Fibonacci Numbers
Fibonacci Numbers
Fibonacci numbers are a sequence of numbers where each number is the sum of the two preceding ones. The Fibonacci numbers are the numbers ordered in a distinct Fibonacci sequence. The sequence of numbers starts with `0` and `1`. Each Fibonacci number is the sum of the previous two Fibonacci numbers. Fibonacci numbers are observed in natural phenomena like the arrangement of leaves on a stem, the spiral shapes of galaxies, and the proportions of the human body. Additionally, Fibonacci numbers have applications in various fields, including computer science, finance, and art.
What Are Fibonacci Numbers?
Fibonacci numbers are the numbers that are sum of the two preceding ones, starting with `0` and `1`. So, the sequence or Fibonacci series begins: `0, 1, 1, 2, 3, 5, 8, 13, 21,` and so on. These numbers were introduced to the Western world by Leonardo of Pisa, also known as Fibonacci, in his book "Liber Abaci" in `1202`. However, the sequence had been described earlier in Indian mathematics.
The Fibonacci sequence can be represented by the recursive formula:
\( F_n = F_{n-1} + F_{n-2} \)
where:
` F_n ` is the `n`th Fibonacci number.
` F_{n-1} ` is the `(n-1)`th Fibonacci number.
` F_{n-2} ` is the `(n-2)`th Fibonacci number.
with
\( F_0 = 0, \)
\( F_1 = 1. \)
Fibonacci numbers have numerous interesting properties and appear in various areas of mathematics, nature, and science.
Meaning of Fibonacci
A Fibonacci number is generally denoted by `F_n`, where `n` is a whole number. The Fibonacci sequence is given by, `F_0 = 0, F_1 = 1, F_2 = 1, F_3 = 2, F_4 = 3, F_5 = 5,` and so on. Fibonacci is a concept that can be represented as numbers, sequences, or series such that each term is the sum of the two terms preceding it and the first two terms are `0` and `1`. The "Fibonacci" not only refers to the mathematician himself but also to the sequence and its implications across multiple disciplines.
Formula of Fibonacci
The general formula to calculate the `n`th number in the sequence of Fib numbers can be given as,
`F_n = F_{n-1} + F_{n-2}`
where,
`n > 1`
`F_{n-1} → (n - 1)`th Fibonacci number
`F_{n-2} → (n - 2)`th Fibonacci number
This formula states that the `n`th Fibonacci number ` F_n ` is the sum of the ` (n-1) `th Fibonacci number and the ` (n-2) `th Fibonacci number, with ` F_0 = 0 ` and ` F_1 = 1 ` being the first two numbers of the series.
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting with `0` and `1`. So, if you know the values of the `(n - 1)`th and `(n - 2)`th Fibonacci numbers, you can calculate the `n`th Fibonacci number by adding them together and we can find the answer of what is the fibonacci sequence number.
Here's an iterative approach using a formula:
`F(n) = \frac{1}{\sqrt{5}} \left( \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n \right)`
This formula is derived from Binet's formula, which provides a direct method to compute the ` n `th Fibonacci number without having to calculate all the preceding Fibonacci numbers.
Both formulas can be used to efficiently compute Fibonacci numbers for any given ` n `. However, the iterative approach using Binet's formula is more computationally efficient for large values of ` n ` compared to the recursive formula.
Fibonacci Numbers Rules
The Fibonacci sequence follows a few key rules:
- Initial Conditions: The sequence starts with two initial terms, typically defined as `F_0 = 0` and `F_1 = 1`.
- Recurrence Relation: Each subsequent term in the sequence is the sum of the two preceding terms. This is often expressed mathematically as `F_n = F_{n-1} + F_{n-2}`, where `n > 1`.
- Non-Negative Integers: The Fibonacci sequence consists of non-negative integers. This means each term in the sequence is a whole number, starting from `0, 1, 1, 2, 3, 5,` and so on.
- Infinite Sequence: The Fibonacci sequence continues indefinitely, producing an infinite series of numbers.
These rules define the essence of the Fibonacci sequence and are crucial for understanding and generating the sequence.
