# The Fibonacci Sequence Explained: Everything You Need To Know (2023)

### Key Points

• The Fibonacci Sequence is a numbers list that follows a pattern starting with 0.
• The Fibonacci Sequence is a series of numbers where each proceeding number is the sum of the two previous numbers
• (Fn) is short for Fibonacci Sequence.

One of the joys of mathematics is the discovery of a numbers list that mirrors patterns found in nature. These instances create a sense of belonging in the universe, a sense of some grand cosmic interconnectedness — it’s practically like magic, but rooted firmly in science and mathematics.

The Fibonacci Sequence is one such example. But what is It? How is it defined, and how is it created? What are some examples of it in nature, and what is it used for?

Let’s discuss all there is to know about these fascinating numbers listed below.

## What is the Fibonacci Sequence? Complete Explanation

The Fibonacci Sequence (Fn) is a numbers list that follows an interesting pattern: Starting with 0, then 1, then 1, then 2, then 3, and so on, each subsequent number in the sequence is the sum of the two preceding numbers added together.

It’s defined by what’s known as the recurrence relation, the formula for which is F0 = 0, F1 = 1, and Fn = Fn – 1 + Fn – 2 for n > 1. (In older iterations, 0 was skipped and the formula began at F1 = F2 = 1, with Fn = Fn – 1 + Fn – 2 being true for n > 2. Fibonacci himself began the formula at F1 = 1 and F2 = 2.)

First discovered in Sanskrit Indian mathematics as far back as 200 BC, the Fibonacci Sequence eventually got its name from the Italian mathematician Leonardo of Pisa — a.k.a. Fibonacci — who detailed the formula in his book Liber Abaci (1202).

In his book, the Fibonacci Sequence was used for describing the growth pattern of the rabbit population, where the sum of the formula was used for hypothesizing about a rabbit’s breeding pattern.

What’s so fascinating about this concept is that the formula often appears out of the blue in mathematics, often unexpectedly and often without trying to find it in the first place. It even appears in nature, such as in the pattern of branching in trees or the placement of a stem’s leaves.

## The Fibonacci Sequence: An Exact Definition

Simply put, the Fibonacci Sequence is a series of numbers where each proceeding number is the sum of the two previous numbers. While the sequence begins with some simple addition, you’ll need a calculator before too long. The first twenty numbers are as follows:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765.

## How Does the Fibonacci Sequence Work?

It works by the rules of a closed-form expression. This means that it’s defined by a linear recurrence that has constant coefficients. As the Fibonacci numbers continue, the ratio between the numbers converges. As they go on, they get incredibly close to the Golden Ratio — however, it’s not an exact match. The Fibonacci Sequence can be proved with a calculator via combinatorial arguments.

## How Do You Create the Fibonacci Sequence?

To create the Fibonacci Sequence, take a calculator and begin by adding 1 + 1 to get 2. Then, add 1 + 2 to get 3. Then add 2 + 3 to get 5. Continue for as long as your calculator can handle it — the numbers get quite big quite fast. Even after the numbers exceed the calculator’s abilities, the sequence theoretically continues infinitely.

## Who Created the Fibonacci Sequence?

The Fibonacci Sequence was first detailed not by mathematicians, but by Sanskrit prosody — a form of ancient poetry used as far back as 1200 BC. In this ancient poetic form, all patterns of long syllables were given two units of duration, while short syllables were given one unit of duration.

Counting these patterns of long and short syllables resulted in the first discovery of a Fibonacci Sequence.

Ancient Indian poet and mathematician, Pingala, told of this formula as far back as 450-200 BC. He referred to it as “misrau cha,” or “the two are mixed.” Scholars took this to mean that the long and short syllables created a unique pattern of Fibonacci numbers.

Talk of this concept didn’t emerge again until Indian mathematicians Virahanka in 700 AD and Hemachandra in 1150 AD.

Despite their early work with the sequence, the creation of the Fibonacci Sequence as we know it today is credited to Fibonacci’s aforementioned book Liber Abaci. It was first credited to Fibonacci by theorist Édouard Lucas in the 19th century — 3,000 years after its initial discovery in Sanskrit prosody.

## What Are the Applications of the Fibonacci Sequence?

There are several applications of this concept, both in mathematics and in nature. For one, the Fibonacci Sequence can be used to describe the totals of the shallow diagonals in Pascal’s Triangle. It can also be used to count {1, 2}-restricted compositions.

The sequence also has a lot of significance in analyzing Euclid’s algorithm to determine computational run-time. In nature, the Fibonacci Sequence appears in a pineapple’s fruitlets, an artichoke’s flowering pattern, a fern’s unfurling method, a pine cone’s arrangement, and a honeybee’s family tree. Many flowers also demonstrate this concept in their blooming formation.

## Examples of the Fibonacci Sequence in the Real World

Beyond these applications of the Fibonacci Sequence in math and nature, there are several other instances of the Fibonacci Sequence in the real world.

### Finance

In the world of finance and economics, there are several appearances of the Fibonacci Sequence. The Fibonacci retracement is used to analyze stock market trading, describing a phenomenon where stock prices fluctuate in a dependable pattern. The concept also appears in the economic growth model of Brock-Mirman.

### Music

Composer Joseph Schillinger created compositions using the Fibonacci Sequence as applied to melodies, with the intervals between notes being determined by the formula. Schillinger’s compositions are distinct from Golden Ratio music, which follows a similar idea down a different path.

### Converting Miles to Kilometers

Because the conversion factor from miles to kilometers is very close to the Golden Ratio, the Fibonacci Sequence can be used to get a general idea of miles to kilometers over longer distances (so long as each proceeding Fibonacci number is replaced by its successor).

