What is the Fibonacci sequence? What are some examples in everyday life?

The Fibonacci sequence refers to such a sequence of numbers 、...

The Fibonacci number is found in the arrangement of leaves, branches, stems, etc., of plants. For example, if you take a leaf on the branch of a tree, write it as a number 0, and then count the leaves in order (assuming there is no loss) until you reach the position directly opposite the leaf of the tree, the number of leaves in between is probably a Fibonacci number. The arrival of a leaf from one position to the next directly opposite position is called a loop.

The number of turns a leaf rotates in a cycle is also a Fibonacci number. The ratio of the number of leaves to the number of rotations of the leaves in a cycle is called the leaf order (derived from the Greek word, meaning the arrangement of leaves) ratio. Most leaf ratios are presented as Fibonacci ratios.

The Fibonacci sequence refers to such a sequence of numbers:

This sequence starts with the third term, each of which is equal to the sum of the first two terms.

Its general formula is: [(1 5) 2] n 5 1 5) 2] n 5 [ 5 for root number 5].

It is very interesting that such a series of numbers that are completely natural numbers is actually expressed in irrational numbers.

The sequence has a lot of wonderful properties.

For example, as the number of items in the series increases, the ratio of the previous term to the latter term is closer to the ** segmentation.

There is also a property, starting with the second term, that the square of each odd-numbered term is 1 more than the product of the preceding and following terms, and that the square of each even-numbered term is 1 less than the product of the preceding and subsequent terms

If you see a question where someone cuts an 8x8 square into four pieces and puts it together into a 5x13 rectangle, and pretends to be surprised and asks you: Why 64 65?

In fact, it is taking advantage of this property of the Fibonacci sequence that is the adjacent three items in the sequence, in fact, the area of the two pieces in the front and back is indeed 1, but there is a slender slit in the back diagram, which is not easy for ordinary people to notice.

If you pick two numbers as the start, such as 5, and then add the two items together to form6……You will find that with the development of the series, the ratio of the two terms before and after is getting closer and closer to the split, and the difference between the square of a certain term and the product of the two terms is also alternately different by a certain value.

Fibonacci alias.

The Fibonacci sequence was introduced by the mathematician Leonardo Fibonacci using rabbit breeding as an example, so it is also called the "rabbit sequence".

1. The Fibonacci number can be found in the arrangement of leaves, branches, stems, etc. of the plant. For example, if you pick a leaf on the branch of a tree, mark it as 0, and then count the leaves in order until you reach the position directly opposite those leaves, the number of leaves in between is probably a Fibonacci number. The arrival of a leaf from one position to the next directly opposite position is called a loop.

2. The growth of trees. Because the new shoots often need a "rest" period for themselves to grow before they can germinate new shoots. So, a sapling grows a new branch at intervals, such as a year; In the second year, the new branches "rest", and the old branches still germinate; After that, the old branches germinate at the same time as the branches that have been "resting" for one year, and the new branches that are born in the current year are "resting" the following year.

In this way, the number of branches of a tree in each year constitutes the Fibonacci sequence.

Split relationship with **.

Interestingly, such a completely natural number.

The series of numbers, the general term formula.

But it is an irrational number.

to express. And when n tends to infinity, the ratio of the former term to the latter term is getting closer and closer to the ** split, or the decimal part of the ratio of the latter term to the previous term is getting closer and closer.

1÷1=1，1÷2=，2÷3=。。3÷5=，5÷8=。Vertical scrambling.

The further back you go, the closer these ratios are to the ** ratio.

Prove. a［n+2］＝a［n+1］＋a［n］。Divide a n+1 on both sides at the same time to get :

a［n+2］／a［n+1］＝1+a［n］／a［n+1］。If the limit of a n+1 a n exists, let its limit be x, then lim n-"; a［n+2］／a［n+1］）＝lim［n-》；Zaosun (a n+1 a n ) x. So x=1+1 x.

i.e. x = x+1. So the limit is the ** split ratio.

Fibonacci is used in **or foreign exchange trading to judge** the point of retracement, this indicator is often used in actual combat, many times the key points of Fibonacci really play a certain role, of course, many times will not be accurate. Now there are many people who use it in trading, and many people will refer to these ** split points to operate, so it has also strengthened one of its functions. Asa Forex Community, a variety of indicators to learn and improve analytical skills.

The sixteen properties of the Fibonacci sequence are:

Property 1: Modulo periodicity.

The result of modulo division of a certain number in a series will show a certain periodicity, because a certain number in the sequence depends on the first two numbers, once there are two connected numbers that are modulo division results equal to the modulo division result of the first term 0, then it represents the beginning of a new period, if modulo divides n, then the elements in each cycle will not exceed n n.

Nature 2: Segmentation.

As i increases, fi fi-1 is close to.

Nature 3: Squared and front and back.

From the second term onwards, the square of each odd-numbered term is one more than the product of the preceding and following terms, and the square of each even-numbered term is one less than the product of the preceding and subsequent terms.

Property 4: The n+2 term of the Fibonacci sequence represents the number of all subsets of the set that do not contain adjacent positive integers.

Nature 5: Summation.

Features of the Fibonacci sequence:

Fibonacci numbers have a lot of interesting and surprising properties, and here I will illustrate and prove two of them. Both proofs will use mathematical induction.

1.Mathematical induction.

If you're new to mathematical induction, think of it this way. Imagine that I have a never-ending set of dominoes, and I'm going to stand them all up and form a string of dominoes that will knock each other over forever. To make sure this happens, here's what I need to know:

The first domino was knocked out.

2.Touching any domino will cause the next domino to be knocked down.

In a similar way, we can prove that it is true for all numbers n by proving the following facts:

1.n = 1 (called the beginning of induction).

2.If n = k holds, then n = k + 1 also holds. (This is known as the inductive step.) That is, if all n k is true, then n = k + 1 is also true. )

The Fibonacci sequence refers to such a sequence of numbers 、...

Fibonacci numbers are found in the sensitive arrangement of leaves, branches, and stems of plants. For example, if you select a leaf on the branch of a tree, mark it as a number of 0, and then count the leaves in order (assuming there is no loss) until you reach the position directly opposite the leaf of the branch, then the number of leaves in between is probably a Fibonacci number. The arrival of a leaf from one position to the next directly opposite position is called a loop.

The number of turns a leaf rotates in a cycle is also a Fibonacci number. In a cycle, the ratio of the number of leaf bends to the number of rotations of the leaves is called the leaf order (derived from the Greek word, meaning the arrangement of leaves) ratio. Most leaf ratios are presented as Fibonacci ratios.

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