# How are Fibonacci numbers expressed in nature

### Fibonacci numbers

**Fibonacci sequences:**These are consequences (a

_{n})

_{n}real numbers, which are defined as follows:

Be a_{0}, a_{1} real numbers.

Define inductively for n ≥ 2

_{n + 1}= a

_{n}+ a

_{n-1}for n ≥ 1

Expressed differently:

Now be a_{0} = 0, a_{1} = 1.

Question: For example, how big is a_{40}? or also a_{300} ?

- It is a

_{40}= 102.334.155 = 3 x 5 x 7 x 11 x 41 x 2161

And a

_{300}= 222232244629420445529739893461909967206666939096499764990979600

For n ≥ 1 let

v (n) = |

So this is a vector in

**R.**

^{2}.

Here are the first 8 vectors v (1), ..., v (8):

They are all close to the straight line with an incline

(You can formulate this as follows: the given value is just lim a

_{n-1}/ a

_{n})

and indeed they "commute" around this straight line - why?

We operate here with the matrix

**R.**

^{2}. The following applies:

The following calculation rules apply to the eigenvalues:

It is advisable to also consider the following vector:

So it holds for all n ≥ 0:

because

- v (0)
_{+}is an eigenvector with eigenvalue φ_{+}, - v (0)
_{-}is an eigenvector with eigenvalue φ_{-},

If we only consider the first coordinate (the x-coordinate) of v (n), we get:

This is called the **Binet's formula** (he published it in 1842, but it was already known to Euler and D. Bernoulli).

- Here, for clarification, the vectors v (0), ..., v (4):

The two inclined straight lines are the natural lines of the matrix A.

For each of the vectors v (0), ..., v (4) it is shown how it is the sum of a vector in the eigenline g

_{+}=

**R.**b

_{+}and a vector in the eigenline g

_{-}=

**R.**b

_{-}writes:

_{+}+ v (n)

_{-}with v (n)

_{+}in g

_{+}and v (n)

_{-}in g

_{-}

_{+}, the red is (parallel to) v (n)

_{-}.) And the following applies:

_{+}= λ

_{+}v (n)

_{+}and v (n + 1)

_{-}= λ

_{-}v (n)

_{-}

_{-}converges to 0, so v (n) differs from v (n)

_{+}for large n only very little.

Since one knows that v (n) is an integer vector, one can in this way v (n) **exactly** to calculate! We are only interested in the x-coordinate of v (n), because this is the i-th Fibonacci number a_{n}.

### More information

- Is a libertarian ideal
- How is Btech Biotech doing at VIT
- What determines the value of bitcoins
- Are wireframes UX or UI
- The minimum wage increases incentives for automation
- Are aircraft that burn biofuel run more efficiently?
- How can I just fill nail holes?
- Why are data types important in SQL
- How do I keep my individuality
- Is ego death real
- Why does my dog poop blood
- How Quora Becomes a Dating Site
- What is the origin of hairdressing quartets
- Anyone who has discovered the universe is expanding
- What is a matrix scheme
- Where can I learn to steer properly
- Why did Hitler keep his mustache
- Why should I study business in college?
- Which film has inspired the people of the world?
- How do I get a writing agent
- How much revenue does Healthcaremagic
- What is heterogeneity
- Is jogging useful for burning fat?
- Can someone upload ICSE English Paper 2