Fibonacci Sequence | Formula, Spiral, Properties, Examples

Fibonacci sequence is a series of numbers where each number is the sum of the two numbers that come before it. The numbers in the Fibonacci sequence are known as Fibonacci numbers and are usually represented by the symbol Fₙ. Fibonacci sequence numbers begins with the following 14 integers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233.

Fibonacci sequence in nature can be seen many places, such as in the growth of trees. As the tree grows, the trunk grows and spirals outward. The branches also follow the Fibonacci sequence, starting with one trunk that splits into two, then one of those branches splits into two, and so on.

Let’s learn about Fibonacci Sequence in detail, including Fibonacci sequence formula, properties, and examples.

Fibonacci Sequence

Fibonacci Sequence is a series of numbers in which each number, starting with 0 and 1, is generated by adding the two preceding numbers. It forms the sequence of 0, 1, 1, 2, 3, 5, 8, 13, 21,… Each number in the Fibonacci series is the sum of the two numbers before it.

Fibonacci sequence is a special sequence of numbers that starts from 0 and 1 and then the next terms are the sum of the previous terms and they go up to infinite terms. This sequence is represented as, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

Fibonacci Sequence Formula

Fibonacci formula is used to find the nth term of the sequence when its first and second terms are given.

The nth term of the Fibonacci Sequence is represented as F_n. It is given by the following recursive formula,

F_n = F_n-1 + F_n-2

where,

• n > 1
• First term is 0 i.e., F₀ = 0
• Second term is 1 i.e., F₁ = 1

Using this formula, we can easily find the various terms of the Fibonacci Sequence. Suppose we have to find the 3rd term of this Sequence then we would require the 2nd and the 1st term according to the given formula, then the 3rd term is calculated as,

• F₃ = F₂ + F₁ = 1 + 0 = 1

Thus, the third term in the Fibonacci Sequence is 1, and similarly, the next terms of the sequence can also be found as,

• F₄ = F₃ + F₂ = 1 + 1 = 2
• F₅ = F₄ + F₃ = 2 + 1 = 3

and so on.

Check: Nth Fibonacci Number

List of first 20 numbers of Fibonacci sequence are represented in the table below.

Terms of Fibonacci Sequence
F0 = 0	F10 = 45
F1 = 1	F11 = 89
F2 = 1	F12 = 134
F3 = 2	F13 = 223
F4 = 3	F14 = 377
F5 = 5	F15 = 610
F6 = 8	F16 = 987
F7 = 13	F17 = 1597
F8 = 21	F18 = 2584
F9 = 34	F19 = 4181

• Fibonacci Sequences have infinite terms.
• By closely observing the table we can say that F_n = F_n-1 + F_n-2 for every n > 1.

Read More,

• Recursive Formula

Fibonacci Spiral

A Fibonacci spiral is a geometric pattern derived from the Fibonacci sequence.

This pattern is created by drawing a series of connected quarter-circles inside a set of squares that have their side according to the Fibonacci sequence. We start the construction of the spiral with a small square, followed by a larger square that is adjacent to the first square. The side of the next square is the sum of the two previous squares, and so on.

Each quarter-circle fits perfectly within the next square in the sequence, creating a spiral pattern that expands outward infinitely.

After studying the Fibonacci spiral we can say that every two consecutive terms of the Fibonacci sequence represent the length and breadth of a rectangle.

Let us now calculate the ratio of every two successive terms of Fibonacci sequence and see the result.

• F₂/F₁ = 1/1 = 1
• F₃/F₂ = 2/1 = 2
• F₄/F₃ = 3/2 = 1.5
• F₅/F₄ = 5/3 = 1.667
• F₆/F₅ = 8/5 = 1.6
• F₇/F₆ = 13/8 = 1.625
• F₈/F₇ = 21/13 = 1.615
• F₉/F₈ = 34/21 = 1.619
• F₁₀/F₉ = 55/34 = 1.617
• F₁₁/F₁₀ = 89/55 = 1.618 (Golden Ratio)

Thus, we see that for the larger term of the Fibonacci sequence, the ratio of two consecutive terms forms the Golden Ratio.

Check: A Fibonacci spiral is a geometric pattern derived from the Fibonacci sequence.

Golden Ratio

The golden ratio is a ratio between two numbers that is approximately 1.618. It is represented by the Greek letter phi “Φ”, and is also known as the golden number, golden proportion, or the divine proportion. We have observed that by taking the ratio of two consecutive terms of the Fibonacci Sequence we get the ratio called the “Golden Ratio“.

Φ = F_n/F_n-1

Golden Ratio Formula

The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. The formula for the golden ratio is ϕ = 1 + (1/ϕ).