Finding Fibonacci Numbers
Let's calculate the Fibonacci numbers using the iterative method and tabulate the calculation for the first ten terms. Below is the fibonacci numbers list showing the first `10` numbers of the Fibonacci sequence.
Here's how each term is calculated:
\( F_0 = 0 \)
\( F_1 = 1 \)
\( F_2 = F_1 + F_0 = 1 + 0 = 1 \)
\( F_3 = F_2 + F_1 = 1 + 1 = 2 \)
\( F_4 = F_3 + F_2 = 2 + 1 = 3 \)
\( F_5 = F_4 + F_3 = 3 + 2 = 5 \)
\(F_6 = F_5 + F_4 = 5 + 3 = 8 \)
\(F_7 = F_6 + F_5 = 8 + 5 = 13 \)
\(F_8 = F_7 + F_6 = 13 + 8 = 21 \)
\(F_9 = F_8 + F_7 = 21 + 13 = 34 \)
And so on. Each Fibonacci number is the sum of the two preceding Fibonacci numbers. Also, In the sequence formed by the Fibonacci numbers, the first and second terms are always `0` and `1`, respectively. Each number in the sequence (after the first two) is the sum of the two preceding numbers. The sequence continues indefinitely. If you need more terms can be calculated accordingly.
Properties of Fibonacci Numbers
A Fibonacci number have several interesting properties. Here are some of them:
`1`. Recursive Definition: Fibonacci numbers are defined recursively, where each number (starting from the third) is the sum of the two preceding numbers: `F(n) = F(n-1) + F(n-2)`.
`2`. Initial Conditions: The first two Fibonacci numbers are `0` and `1`. These serve as the initial conditions for the sequence.
`3`. Golden Ratio: As Fibonacci numbers increase, the ratio of consecutive Fibonacci numbers approaches the golden ratio, `\phi (\text{phi})`, which is approximately `1.61803398875`. The golden ratio has many fascinating properties and is found in various natural phenomena and artistic endeavors.
`4`. Divisibility: Some interesting divisibility properties exist for Fibonacci numbers. For instance, every third Fibonacci number is even, and every fourth Fibonacci number is a multiple of `3`.
`5`. Pisano Period: Fibonacci numbers exhibit a periodic pattern when taken modulo of any positive integer. The period length is known as the Pisano period.
`6`. Lucas Numbers: Lucas numbers are similar to Fibonacci numbers, but they start with `2` and `1` instead of `0` and `1`.
`7`. Binet's Formula: There's a closed-form expression called Binet's formula for directly calculating the nth Fibonacci number using the golden ratio.
`n`th Fibonacci Number and Golden Ratio
The `n`th Fibonacci number `(F(n))` and the golden ratio `(\phi)` are intimately connected. As `n` approaches infinity, the ratio between consecutive Fibonacci numbers converges to the golden ratio. Mathematically, this can be expressed as:
`\lim_{n \to \infty} \frac{F(n+1)}{F(n)} = \phi`
In other words, the larger the value of `n`, the closer the ratio of ` F(n+1) ` to ` F(n) ` gets to the golden ratio, `φ`.
Furthermore, Binet's formula provides a direct relationship between the `n`th Fibonacci number and the golden ratio:
`F(n) = \frac{\phi^n - (1 - \phi)^n}{\sqrt{5}}`
This formula allows for the direct calculation of the `n`th Fibonacci number using the golden ratio.
This property highlights the significance of the golden ratio in the Fibonacci sequence and its relationship to the growth of Fibonacci numbers.
Fibonacci Numbers in Nature
Fibonacci numbers appear abundantly in nature, showcasing the inherent mathematical beauty and efficiency of natural processes. Here are some examples of where Fibonacci numbers can be observed in nature:
`1`. Phyllotaxis: The arrangement of leaves, seeds, or petals around a stem often follows a Fibonacci sequence. This arrangement allows for efficient packing and optimal exposure to sunlight.
`2`. Flower Petals: Many flowers exhibit a number of petals that are a Fibonacci number or a Fibonacci sequence. For example, lilies often have three petals, buttercups have five, delphiniums have eight, and so on.
