Fibonacci Sequence Formula, Applications, With Solved Examples

Feb 4, 2022

Probably most of us have never taken the time to notice the arrangement of a flower. The petals and their arrangement to be precise. Also, we wouldn’t have seen the arrangement of the flowers in cauliflower or patterns in the leaves. If we observe them, we can see a definite pattern present in them. That arrangement is set per the Fibonacci Sequence.

Fibonacci numbers are the digits organized in a specific Fibonacci sequence in mathematics. These numerals were developed to describe positive numbers in a predetermined order sequence. The recurrence relation represents the list of numbers in the Fibonacci series: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…

Leonardo Bonacci was an Italian mathematician, known as Leonardo the Traveler from Pisa, who made up Fibonacci in 1838 by the Franco-Italian historian Guillaume Libri. Liber Abaci solved the problem of the population of rabbits based on some assumptions. The solution resulted in the Fibonacci numbers. The same numbers were used by Pingala, an Indian mathematician, in the 6th century. It was Fibonacci who came up with the concept of the golden ratio, which is a widely seen concept in nature.

What is a Fibonacci number?

A Fibonacci number is a sequence of numbers in which each Fibonacci number is determined by combining the two integers before it. It implies that the following figure in the sequence is the sum of the two preceding ones. For example, let’s use 0 and 1 for the first two integers in the series. So, if we add 0 and 1, we get the third number to be 1, and if we add the second and third numbers, which are 1 and 1, we obtain the fourth figure to be 2, and so on. The Fibonacci number is referred to as “mother nature’s hidden code.” Flowers and veggies like daisies, broccoli, seashells, cauliflower, and sunflowers all have spiral designs that follow the Fibonacci sequence.

What is the Fibonacci sequence?

The numerals in the Fibonacci Sequence are listed below. The Fibonacci formula was used to make this list. The sequence of the Fibonacci numbers is a series of numbers that begins with a one or a zero and continues with a one, following the rule that each number (called a Fibonacci number) equals the addition of the two previous numbers. For example, if you wish to write the Fibonacci sequence as F (n), where the first term is denoted by n, you get the following equation for n = 0, in which the first two factors are written as 0, 1.

Fibonacci Number Series:

0, 1, 1, 2, 3, 5, 8, 13,21,34,55,89,144,233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28567, 46368, 75025, 121393, 196418, 317811, …

Explaining the Fibonacci series with the rabbit puzzle

In the grass, two newborn bunnies are found. After the first month, they are still a single couple.
They mate and create a new pair at the end of the second month, resulting in two pairs in the area.
The first couple gives birth to the second, but the second pair is left unbed, resulting in three pairs after the third month.
The first pair generates a second pair, the second pair produces their first pair, and the third pair does not reproduce, resulting in five pairs.
The cycle continues, and the population of bunnies in the area at the end of the nth month is equivalent to the total sum of older pairs (n-2) and the number of pairs living last month (n-1). This is the number n in the Fibonacci sequence.

What is the Fibonacci sequence formula?

The formula of the Fibonacci number sequence can be expressed as:

F_n=F_n-1+F_n-2

where,

F_n denotes the number or nth term

(n-1)th term is denoted by F_n-1

(n-2)th term is denoted by F_n-2

where n > 1

Properties of the Fibonacci series

Fn is a multiple of every nth integer. Look through the sequence to see if anything else stands out. For example, every third element in the series is a two-digit multiple. Every fourth number is a multiple of three, and every fifth number is a multiple of five.
The Fibonacci series can also be used below zero. The formula for that is

F_n = ((-1)ⁿ⁺¹)F_n

For instance, the fourth term = (-1)5. 3 = -3

In the Fibonacci sequence, take four consecutive values that aren’t “0.” Multiply the exterior number and the interior number together. The difference “1” is obtained by subtracting these values. Take four digits in a row, for instance, 2, 3, 5, and 8. Multiply the surrounding numbers, such as 2(8), and the interior number 3 (5). Subtract these two integers, resulting in 16-15 = 1. As a result, the difference between the products is 1.
Take any three consecutive numbers in the Fibonacci series and add them together. The three numbers are obtained by dividing the result by two. For example, take three consecutive numbers, 1, 2, and 3. When these digits are added together, 1+ 2+ 3 Equals 6. When you divide 6 by 2, you get 3, which is 3.

Fibonacci series applications

The Fibonacci sequence may be found in many places, including the human body, music, or nature.
In music, it’s used to gather numbers and create a beautiful proportion.
In coding systems (distributed systems, interconnecting parallels, and computer algorithms)
In various domains of research, including quantum mechanics, high-energy physics, and cryptography.
Fibonacci spiral is the arrangement of seeds and flower heads in most daisies and sunflowers. Pinecones clearly show the Fibonacci spirals.
The organs of the human body exhibit Fibonacci characteristics. The lengths of bones in hand follow the Fibonacci sequence. The cochlea of the inner ear forms a Golden Spiral.
The Fibonacci numbers are also applied in Pascal’s triangle. Entry is the sum of the two numbers on either side of it, but in the row above. Diagonal sums in Pascal’s triangle are the Fibonacci numbers.
The Golden Ratio, the side of a regular pentagon to its diagonal, is also based on Fibonacci numbers. The five-point symmetry in a starfish is an example of the Golden Ratio.
The Great Pyramids are also built on the concept of the Golden Ratio, which is based on Fibonacci numbers.

Learning Fibonacci with a few examples

Example 1: Starting with 0 and 1, write the first 5 Fibonacci numbers.

Solution: The formula for the Fibonacci sequence is F_n= F_n-1+F_n-2

The first and second terms are 0 and 1, respectively.

F0 = 0 and F1 = 1.

F2 = F0 + F1 = 0+1 = 1 is the third term.

F3 = F2+F1 = 1 + 1 = 2 is the fourth term.

F4 = F3+F2 = 1+2 = 3 is the fifth term.

The Fibonacci sequence’s first five terms are 0,1,1,2,3.

Example 2: Find the Fibonacci series’ next term: 0, 1, 1, 2, 3, 5…

Solution: The preceding two Fibonacci terms are the next term in the series.

As a result, the word that must be used is – 3 + 5 = 8

Example 3: What is the significance of Fibonacci numbers?

Solution: In financial statement analysis, Fibonacci numbers are pretty helpful. In addition, the Fibonacci number series may be utilized to provide valuable proportions or ratios for business people.

Example 4: Find the 12th term of the Fibonacci sequence if the 10th and 11th terms are 34 and 55, respectively.

Solution: Using the Fibonacci Sequence recursive formula, we can say that the 12th term is the sum of the 10th term and the 11th term

12th term = 10th term + 11th term

= 34 + 55

= 89

The 12th term of the Fibonacci sequence is 89.

Example 5: The 14th term in the sequence is 377. Find the next term.

Solution: We know that the 15th term = the 14th term × the golden ratio.

F15 = 377 × 1.618034

≈ 609.99 = 610

The 15th term in the Fibonacci sequence is 610.

Example 6: Calculate the value of the 12th and the 13th term of the Fibonacci sequence, given that the 9th and 10th terms in the sequence are 21 and 34.

Solution: Using the Fibonacci sequence formula, we can say that the 11th term is the sum of the 9th term and 10th term.

11th term = 9th term + 10th term = 21 + 34 = 55

Now, 12th term = 10th term + 11th term = 34 + 55 = 89

Similarly,13th term = 11th term + 12th term = 55 + 89 = 144

The 12th and the 13th term of the Fibonacci sequence are 89 and 144.