We can calculate the golden ratio of Fibonacci sequence using the formula

• F₁₁ = 89
• F₁₀ = 55

The ratio of these two terms are,

F₁₁/F₁₀ = 89/55 = 1.618 (Golden Ratio)

Here the ratio so obtained is called the golden ratio. {Φ = 1.618 (Golden Ratio)}

We can also calculate the Fibonacci number using the golden ratio by the formula:

F_n = (Φⁿ – (1-Φ)ⁿ)/√5

where, Φ is the Golden ratio.

Check: Fibonacci Series

Fibonacci Series in Pascal’s Triangle

Pascal’s triangle is a triangular array of numbers that begins with 1 at the top and 1s running down the two sides of a triangle. Each new number is the sum of the two numbers above it.

Pascal’s triangle contains the Fibonacci sequence, which is an infinite sequence of numbers that are generated by adding the two previous terms in the sequence. The Fibonacci sequence in Pascal’s triangle is 1, 1, 2, 3, 5, 8, 13, 21, and so on.

Fibonacci Numbers in Pascal’s Triangle

To find the Fibonacci series in Pascal’s triangle, you can draw “shallow diagonals” from the top to the bottom of the triangle. The sum of the diagonals of Pascal’s triangle is equal to the corresponding Fibonacci sequence term.

Fibonacci Sequence in Real Life

• The Fibonacci sequence is evident in nature’s patterns, including plant growth, shell spirals, and leaf arrangements.
• It influences architectural design, with structures like the Parthenon and spiral staircases adhering to Fibonacci ratios.
• Artists use Fibonacci proportions to create aesthetically pleasing compositions.
• In financial markets, Fibonacci retracement levels are utilized for technical analysis to identify potential areas of support and resistance.

Important Facts about Fibonacci Numbers

• Fibonacci sequence is a series of numbers where each number is the sum of the two numbers that come before it. For example, 0 + 1 = 1, 1 + 1 = 2, and 1 + 2 = 3.
• Fibonacci sequence is a never-ending series of numbers.
• Fibonacci sequence is described by the mathematical equation: Fn+2 = Fn+1 + Fn.
• Fibonacci sequence was first described in Indian mathematics around 200 BC.
• The sequence is named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci.
• Fibonacci sequence is used in many applications, including computer algorithms and graphs.
• Golden ratio of 1.618 is derived from the Fibonacci sequence.
• Many things in nature have dimensional properties that adhere to the golden ratio of 1.618.
• November 23 is Fibonacci Day as it forms the first 4 digits of Fibonacci numbers 11/23.

Fibonacci Sequence Properties

Important properties of Fibonacci Sequence are:

• We can easily calculate the Fibonacci Numbers using the Binet Formula:

F_n = (Φⁿ – (1-Φ)ⁿ)/√5

where Φ is called Golden Ratio and its value is, Φ ≈ 1.618034.

Using this formula we can easily calculate the nth term of the Fibonacci sequence as, for

F3₄ = (Φ⁴ – (1-Φ)⁴)/√5 = ({1.618034}⁴– (1-1.618034)⁴)/√5 = 3

• For larger terms the ratio of two consecutive terms of the Fibonacci Sequence converges to the Golden Ratio.

This can be understood by the table added below,

Fibonacci Sequence Property and Golden Rule
Term of Fibonacci Sequence (A)	Next Term of Fibonacci Sequence (B)	Ratio of two consecutive terms (B/A)
2	3	1.5
3	5	1.6
5	8	1.6
144	233	1.6180556
233	377	1.61802575
377	610	1.61803714

Thus, it is evident that as the number becomes larger their ratio converges close to the Golden Ratio (1.618034).

• Multiplying a term of Fibonacci Sequence with Golden Ratio gives the next term of the Fibonacci sequence as,

F₇ in Fibonacci Sequence is 13 then F₈ is calculated as,

F₈ = F₇(1.618034) = 13(1.618034) = 21.0344 = 21 (approx.)

Thus, the F₈ in the Fibonacci Sequence is 21.

• We can also calculate the Fibonacci Sequence for below zero numbers as,

F_-n = (-1)ⁿ⁺¹F_n

For example, F_-2 = (-1)²⁺¹F₂ = -1

• Fibonacci Numbers are used to define other mathematical concepts such as Pascal Triangle and Lucas Number.

Check: Fibonacci Sequence Formula

Fibonacci Sequence Examples

We have solved some questions on Fibonacci Sequence to help you consolidate your concepts.

Example 1: Find the 7th term of the Fibonacci sequence if the 5th and 6th terms are 3 and 5 respectively.

