Fibonacci Sequence: Lesson for Kids

Fibonacci sequence is a type series where each number is the sum of the two numbers before it. It starts from 0 and 1 usually. The Fibonacci sequence is given by 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. The numbers in the Fibonacci sequence are also called Fibonacci numbers.

In this article, we will study the meaning Fibonacci Sequence, Fibonacci Sequence Formula, Fibonacci Sequence List and Golden Ratio to Calculate Fibonacci Numbers etc

What is Fibonacci Sequence?

Fibonacci sequence, also known as Fibonacci numbers, is defined as the sequence of numbers in which each number in the sequence is equal to the sum of two numbers before it. The Fibonacci Sequence is given as:

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

Here, the third term “1” is obtained by adding the first and second term. (i.e., 0+1 = 1).

Similarly, “2” is obtained by adding the second and third term (1+1 = 2) and “3” is obtained by adding the third and fourth term (1+2) and so on.

Fibonacci Sequence Formula

The Fibonacci sequence of numbers “F(n)” is defined using the recursive relation with the seed values F(0) = 0 and F(1)= 1:

F(n) = F(n - 1) + F(n - 2)

Here, the sequence is defined using two different parts, such as kick-off and recursive relation.

• The kick-off part is F(0) = 0 and F(1) = 1.
• The recursive relation part is F(n) = F(n - 1) + F(n - 2)

Fibonacci Sequence List

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.

Here is a list of the first few numbers in the fibonacci sequence:

Fibonacci Spiral

The Fibonacci spiral is a geometric pattern that is based on the Fibonacci sequence, a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. The sequence goes 0, 1, 1, 2, 3, 5, 8, 13, and so on.

To create a Fibonacci spiral follow the steps given below:

• Draw Squares: Start by drawing squares whose side lengths correspond to the Fibonacci sequence (e.g., 1, 1, 2, 3, 5, 8, etc.).
• Arrange the Squares: Place the squares next to each other, beginning with the smallest and continuing in sequence, so that each square's side length matches the next number in the Fibonacci sequence.
• Draw Quarter Circles: Inside each square, draw a quarter circle connecting two opposite corners. These quarter circles should be continuous as you move from one square to the next.
• Continue the Pattern: Repeat the process, adding more squares and quarter circles, gradually forming a spiral that expands outward.
• Observe the Spiral: The resulting pattern resembles a nautilus shell and is known as the Fibonacci spiral. This spiral is commonly seen in nature, such as in the arrangement of leaves, flowers, and shells, due to its efficient packing and growth properties.

The Fibonacci spiral is an example of how mathematical patterns can be found in the natural world.

Golden Ratio to Calculate Fibonacci Numbers

Golden Ratio (φ), approximately 1.6180339887, is closely related to the Fibonacci sequence. To find the nth Fibonacci number using the Golden Ratio, you can use

Binet's Formula

Where:

• φ is the Golden Ratio, ,
• √5 is the square root of 5.

This formula provides an exact value for Fibonacci numbers, though for large (n), it's often easier to use iterative methods or approximation due to rounding issues with irrational numbers.

Tips and Tricks of Fibonacci Sequence and numbers

The various tips and tricks of fibonacci sequence are as follows:

• Start with Simple Numbers: The sequence begins with 0 and 1. Each subsequent number is the sum of the two preceding ones. So, the next few numbers are 1, 2, 3, 5, 8, 13, and so on.

• Pattern in Nature: You can find Fibonacci numbers in nature. For example, the number of petals on some flowers or the arrangement of leaves on a stem often follow the Fibonacci sequence.

• Sum of Previous Numbers: Each number in the sequence is the sum of the two before it. So, to find the 6th number, add the 4th and 5th numbers (3 + 5 = 8).

• Odd and Even Pattern: The sequence alternates between odd and even numbers, but not in a simple pattern. For instance, the 3rd number is odd (1), the 4th number is odd (2), the 5th number is odd (3), and so forth.

• Golden Ratio Connection: As you go further in the sequence, the ratio between consecutive Fibonacci numbers gets closer to the Golden Ratio (about 1.618). This ratio is often seen in art and architecture.

Uses of Fibonacci Sequence

The uses of fibonacci sequence are as follows:

• Humans are bad at estimation but good at comparison.
• Natural rhythm.
• The sequence appears in various natural patterns, such as arrangements of leaves, the branching of trees.
• Teaching the Fibonacci sequence can help students understand mathematical concepts, and sequences and problem solving etc.
• The Fibonacci sequence has interesting mathematical properties, such as the Golden ratio, which is the limit of the consecutive Fibonacci numbers.
• Faster estimation using poker game for story pointing.
• Law of large numbers.

Fibonacci Sequence Solved Examples

Example 1: Find the Fibonacci number when n=5, using recursive relation.

Solution:

The formula to calculate the Fibonacci Sequence is: F(n) = F(n-1) + F(n-2)

Take: F(0)=0 and F(1)=1

Using the formula, we get

• F(2) = F(1)+F(0) = 1+0 = 1
• F(3) = F(2)+F(1) = 1+1 = 2
• F(4) = F(3)+F(2) = 2+1 = 39
• F(5) = F(4)+F(3) = 3+2 = 5

Therefore, the fibonacci number is 5.

Example 2: Find the Fibonacci number using the Golden ratio when n=6.

Solution:

The formula to calculate the Fibonacci number using the Golden ratio is X(n) = [φn – (1-φ)n]/√5

We know that φ is approximately equal to 1.618.

n= 6

Now, substitute the values in the formula, we get

• X(n) = [φn – (1-φ)n]/√5
• X(6) = [1.6186 – (1-1.618)6]/√5
• X(6) = [17.942 – (0.618)6]/2.236
• X(6) = [17.942 – 0.056]/2.236
• X(6) = 17.886/2.236
• X(6) = 7.999
• X(6) = 8 (Rounded value)

The Fibonacci number in the sequence is 8 when n = 6.

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