectural masterpieces, such as the Parthenon and Leonardo da Vinci's "Vitruvian Man," integrate this ratio to enhance aesthetic harmony and balance.
3. Finance: Fibonacci retracement levels are used in technical analysis to predict future movements in stock prices based on past high and low points.
4. Computer Science: Algorithms and data structures, such as Fibonacci heaps and Fibonacci search techniques, utilize the properties of Fibonacci numbers to optimize performance.
5. Mathematics: It is used in number theory and to solve problems involving recursive sequences and relationships. It also has connections to the golden ratio, which appears in various mathematical contexts.
6. Computer Science: Fibonacci numbers are used in algorithms and data structures to improve efficiency. For example, the Fibonacci search technique and Fibonacci heaps optimize search and priority queue operations.

7.0Examples on Fibonacci Sequence

Example 1: Calculate F₇. 

Solution: Using the recursive definition:

$F_0=0,F_1=1$

$F_n=F_{n-1}+F_{n-2}\text{ for }n\ge 2$

Calculate F₇:

${\mathrm{F}}_2={\mathrm{F}}_1+{\mathrm{F}}_0=1+0=1$

${\mathrm{F}}_3={\mathrm{F}}_2+{\mathrm{F}}_1=1+1=2$

${\mathrm{F}}_4={\mathrm{F}}_3+{\mathrm{F}}_2=2+1=3$

${\mathrm{F}}_5={\mathrm{F}}_4+{\mathrm{F}}_3=3+2=5$

${\mathrm{F}}_6={\mathrm{F}}_5+{\mathrm{F}}_4=5+3=8$

${\mathrm{F}}_7={\mathrm{F}}_6+{\mathrm{F}}_5=8+5=13$

So, F₇ = 13.

Example 2: Calculate F₆ using Binet's Formula correctly.

Solution: Binet's Formula for the Fibonacci sequence is:

$F_n=\frac{{\varphi }^n-{\psi }^n}{\sqrt{5}}{\mathrm{F}}_{\mathrm{n}}=\frac{{\varphi }^n-(1-\varphi {)}^n}{\sqrt{5}}$

where $\varphi =\frac{1+\sqrt{5}}{2}\approx 1.618$ (the golden ratio) and $1-\varphi =\frac{1-\sqrt{5}}{2}\approx -0.618$ .

Calculation for F₆:

1. Calculate ${\varphi }^6$ and $(1-\varphi {)}^6$ :

${\varphi }^6\approx 1.618^6\approx 17.944$

$(1-\varphi {)}^6\approx (-0.618{)}^6\approx 0.088$

2. Apply Binet's Formula:

${\mathrm{F}}_6=\frac{{\varphi }^6-(1-\varphi {)}^6}{\sqrt{5}}$

$=\frac{17.944-0.088}{\sqrt{5}}$

$=\frac{17.856}{2.236}$

= 7.984

3. Round to the nearest whole number:

${\mathrm{F}}_6\approx 8$

Therefore, F₆ calculated using Binet's Formula is approximately 8. This calculation demonstrates the accurate application of Binet's Formula for finding Fibonacci numbers.

8.0Fibonacci Sequence Practice Problems

1. Calculate the 15th Fibonacci number using the recursive definition.
2. Prove that the sum of the first n Fibonacci numbers is F_n+2 – 1.
3. Show that every third Fibonacci number is even, i.e., F_3k is even for all integers $\mathrm{k}\ge 0.$

9.0Sample Questions on Fibonacci Sequence

Q1. What is the formula for the Fibonacci sequence?

Ans: The Fibonacci sequence can be defined by the recurrence relation:

F(n)=F(n-1)+F(n-2)

with initial conditions:

$F(0)=0,F(1)=1$

Q2. What is Binet's formula?

Ans: Binet's formula provides a closed-form expression to compute the nth Fibonacci number without recursion:

${\mathrm{F}}_{\mathrm{n}}=\frac{{\varphi }^n-(1-\varphi {)}^n}{\sqrt{5}}$

where $\varphi =\frac{1+\sqrt{5}}{2}$ (the golden ratio) and $1-\varphi =\frac{1-\sqrt{5}}{2}$

Q3. How is the Fibonacci sequence related to the golden ratio?

Ans: As the Fibonacci sequence progresses, the ratio of consecutive Fibonacci numbers

$\left(\frac{F(n)}{F(n-1)}\right)$ converges towards the golden ratio $\varphi \approx 1.618$. This relationship underlies many patterns in nature and art.