LESSON 1.3 Fibonacci Sequence Flashcards
What is the Fibonacci sequence?
The Fibonacci sequence is a sequence of numbers where each number is the sum of the two preceding ones, starting with 1 and 1. The sequence begins: 1, 1, 2, 3, 5, 8, 13, and so on.
Who is credited with the invention of the Fibonacci sequence?
The Fibonacci sequence was invented by the Italian mathematician Leonardo Pisano Bigollo, known as Fibonacci or “son of Bonacci”.
How is the Fibonacci sequence formally defined?
It is defined by starting with F1 = 1 and F2 = 1, and for n > 2, Fn is defined as Fn = Fn-1 + Fn-2.
What is a recursive definition in the context of the Fibonacci sequence?
A recursive definition starts with initial values and defines each subsequent term as a function of the previous terms, as seen in how the Fibonacci sequence is generated.
What is a golden rectangle?
A golden rectangle is a rectangle whose side ratio (length to width) equals the golden ratio ϕ.
How has the Fibonacci sequence influenced various fields, according to George Dvorsky (2013)?
The Fibonacci sequence has captivated mathematicians, scientists, artists, and designers for centuries due to its interesting properties and its visibility in nature.
In which natural patterns can Fibonacci numbers be observed?
Fibonacci numbers can be observed in the arrangement of flower petals, the spirals of sunflower seeds, pinecones, broccoli florets, and the arrangement of leaves around plant stems.
How do the growth points of plants illustrate the Fibonacci sequence?
As a trunk grows and produces branches, the number of growth points follows the Fibonacci sequence: starting with one trunk, then adding branches, resulting in totals of 2, 3, and 5 growth points.
How to formally define the Fibonacci sequence?
To formally, define the Fibonacci sequence, we start by defining F1 = 1 and F2 = 1. For n > 2, we
define
Fn : = Fn−1 + Fn−2:
EXAMPLE:
- ( F0 = 0 )
- ( F1 = 1 )
- ( F2 = F1 + F0 = 1 + 0 = 1 )
- ( F3 = F2 + F1 = 1 + 1 = 2 )
- ( F4 = F3 + F2 = 2 + 1 = 3 )
- ( F5 = F4 + F3 = 3 + 2 = 5 )
What happens to the ratio Fn/Fn-1 as n becomes larger in the Fibonacci sequence?
As n gets larger, the ratio Fn/Fn-1 approaches the golden ratio ϕ, which is approximately 1.618.
How is the golden ratio formally defined?
As the limit of the ratio of consecutive Fibonacci numbers:
ϕ = lim(n)→∞ Fn/Fn-1
This means that as n becomes larger, the ratio of two consecutive Fibonacci numbers approaches the golden ratio.
What is the exact value of the golden ratio as a mathematical constant?
ϕ = 1+√5/2
What is the exact value of the golden ratio?
Approximately 1.6180339887.
How to express nth Fibonacci number using binet formula?
Fn = ϕ^n - (1 - ϕ)^n / √5
where in ϕ = 1 + √5 / 2
What is the symbol lim(n)→∞ used to represent?
The symbol lim(n)→∞ represents “the limit as n approaches infinity,” which is a concept from calculus.