Solution:

Using the Fibonacci sequence recursive formula,

7th term = 6th term + 5th term

F₆ = 3 + 5 = 8

Thus, the 7th term of the Fibonacci Sequence is F₆ = 8

Example 2: If F₉ in the Fibonacci sequence is 34. Find the next term(F₁₀)

Solution:

We know that,

F_n = F_n-1 × Φ

where, Φ is golden ration and its value is 1.618034

F₉ = 34 × Φ

= 34 × (1.618034)

= 55.0131

= 55

Thus, the F₉ term in the Fibonacci Sequence is 55.

Example 3: Find the 10th term of the Fibonacci sequence if the 8th and 9th terms are 13 and 21 respectively.

Solution:

Using the Fibonacci sequence recursive formula,

10th term = 9th term + 8th term

F₉ = 13 + 21 = 34

Thus, the 10th term of the Fibonacci Sequence is F₉ = 34

Example 4: If F₁₂ in the Fibonacci sequence is 144. Find the next term(F₁₃)

Solution:

We know that,

F_n = F_n-1 × Φ

where, Φ is golden ration and its value is 1.618034

F₁₃ = 144 × Φ

= 144 × (1.618034)

= 232.996

= 233

Thus, the F₁₃ term in the Fibonacci Sequence is 233.

Practice Problems on Fibonacci Sequence

1. What is the next number in the Fibonacci sequence: 0, 1, 1, 2, 3, 5, …?

1. 6
2. 7
3. 8
4. 9

2. What is the sum of the first five Fibonacci numbers: 0, 1, 1, 2, 3?

1. 5
2. 7
3. 8
4. 10

3. In the Fibonacci sequence, if F(6) represents the 6th term, what is the value of F(6)?

1. 5
2. 8
3. 13
4. 21

4. What is the common ratio between consecutive Fibonacci numbers as you move further along the sequence?

1. 1.414
2. 1.618
3. 2.0
4. 3.142

5. Which Fibonacci number is known as the “golden ratio,” often denoted by the Greek letter phi (φ)?

1. 1.317
2. 1.318
3. 0.617
4. 1.618

6. What is the only even number in the first ten Fibonacci numbers?

1. 2
2. 5
3. 8
4. 13

7. If F(0) = 0 and F(1) = 1, what is the value of F(2)?

1. A) 0
2. B) 1
3. C) 2
4. D) 3

8. Which Fibonacci property leads to the appearance of Fibonacci sequence in nature, such as in the arrangement of leaves or seeds?

1. Prime property
2. Golden ratio property
3. Exponential growth property
4. Palindrome property

9. If the Fibonacci sequence starts with F(0) = 1 and F(1) = 2, what is the third term, F(2)?

1. 2
2. 3
3. 4
4. 5

10. What is the relationship between consecutive Fibonacci numbers as you move further along the sequence?

1. Addition
2. Subtraction
3. Multiplication
4. Division

Conclusion of Fibonacci Sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Named after Leonardo of Pisa, commonly known as Fibonacci, the sequence has fascinated mathematicians, scientists, and artists for centuries due to its intriguing properties and widespread occurrence in nature.

Fibonacci Sequence – FAQs

What is Fibonacci Sequence?

Fibonacci Sequence is the sequence of the number that is generated by adding the last two numbers of the term when the first term and the second term of the sequence are, 0 and 1.

What is Fibonacci Sequence Formula?

Formula for generating the Fibonacci Sequence is F_n = F_n-1 + F_n-2 where n > 1.

What is the sum of Fibonacci Sequence?

In Fibonacci Sequence after the first two terms each new term is the sum of the previous two terms. The following first 14 integers of the Fibonacci Sequence are, 0, 1, 1, 2, 3,5, 8, 13, 21, 34, 55, 89, 144, 233,…

What is Fibonacci Spiral?

A geometric pattern observed in the nature derived from the Fibonacci sequence is called the Fibonacci Spiral. This pattern is observed in the nature in various aspects.

How is Fibonacci Sequence Related to the Golden Ratio?

By closely observing the Fibonacci Sequence we see that the ratio of two consecutive terms of the Fibonacci Terms converges to the Golden Ratio.

What is formula of Fibonacci Sequence for nth term?

Formula to find the nth term of the Fibonacci Sequence is, F_n = F_n-1 + F_n-2 where n >1

Who discovered Fibonacci Sequence?

Fibonacci sequence was first discovered by the famous Italian mathematician “Leonardo Fibonacci” in the early 13th century. But in Indian literature, the Fibonacci sequence was mentioned in early 200 BC literature.

What is the application of Fibonacci Sequence?

Fibonacci sequence is used in fields like art, architecture, and nature due to its occurrence in patterns such as the Golden Ratio. It is also used in finance for predicting market trends and in computer science for algorithm design.