`3`. Seed Heads: Sunflower seed heads display a spiral pattern that follows the Fibonacci sequence. The seeds are arranged in logarithmic spirals that correspond to consecutive Fibonacci numbers.
`4`. Pinecones and Pineapples: The spirals on pinecones, pineapples, and other similar structures often follow Fibonacci numbers. This arrangement allows for efficient packing of seeds or reproductive structures.
`5`. Spiral Shells: Some mollusk shells, such as snail shells and nautilus shells, exhibit spirals that adhere to the Fibonacci sequence. These spirals help in growth and structural integrity.
`6`. Animal Reproduction: In some species, such as rabbits, the number of offspring in each generation can approximate Fibonacci numbers if certain conditions are met.
`7`. Human Body: While not as obvious as in plants, some proportions in the human body, such as the lengths of bones in fingers, exhibit ratios close to the golden ratio, which is derived from the Fibonacci sequence.
Real-Life Applications of Fibonacci Numbers
`1`. Financial Markets: Fibonacci numbers are widely used in technical analysis in financial markets. Traders use Fibonacci retracement levels to identify potential support and resistance levels in stock prices, currency pairs, and other financial instruments. The ratios derived from Fibonacci numbers, such as `0.382, 0.50,` and `0.618`, are commonly used as retracement levels.
`2`. Computer Science: Fibonacci numbers are used in various algorithms and data structures. For example, they are used in dynamic programming algorithms, such as the Fibonacci sequence itself, where previous results are stored and reused to compute larger Fibonacci numbers more efficiently. Fibonacci heaps are a type of data structure used in priority queue implementations.
`3`. Art and Design: Fibonacci numbers and the Fibonacci sequence appear in art and design due to their aesthetically pleasing proportions. Artists, architects, and designers often use Fibonacci ratios and spirals to create visually appealing compositions. Examples include the proportions of the Parthenon in Athens and the layout of the spiral shells of certain mollusks.
`4`. Biology: Fibonacci numbers and the Fibonacci sequence appear in various biological phenomena. For instance, the arrangement of leaves on a stem, the branching patterns of trees, and the spirals of seeds in sunflowers and pinecones often follow Fibonacci sequences. These patterns optimize space utilization and allow for efficient packing of structures.
`5`. Music: Fibonacci numbers are used in music composition and theory. Some composers have used Fibonacci sequences to structure musical compositions, determining the lengths of musical sections or the arrangement of musical motifs. Additionally, the Fibonacci sequence has been associated with the mathematical properties of musical scales and harmonies.
`6`. Cryptography: Fibonacci numbers can be used in cryptographic algorithms, particularly in generating pseudorandom numbers for encryption and decryption processes. The properties of Fibonacci sequences, such as their unpredictability and pseudo-randomness, make them useful in certain cryptographic applications.
Solved Examples
Example `1`: It is given that the `5`th Fibonacci number is `3` and the `6`th Fibonacci number is `5`. Find the `7`th and `8`th Fibonacci numbers.
Solution:
`1`. To find the `7`th Fibonacci number:
- It is the sum of the `5`th and `6`th Fibonacci numbers: `3 + 5 = 8`.
`2`. To find the `8`th Fibonacci number:
- It is the sum of the `6`th and `7`th Fibonacci numbers: `5 + 8 = 13`.
So, the `7`th Fibonacci number is `8`, and the `8`th Fibonacci number is `13`.
Example `2`: Assume that the `4`th Fibonacci number is `3` and the `5`th Fibonacci number is `5`. Find the `6`th and `7`th Fibonacci numbers.
Solution:
We can find the `6`th and `7`th Fibonacci numbers as follows:
`1`. To find the `6`th Fibonacci number:
- It is the sum of the `4`th and `5`th Fibonacci numbers: `3 + 5 = 8`.
`2`. To find the `7`th Fibonacci number:
- It is the sum of the `5`th and `6`th Fibonacci numbers: `5 + 8 = 13`.
So, the `6`th Fibonacci number is `8`, and the `7`th Fibonacci number is `13`.
Example `3`: Find the value of the `12`th and the `13`th Fibonacci numbers. The `9`th and `10`th terms in the sequence are `21` and `34`.
