Fibonacci Sequence and Golden Ratio

Fibonacci Sequence

• Named after Leonardo of Pisa (Fibonacci).

• Introduced in his book Liber Abacci in 1202.

• Fibonacci used it to solve a problem about rapid population growth.

• Sequence starts with 0 and 1.

• Each number is the sum of the two preceding ones.

• Roots go back to ancient India.

• Mathematicians like Virahanka described similar sequences as early as 200 BC, using them in poetry to analyze rhythm.

Definition

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers.

Fibonacci Numbers

• 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, and so on.

Mathematical Representation

• Each number in the sequence is a term, represented by $f$.

• $n$ reflects the number's position in the sequence, starting with 0.

• Example: The sixth term is $f(5)$, and the seventh term is $f(6)$.

Equations Defining Fibonacci Sequence

1. $f(0) = 0$ (applies only to the first integer)

2. $f(1) = 1$ (applies only to the second integer)

3. $f(n) = f(n-1) + f(n-2)$ (applies to all other integers)

   • The first two equations state that the term in the first position equals 0 and the term in the second position equals 1.

   • The third equation is a recursive formula.

     • Each number of the sequence is defined using the preceding numbers.

     • To define the fifth number $f(4)$, the terms $f(2)$ and $f(3)$ must already be defined.

Fibonacci Sequence Rule

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

• $f(n)$ is the nth term in the sequence.

• $f(n-1)$ is the previous term.

• $f(n-2)$ is the term before that.

Example: Find $f(7)$

$f(7) = f(6) + f(5) = 8 + 5 = 13$

Thus, the seventh Fibonacci number is 13.

Golden Ratio

Definition

• Often symbolized as the Greek letter phi ($\phi$).

• Irrational number approximately equal to 1.618.

• Exists in nature, architecture, art, and design.

Relation to Fibonacci Sequence

• Reflects the ratio of two consecutive Fibonacci numbers as the sequence progresses towards infinity.

Formula

$\frac{f(n)}{f(n-1)} = \phi$

Examples

• $f(6) = 8$ and $f(5) = 5$

• $\frac{8}{5} = 1.6$

• As $n$ increases, the ratio converges to 1.618.

• The ratio of succeeding consecutive terms will also be approximately 1.618.

Significance

• Multiplying a term by 1.618 gives the next term in the Fibonacci sequence.

  • If you multiply 8 by 1.618, you get approximately 13.

  • If you multiply 13 by 1.618, you get approximately 21.

• Multiplying a term by 0.618 gives the previous term.

  • If you multiply 8 by 0.618, you get approximately 5.

Golden Rectangle

• The ratio of its length to its width is 1.618.

Relation to the Golden Spiral

• When you take consecutive Fibonacci numbers and divide the larger one by the smaller one, the result approaches 1.618 (the golden ratio).

• Expressed as: $\frac{a+b}{a} = \frac{a}{b} = \phi$

  • For example, $\frac{34+21}{34} = \frac{34}{21} \approx 1.618$

• Dividing the rectangle into smaller sections following the ratio results in the golden spiral.

• Visible in seashells, galaxies, and avian hurricanes.

Quadratic Equation

• The golden ratio can also be found using the quadratic equation:

$x^2 - x - 1 = 0$

• Solving for $x$:

$x = \frac{1 \pm \sqrt{5}}{2}$

• This results in approximately 1.618, which is $\phi$

Uses of Golden Ratio to the Fibonacci Sequence

Finding Succeeding and Previous Terms

• To find the succeeding term: multiply the previous number by the golden ratio (1.618).

  • Instead of adding the first and second number, you can multiply the last number by the golden ratio to find the next number.

  • For example: $144 \times 1.618 \approx 233$, which is the succeeding term after 144.

Finding the nth Term Without Intervening Terms

• Formula:

$f(n) = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}}$

• Substitute the given data into the formula to find the nth term.

Summary of Formulas

1. To determine the next term:

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

2. To determine the ratio of the previous term to the succeeding term:

$\frac{f(n)}{f(n-1)} \approx 1.618$

3. To derive the golden ratio from the quadratic equation:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}$

$x = \frac{1 \pm \sqrt{5}}{2}$

• Golden ratio:

$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.618$

4. To derive the golden ratio from a rectangle:

$\frac{a+b}{a} = \frac{a}{b} = \phi$

5. To determine the nth term:

$f(n) = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}}$