Fibonacci Numbers

Fibonacci numbers are a sequence of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1. This sequence is important because it appears in many different areas of mathematics and nature, such as branching in trees, the arrangement of leaves on a stem, and the flowering of an artichoke.

1.0Fibonacci Numbers

A Fibonacci number is part of a series where each number is the sum of the two preceding ones. Starting with 0 and 1, the sequence continues by adding these two to get the third number, which is also 1. The fourth number is obtained by the addition of  the second and third numbers (1 and 1), resulting in 2. This process repeats, producing the sequence: 0, 1, 1, 2, 3, 5, 8, and so on. Consequently, this sequence is known as the Fibonacci Sequence.

Fibonacci numbers can also be derived from Pascal's Triangle, as illustrated in the figure below.

In Pascal's Triangle, the sum of the elements along each diagonal, highlighted by colored lines, represents the Fibonacci Number.

2.0Fibonacci Number Series List

The Fibonacci number series is a sequence where each number is the sum of the two preceding ones. Starting with 0 and 1, the series progresses as follows:

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, 514229, 832040, …

Each term in this list is obtained by adding the two previous terms. This fundamental property defines the entire series and highlights its recursive nature.

3.0Rules for Fibonacci Numbers

Addition Rule: Each Fibonacci number is the sum of the two preceding numbers. Mathematically, this is expressed as:

F(n) = F (n – 1) + F (n – 2)

where F (0) = 0 and F (1) = 1.

Where F(n) is termed as the n^th term, F (n – 1) is the (n – 1)^th term and F (n – 2) is the (n – 2)^th term.

4.0Properties of Fibonacci Numbers

1. Sum of Three Consecutive Numbers:

Take any three consecutive Fibonacci numbers, add them, and then divide the sum by 2. The result will be the middle number. For example, for 1, 2, and 3:

$1+2+3=6\text{ and }\frac{6}{2}=3$

Here, 3 is the middle number.

2. Product of Four Consecutive Numbers:

Take four consecutive Fibonacci numbers (other than 0), multiply the outer numbers, and then multiply the inner numbers. Subtract the second product from the first, and the result will always be 1. For example, for 2, 3, 5, and 8:

$2\times 8=16\text{ and }3\times 5=15\text{ so }16-15=1$

5.0Fibonacci Number Formula

The Fibonacci sequence can be generated using different formulas and methods, each with its own significance and application. Here are the primary formulas for Fibonacci numbers:

1. Recursive Formula:

$F(n)=F(n-1)+F(n-2)$  

with the initial conditions:

$F(0)=0,F(1)=1$

This is the most fundamental definition of Fibonacci numbers, where every number is the sum of the two preceding ones.

2. Closed-Form Formula (Binet's Formula):

$\mathrm{F}(\mathrm{n})=\frac{{\varphi }^n-(1-\varphi {)}^n}{\sqrt{5}}$

where $\varphi =\frac{1+\sqrt{5}}{2}$  is the golden ratio, and $1-\varphi =\frac{1-\sqrt{5}}{2}$ . This formula provides a direct way to compute the nth Fibonacci number without recursion.

6.0Fibonacci Number Pattern

The Fibonacci sequence exhibits several intriguing patterns:

1. Multiples Pattern:

• Every third number in the sequence, starting with 2, is a multiple of 2.
• Every fourth number, starting with 3, is a multiple of 3.
• Every fifth number, starting with 5, is a multiple of 5, and so on.

2. Parity Pattern:

• The addition of two odd numbers always results in an even number.
• The sum of an even and an odd number is always odd.
• Consequently, the Fibonacci sequence follows an even, odd, odd, even, odd, odd pattern.

7.0Fibonacci Sequence and Golden Ratio

The ratio of two consecutive Fibonacci numbers $\varphi =\frac{1+\sqrt{5}}{2}$ approximately 1.618…, known as the Golden Ratio, or phi (φ), which is an irrational number.

