Fibonacci Sequence

The Fibonacci sequence was first found by an Italian named Leonardo Pisano Bogollo (Fibonacci). Fibonacci numbers are a sequence of whole numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, … This infinite sequence is called the Fibonacci sequence. Here each term is the sum of the two preceding ones, starting from 0 and 1. This has been termed “nature’s secret code”. We can spot the Fibonacci sequence in the spiral patterns of sunflowers, daisies, broccoli, cauliflowers, and seashells. Let us learn about the Fibonacci sequence and its interesting properties.

What is Fibonacci Sequence?

The Fibonacci sequence, in simple terms, says that every number in the Fibonacci sequence is the sum of two numbers preceding it in the sequence. The first 20 Fibonacci numbers are given as follows:

The Fibonacci sequence is represented as the spiral shown below. The spiral represents the pattern of the Fibonacci numbers. This spiral starts with a rectangle whose length and width form the golden ratio(≈1.618). This rectangle is partitioned into two squares. Then the squares are further partitioned. Connecting the corners of the boxes, the spiral is drawn inside these squares. The larger the numbers in the Fibonacci sequence, the ratio becomes closer to the golden ratio.

The puzzle of rabbits explains the wonder behind this Fibonacci sequence.

• Two newborn rabbits are left in the field. They are still one pair at the end of the first month.
• They mate and produce a new pair, so there are 2 pairs in the field, at the end of the second month.
• The first pair produces the second pair, but the second pair is left without breeding, so 3 pairs in all at the end of the third month.
• The original pair produces another pair, the second pair produces their first pair and the third pair remains without breeding, making 5 pairs.
• The sequence continues in this pattern and at the end of the n^th month, the number of rabbits in the field is equal to the sum of the number of mature pairs (n-2)^th month and the number of pairs alive last month(n-1)^th month. This happens to be the n^th Fibonacci number.

Fibonacci Sequence Formula

The Fibonacci sequence formula for “FnFn” is defined using the recursive

Fibonacci Sequence Properties

The interesting properties of the Fibonacci sequence are as follows:

1) Fibonacci numbers are related to the golden ratio. Any Fibonacci

Applications of Fibonacci Sequence

The Fibonacci sequence can be found in a varied number of fields from nature, to music, and to the human body.

• used in the grouping of numbers and the brilliant proportion in music generally.
• used in Coding (computer algorithms, interconnecting parallel, and distributed systems)
• in numerous fields of science including high energy physical science, quantum mechanics, Cryptography, etc.

You can use the Fibonacci calculator that helps to calculate the Fibonacci Sequence. Look at a few solved examples to understand the Fibonacci formula better.

Examples of Fibonacci Sequence

What Is The Fibonacci Sequence in Nature?

What Is the Recursive Formula for the Fibonacci Sequence?

Why Is Fibonacci Sequence Important?

Fibonacci Number Formula

Fibonacci Sequence

The Fibonacci Sequence is the series of numbers:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

The next number is found by adding up the two numbers before it:

• the 2 is found by adding the two numbers before it (1+1),
• the 3 is found by adding the two numbers before it (1+2),
• the 5 is (2+3),
• and so on!

Example: the next number in the sequence above is 21+34 = 55

It is that simple!

Here is a longer list:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, …

Can you figure out the next few numbers?

Makes A Spiral

When we make squares with those widths, we get a nice spiral:

Do you see how the squares fit neatly together?
For example 5 and 8 make 13, 8 and 13 make 21, and so on.

The Rule

The Fibonacci Sequence can be written as a “Rule” .

First, the terms are numbered from 0 onwards like this:

n =	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	…
xn =	0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	…

So term number 6 is called x₆ (which equals 8).

Using The Golden Ratio to Calculate Fibonacci Numbers

And even more surprising is that we can calculate any Fibonacci Number using the Golden Ratio:

Some Interesting Things

Terms Below Zero

The sequence works below zero also, like this:

History

Fibonacci was not the first to know about the sequence, it was known in India hundreds of years before!

About Fibonacci The Man

His real name was Leonardo Pisano Bogollo, and he lived between 1170 and 1250 in Italy.

“Fibonacci” was his nickname, which roughly means “Son of Bonacci”.