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## The Fibonacci Sequence Explained: Everything You Need To Know FAQs (Frequently Asked Questions)

What is the Fibonacci Sequence?

The Fibonacci Sequence is a series of numbers where each proceeding number is the sum of the two previous numbers.

What are examples of the Fibonacci Sequence in nature?

The sequence can be seen in tree branch growth, pineapple fruitlet growth, floral blooming patterns, and even in the breeding pattern of rabbits.

How is Fibonacci used in real life?

The Fibonacci Sequence has many uses in real life, namely in the realms of computer science and finance.

Why is Fibonacci important?

The Fibonacci Sequence is important because of its close relation to the Golden Ratio and its frequent (sometimes eerie) recurrence across mathematics and in nature.

Is Fibonacci the Golden Ratio?

The Fibonacci Sequence is not the Golden Ratio, though the two are very closely related.

What is the formula of Fibonacci?

The formula for Fibonacci is as follows: F0 = 0, F1 = 1, and Fn = Fn – 1 + Fn – 2 for n > 1.

Are all spirals Fibonacci?

While many spirals are representatives of the Fibonacci Sequence, not all spirals fit this bill. Most are equiangular spirals.

Who discovered the Fibonacci Sequence?

While the sequence is largely credited to Italian mathematician Leonardo of Pisa, the discovery of the Fibonacci Sequence actually dates back to Sanskrit prosody as far back as 1200 BC.

What is the difference between the Fibonacci Sequence and the Golden Ratio?

While both the Fibonacci Sequence and the Golden Ratio result in a ratio of about 1.618, the Fibonacci Sequence differs from the Golden Ratio because the former’s reduction of intervals is not constant like the latter.

## FAQs

### The Fibonacci Sequence Explained: Everything You Need To Know? ›

The Fibonacci sequence is the series of numbers where each number is the sum of the two preceding numbers. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, …

What does the Fibonacci sequence explain? ›

The Fibonacci sequence is a set of integers (the Fibonacci numbers) that starts with a zero, followed by a one, then by another one, and then by a series of steadily increasing numbers. The sequence follows the rule that each number is equal to the sum of the preceding two numbers.

How is the Fibonacci sequence in everything? ›

A perfect example of this is the nautilus shell, whose chambers adhere to the Fibonacci sequence's logarithmic spiral almost perfectly. This famous pattern shows up everywhere in nature including flowers, pinecones, hurricanes, and even huge spiral galaxies in space.

What is the pattern of 1 1 2 3 5 8? ›

The Fibonacci sequence is a famous group of numbers beginning with 0 and 1 in which each number is the sum of the two before it. It begins 0, 1, 1, 2, 3, 5, 8, 13, 21 and continues infinitely.

How do you find the answer of a Fibonacci sequence? ›

Fibonacci Sequence = 0, 1, 1, 2, 3, 5, 8, 13, 21, …. “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.

What is so magical about Fibonacci numbers? ›

Fibonacci numbers appear in so many contexts in our lives and surroundings, for example, the number of the petals in a flower, the seed heads of a flower, paintings and a lot more. In fact, the beauty of a human face is based on Golden Ratio whose nth power forms the nth Fibonacci number.

Why is Fibonacci so important? ›

Fibonacci is remembered for two important contributions to Western mathematics: He helped spread the use of Hindu systems of writing numbers in Europe (0,1,2,3,4,5 in place of Roman numerals). The seemingly insignificant series of numbers later named the Fibonacci Sequence after him.

How do you use Fibonacci numbers in real life? ›

Many things in nature have dimensional properties that adhere to the golden ratio of 1.618, a quotient derived from the Fibonacci sequence. When applied to finance and trading, investors apply the Fibonacci sequence through four techniques including retracements, arcs, fans, and time zones.

Is life a Fibonacci sequence? ›

Rather, each step at life is a direct outcome of a few immediate past steps and all of these past steps combine together to yield a much bigger outcome. In other words, life is really a Fibonacci sequence that starts with 0.

How accurate are Fibonacci? ›

How Accurate Are Fibonacci Retracements? Some experts believe that Fibonacci retracements can forecast about 70% of market movements, especially when a specific price point is predicted. However, some critics say that these are levels of psychological comfort rather than hard resistance levels.

### What is Fibonacci of 7? ›

The notation that we will use to represent the Fibonacci sequence is as follows: f1=1,f2=1,f3=2,f4=3,f5=5,f6=8,f7=13,f8=21,f9=34,f10=55,f11=89,f12=144,…

What is the pattern of 1 8 27 64? ›

Cube Number Pattern

We get cubes when we multiply a number by itself thrice. An example of a cube number pattern is 1, 8, 27, 64, 125, 216… Here, the cubes of consecutive numbers from 1 to 6 form the sequence.

What are the next three numbers in this pattern 1 2 4 7 11 ___ ___ ___? ›

1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, ... Its three-dimensional analogue is known as the cake numbers.

What is the golden ratio explained simply? ›

You take a line and divide it into two parts – a long part (a) and a short part (b). The entire length (a + b) divided by (a) is equal to (a) divided by (b). And both of those numbers equal 1.618. So, (a + b) divided by (a) equals 1.618, and (a) divided by (b) also equals 1.618.

What is the difference between the golden ratio and the Fibonacci sequence? ›

The golden ratio is derived by dividing each number of the Fibonacci series by its immediate predecessor. In mathematical terms, if F(n) describes the nth Fibonacci number, the quotient F(n)/ F(n-1) will approach the limit 1.618... for increasingly high values of n. This limit is better known as the golden ratio.

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