For a given spiral, the Golden Ratio follows this property:

Let the Fibonacci numbers be a, b, c, d, then…

$\frac{\mathrm{a}+\mathrm{b}}{\mathrm{a}}\approx \frac{\mathrm{b}+\mathrm{c}}{\mathrm{b}}\approx \frac{\mathrm{c}+\mathrm{d}}{\mathrm{c}}\approx 1.618$

It follows a constant angle close to the Golden Ratio and is commonly known as the Golden Spiral. In geometry, this ratio forms a Golden Rectangle, where the ratio of its length to its width is the Golden Ratio. This concept frequently appears in various works of art and architecture.

nth Fibonacci Number and the Golden Ratio

The relationship between the nth Fibonacci number and the Golden Ratio is a fascinating aspect of mathematics. The Golden Ratio, commonly represented by the Greek letter \phi  (phi), is approximately 1.6180339887. It can be expressed mathematically as: $\varphi =\frac{1+\sqrt{5}}{2}$

Binet's Formula

The nth Fibonacci number F(n) can be approximated using Binet's formula, which directly involves the Golden Ratio:

$\mathrm{F}(\mathrm{n})=\frac{{\varphi }^n-(1-\varphi {)}^n}{\sqrt{5}}$

Convergence to the Golden Ratio

As n increases, the ratio of successive Fibonacci numbers converges to the Golden Ratio:

${\lim }_{n\to \infty }\frac{\mathrm{F}(\mathrm{n}+1)}{\mathrm{F}(\mathrm{n})}=\varphi$

This means that for large n, the ratio $\frac{F(n+1)}{F(n)}$ gets closer and closer to $\varphi$ .

Geometric Interpretation

In geometry, the Golden Ratio is used to construct the Golden Rectangle, where the ratio of the longer side to the shorter side is \phi. When a square is removed/cut from a Golden Rectangle, the leftover rectangle is also a Golden Rectangle. This recursive property forms the basis of the Golden Spiral, which appears in various natural and human-made structures.

8.0Fibonacci Numbers Solved Examples 

Example 1: Find the sum of the first 10 Fibonacci numbers. 

Solution: As we know; the sum of the Fibonacci sequence 

$={\sum }_{\ell =0}^n{F}_i=F_{n+2}-F_2$

$=Fn+2$–1, Where F_n is the n^th Fibonacci

Number, and the sequence starts from F₀,

Thus the sum of the first 10 Fibonacci numbers;

= (10 + 2)^th term – 2^nd term.

= 12^th term – 1

= 89 – 1 = 88.

Example 2: Find the 6^th Fibonacci number.

Solution: As we know,

The n^th Fibonacci number is F(x_n) = F(x_n–1) + F(x_n–2), for n > 2

Then the 6^th Fibonacci number is 

F(x₆) = F(x_6–1) + F(x_6–2), for n = 6

Example 3: Find the next number when F₁₁ = 55.

Solution:

Here, F₁₂ = F₁₁ × Golden Ratio.

= 55 × 1.6180

= 88.99 ≈ 89.

Hence F₁₂ = 89.

9.0Fibonacci Numbers Practice Problems

1. Find F (7).

2. Prove that the sum of the first n Fibonacci numbers is F (n + 2) –1.

3. Show that F(n) × F (n + 1) – F (n – 1) × F (n + 2) = (–1) ^n.

10.0Sample Questions on Fibonacci Numbers

Q1. What is the formula to find Fibonacci numbers?

Ans: The nth Fibonacci number can be found using the recursive formula:

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

with initial values F(0) = 0 and F(1) = 1.

Alternatively, Binet's formula provides a direct way to calculate Fibonacci numbers:

$F(n)=\frac{{\varphi }^n-(1-\varphi {)}^n}{\sqrt{5}}$where $\varphi$ is the Golden Ratio.

Q2. How is the Golden Ratio related to Fibonacci numbers?

Ans: The ratio of successive Fibonacci numbers converges to the Golden Ratio ($\varphi$), approximately 1.6180339887. This relationship is expressed as:

${\lim }_{n\to \infty }\frac{F(n+1)}{F(n)}=\varphi$