Solution:
Given that the `9`th Fibonacci number is `21` and the `10`th Fibonacci number is `34`:
`1`. To find the `11`th Fibonacci number:
- It is the sum of the `9`th and `10`th Fibonacci numbers: `21 + 34 = 55`.
`2`. To find the `12`th Fibonacci number:
- It is the sum of the `10`th and `11`th Fibonacci numbers: `34 + 55 = 89`.
`3`. To find the `13`th Fibonacci number:
- It is the sum of the `11`th and `12`th Fibonacci numbers: `55 + 89 = 144`.
So, the `12`th Fibonacci number is `89`, and the `13`th Fibonacci number is `144`.
Practice Problems
Q`1`. Given that ` F_{10} = 55 ` and ` F_{11} = 89 `, find ` F_{13} `.
- ` F_{13} = 233 `
- ` F_{13} = 610 `
- ` F_{13} = 34 `
- ` F_{13} = 987 `
Answer: a
Q`2`. Given that ` F_{6} = 8 ` and ` F_{7} = 13 `, find ` F_{9} `.
- ` F_{9} = 234 `
- ` F_{9} = 34 `
- ` F_{9} = 134 `
- ` F_{9} = 233 `
Answer: b
Q`3`. Given that ` F_{15} = 610 ` and ` F_{16} = 987 `, find ` F_{17} `.
- ` F_{17} = 233 `
- ` F_{17} = 610 `
- ` F_{17} = 1597 `
- ` F_{17} = 987 `
Answer: c
Q`4`. Given that ` F_{11} = 89` and ` F_{12} = 144 `, find ` F_{13} `.
- ` F_{13} ` is `1597`
- ` F_{13} ` is `6765`
- ` F_{13} ` is `333`
- ` F_{13} ` is `233`
Answer: d
Q`5`. Given that ` F_{20} = 6765 ` and ` F_{21} = 10946 `, find ` F_{22} `.
- ` F_{22} ` is `17711`
- ` F_{22} ` is `1772`
- ` F_{22} ` is `11411`
- ` F_{22} ` is `11711`
Answer: a
Frequently Asked Questions
Q`1`. What are Fibonacci numbers?
Answer: Fibonacci numbers are a sequence of numbers where each number is the sum of the two preceding ones. The sequence starts with `0` and `1`, and the subsequent numbers are found by adding the last two numbers together `(0, 1, 1, 2, 3, 5, 8, 13,` and so on`)`.
Q`2`. Who discovered Fibonacci numbers?
Answer: Fibonacci numbers were first introduced to the Western world by Leonardo of Pisa, also known as Fibonacci, in his book "Liber Abaci" (The Book of Calculation) in `1202`. However, the sequence had been described earlier in Indian mathematics.
Q`3`. What is the mathematical representation of the Fibonacci sequence?
Answer: The `n`th term in a Fibonacci sequence is often denoted as ` F_n `, where ` n ` represents the position of the term in the sequence. So, ` F_0 = 0 `, ` F_1 = 1 `, and for ` n \geq 2 `, ` F_n = F_{n-1} + F_{n-2} `.
Q`4`. What are some properties of Fibonacci numbers?
Answer:
- Each Fibonacci number is the sum of the two preceding ones.
- The ratio of consecutive Fibonacci numbers converges to the golden ratio (` \phi \approx 1.6180339887 `).
- Fibonacci numbers have connections to various natural phenomena and mathematical concepts, including geometry, fractals, and number theory.
Q`5`. Where do we find occurences of Fibonacci numbers in real life?
Answer: They appear in nature, such as the arrangement of petals in flowers, branching in trees, and the spirals of shells and galaxies.
Q`6`. Are there other Fibonacci-like sequences?
Answer: Yes, there are many variations and generalizations of the Fibonacci sequence, such as Lucas numbers, Pell numbers, and Tribonacci numbers, each with its own unique properties and applications.
Q`7`. Is there a formula to directly calculate the `n`th Fibonacci number?
Answer: Yes, there are various formulas, including Binet's formula, which directly calculates the `n`th Fibonacci number using the golden ratio. Another approach is using matrix exponentiation, which provides a way to compute Fibonacci numbers efficiently in logarithmic time.