Fibonacci Sequence Formula | Formula, Examples & Problems (2024)

Fibonacci Sequence Formula: Fibonacci sequence, the sequence of numbers 1, 1, 2, 3, 5, 8, 13, 21, …, each of which, after the second, is the sum of the two previous numbers; that is, the nth Fibonacci numberF_n= F_n−1+ F_n−2.

In the Fibonacci sequence, each number in the series is calculated by adding the two numbers before it. Generally, the first two terms of the Fibonacci series are 0 and 1. The Fibonacci sequence was known in India hundreds of years before Leonardo Pisano Bigollo knew about it. November 23rd is celebrated as Fibonacci Day, as it has the digits “1, 1, 2, 3” which is part of the sequence

In this article, we will learn about the Fibonacci Sequence, along with its formula, examples, golden ratio, etc.

What is the Fibonacci Sequence?

Fibonacci sequence is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946,….

Fibonacci’s sequence is useful for its operations in advanced mathematics and statistics, computer science, economics, and nature.

Fibonacci Sequence Formula

Fibonacci Sequence Formula is,

F_n = F_n-1 + F_n-2

where,

• F_n is term number “n”
• F_n−1 is the previous term (n−1)
• F_n−2 is the term before that (n−2)

To calculate the 5th Fibonacci number, add the 4th and 3rd Fibonacci numbers and so on.

Golden Ratio

Ratio of two consecutive numbers is called Golden Ratio. It is denoted by “φ“. The golden ratio is generally seen in nature, and when applied in a design, it fosters natural-seeming works that are pleasing to the eye. There are numerous operations of the golden ratio in the field of architecture. For illustration, the Great Pyramid of Egypt and the Great Mosque of Kairouan is many of the architectural miracles in which the notion of the golden ratio has been applied.

For example:

Note: Golden Ratio can be calculated from Any Fibonacci sequence, it does not necessarily have to start with 2 and 3.

Calculating the Fibonacci sequence

Any Fibonacci number can be calculated by using this formula,

x_n = (φⁿ − (1−φ)ⁿ)/√5

• x_n denotes Fibonacci number to be calculated
• φ is Golden ratio that is 1.618034

For example: If you want to calculate the 7th term:

x₇ = ((1.618034)⁷-(1-1.618034)⁷)/√5

x₇ = 13.0000007

x₇ = 13(rounded off)

Next Fibonacci number can also be calculated using Golden Ratio. Multiplying a Fibonacci number with a golden ratio will give the next Fibonacci number of the sequence. But that only works for numbers greater than 1.

Example: 13 × 1.618034 = 21.034442 = 21(rounded off)

Fibonacci Sequence Examples

Problem 1: Calculate the 9th Fibonacci number if given golden ratio is 1.618034.

Solution:

We can calculate the 9th Fibonacci number by using the formula:

x_n = (φⁿ − (1−φ)ⁿ)/√5

x₉ = ((1.618034)⁹-(1-1.618034)⁹)/√5

x₉ = (76.0131604-(-0.0131556197))/√5 = 34.0000021

x₉ = 34

Problem 2: Find the next Fibonacci number of answers calculated in the above question.

Solution:

Next Fibonacci number of 34 can be easily found by multiplying it by the Golden ratio that is 1.618034.

x₁₀ = 34×1.61803 = 55.01302

x₁₀ = 55(rounded off)

Problem 3: If the 5th and 6th terms of a Fibonacci sequence are 3 and 5 respectively, find the 7th term of the sequence.

Solution:

With the use of the Fibonacci Sequence formula, we can easily calculate the 7th term of the Fibonacci sequence which is the sum of the 5th and 6th terms.

seventh term = 5th term + 6th term

= 3+5

= 8

The 7th term of the Fibonacci sequence is 8.

Problem 3: The first 4 numbers in the Fibonacci sequence are given as 1,1,2,3.

(a) What is the eighth term of the Fibonacci sequence?

(b) What is the eleventh term of the Fibonacci sequence?

Solution:

By the use of the Fibonacci number formula, we can calculate the rest of the Fibonacci numbers like 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.

(a) Therefore, the 8th term will be 21.

(b) 11th term will be 89.

Problem 4: Find the next 3 terms for each of the following Fibonacci-style sequences.

(a) x, 4x, 5x, 9x,…

(b) 3a, 3a+b, 6a+b, 9a+2b….

Solution:

With use of the Fibonacci Sequence formula, we can easily calculate the rest of the terms

(a)

Fifth term = 5x+9x = 14x,

Sixth term = 9x+14x = 23x,

Seventh term = 14x+23x = 37x

(b)

Fifth term = 6a+b+9a+2b = 15a+3b,

Sixth term = 9a+2b+15a+3b = 24a+5b,

Seventh term = 15a+3b+24a+5b = 39a+8b

Problem 5: John wants to generate a Fibonacci series with the first term as 3 and the second term as 4.

(a) Find the 3rd and 4th terms.

(b) He thinks that the sum of the first ten terms is equal to eleven times the seventh term of his sequence. Check if he is correct.

Solution:

Using 3 and 4 as first and second terms, we can calculate the rest of the terms by simply adding the last two terms.

(a)

First term = 3

Second term = 4

Third Term = 3+4 = 7

Forth term = 4+7 = 11

(b)

On calculating the first ten terms of the series: 3,4,7,11,18,29,47,76,123,199.

Sum of first ten terms = 3+4+7+11+18+29+47+76+123+199 = 517

7th term = 47

Eleven times the 7th term = 11*47 = 517

As we can see that the sum of the first ten terms is equal to eleven times the seventh term of his sequence. Therefore, John was correct.

Problem 6: What is the first three-digit square number that appears on the list of Fibonacci numbers, if the first 4 terms are 0,1,1,2.

Solution:

With the use of the Fibonacci Sequence formula, we can easily calculate the rest of the terms:

0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,…

As we can see the first three-digit number which is a square that appears on the list of Fibonacci numbers is 144(square of 12).

Practice Problems on Fibonacci Sequence Formula

1. Find the sum of the first 7 terms of the Fibonacci sequence.

2. If the 6th term in the Fibonacci sequence is 8, find the 7th and 8th terms.

3. Find the ratio of the 9th term to the 8th term in the Fibonacci sequence. Simplify your answer if possible.

4. A pair of rabbits mates every month and produces a new pair every month from the second month onwards. If one pair of rabbits starts the sequence, how many pairs of rabbits will there be at the end of 6 months?

FAQs on Fibonacci Sequence Formula

What is Fibonacci sequence in simple words?

Fibonacci sequence is a special sequence where each number is the sum of the two preceding number. It starts from 0 and 1 usually. The Fibonacci sequence is: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on.

Why is the Fibonacci sequence so important?

Fibonacci sequence is important to biologists and physicists because they are frequently observed in various natural objects and phenomena.

What is Fibonacci sequence in daily life?

Fibonacci sequence in daily life is observed on branches on a tree, sequences in music, number of petals on a flower, human anatomy, shape of a spiral, and more.

What is the fifth term of Fibonacci Sequence?

Fifth term in the Fibonacci sequence is 5.