As well as being famous for the Fibonacci Sequence, he helped spread Hindu-Arabic Numerals (like our present numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). That has saved us all a lot of trouble! Thank you Leonardo.

Fibonacci Day

Fibonacci Day is November 23rd, as it has the digits “1, 1, 2, 3” which is part of the sequence. So next Nov 23 let everyone know!

The Origin of the Fibonacci Sequence

The Fibonacci sequence first appears in ancient Sanskrit texts as early as 200 BC, but the sequence wasn’t widely known to the western world until 1202 when Italian mathematician Leonardo Pisano Bogollo published it in his book of calculations called Liber Abaci. Leonardo also went by the moniker Leonardo of Pisa, but it wasn’t until 1838 that historians gave him the nickname Fibonacci (roughly translating to “son of Bonacci”). In addition to popularizing the Fibonacci sequence, Fibonacci’s book Liber Abaci advocated for the use of Hindu-Arabic numerals (1, 2, 3, 4, etc.) and helped replace the Roman numeral system (I, II, III, IV, etc.) throughout Europe.

In Liber Abaci, the Fibonacci sequence was actually used to answer a hypothetical math problem involving rabbit population growth: If a single pair of rabbits mate at the end of every month, then birth a new pair of rabbits one month after they mate, and all new pairs of rabbits follow that same pattern, how many pairs or rabbits will exist in one year? Here’s how you’d start answering this problem:

• Begin with 1 pair of rabbits.
• At the end of the first month, there is still only 1 pair of rabbits since they’ve mated, but have not yet given birth.
• At the end of the second month, there are 2 pairs of rabbits since the first pair have now birthed a second pair.
• At the end of the third month, there are 3 pairs of rabbits. This is because the first pair have birthed a third pair, but the second pair have only mated.
• At the end of the fourth month, there are now 5 pairs of rabbits. This is because the first pair have birthed another pair, and the second pair have now birthed their first pair.

As you can see, this 1, 1, 2, 3, 5 pattern follows the Fibonacci sequence. If you continue for 12 months, the number of pairs will equal 144.

Fibonacci Number Formula

To calculate each successive Fibonacci number in the Fibonacci series, use the formula

where 𝐹 is 𝑛th Fibonacci number in the sequence, and the first two numbers, 𝐹0 and 𝐹1 , are set at 0 and 1 respectively.

The only problem with this formula is that it’s a recursive formula, meaning it defines each number of the sequence using the preceding numbers. So if you wanted to calculate the tenth number in the Fibonacci sequence, you’d need to first calculate the ninth and eighth, but to get the ninth number you’d need the eighth and seventh, and so on.

To find any number in the Fibonacci sequence without any of the preceding numbers, you can use a closed-form expression called Binet’s formula:

In Binet’s formula, the Greek letter phi (φ) represents an irrational number called the golden ratio: (1 + √ 5)/2, which rounded to the nearest thousandths place equals 1.618.

Fibonacci Sequence and the Golden Ratio

The golden ratio (or golden section) is an irrational number that results when the ratio of two numbers is the same as the ratio of their sum to the larger of the two numbers. The Fibonacci sequence is closely connected to the golden ratio because as the Fibonacci numbers increase, the ratio of any two consecutive Fibonacci numbers gets closer and closer to the golden ratio.

Fibonacci Sequence in Nature

There’s considerable misinformation about where you may find the Fibonacci sequence and golden ratio in the real world; despite what you may read, the golden ratio was not used to build the pyramids at Giza, and the nautilus seashell does not grow new cells based on the Fibonacci sequence.

But these mathematical properties behind the Fibonacci sequence and golden ratio appear throughout nature in a number of ways. For example, you can find the golden ratio in the spiral arrangement of leaves (called a phyllotaxis) on some plants, or in the golden spiral pattern of pinecones, cauliflower, pineapples, and the arrangement of seeds in sunflowers. Additionally, the number of petals on a flower is typically a Fibonacci number.

Further, a honeybee drone’s family tree follows the Fibonacci sequence. This is because a male drone hatches from an unfertilized egg and only has one parent, while female bees have two parents. This results in a drone’s family tree consisting of one parent, two grandparents, three great-grandparents, five great-great-grandparents, and so on throughout the Fibonacci sequence.

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