Excuse me, do you know where to start writing high school math papers? Help people in a hurry. I'm here to feel everyone first.

I used to be a representative in math class, and it was not difficult to write things, such as the study of Fibonacci series.

Fibonacci series,

Also known as golden section series, it refers to such a series: 1, 1, 2, 3, 5, 8, 13, 2 1, ... Mathematically, Fibonacci series is defined recursively as follows: F0=0, F65438+. =2, n∈N*) Fibonacci sequence has direct applications in modern physics, quasicrystal structure, chemistry and other fields. Therefore, the American Mathematical Society published a mathematical magazine named Fibonacci Series Quarterly from 1963 to publish the research results in this field.

definition

Fibonacci series refers to 0, 1, 1, 2, 3, 5, 8, 13, 2 1, 34, 55, 89, 144, 233, 377.

Specifically, item 0 is 0, and item 1 is item 1.

This series begins with the second term, and each term is equal to the sum of the first two terms.

The inventor of Fibonacci sequence is Italian mathematician Leonardo Fibonacci.

recurrence formula

Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8, 13, 2 1, 34, 55, 89, 144, ...

If F(n) is the nth term of the series (n∈N*), then this sentence can be written as follows:

Obviously, this is a linear recursive sequence.

General term formula

(As mentioned above, it is also called "Binet.Alfred formula", which is an example of using irrational numbers to represent rational numbers. )

Note: at this time, a 1= 1, a2= 1, An = A (n- 1)+A (n-2) (n >: =3, n∈N*).

Derivation of general term formula

Method 1: The characteristic equation (linear algebraic solution) was used.

The characteristic equation of linear recursive sequence is:

X^2=X+ 1

solve

X 1=( 1+√5)/2，X2=( 1-√5)/2。

Then f (n) = c1* x1n+C2 * x2n.

∫F( 1)= F(2)= 1

∴c 1*x 1+c2*x2=c 1*x 1^2+c2*x2^2= 1

The solution is c1=1√ 5, C2 =-1√ 5.

∴f(n)=( 1/√5)*{n+fn=fn,fn-fn=f[0, 1]n=f[ 1, 1](n- 1)，

n

1

2

three

four

five

six

seven

eight

nine

10

…

[Mathematics] Function

1

four

five

nine

14

23

37

60

97

157

…

[Mathematics] Function

1

three

four

seven

1 1

18

29

47

76

123

…

Fn-Fn

1

1

2

three

five

eight

13

2 1

34

…

Fn+Fn

2

seven

nine

16

25

4 1

66

107

173

280

…

(2) Any Fibonacci-Lucas sequence can be obtained by the sum of the finite terms of Fibonacci sequence, such as

n

1

2

three

four

five

six

seven

eight

nine

10

…

F[ 1， 1](n)

1

1

2

three

five

eight

13

2 1

34

55

…

F[ 1， 1](n- 1)

1

1

2

three

five

eight

13

2 1

34

…

F[ 1， 1](n- 1)

1

1

2

three

five

eight

13

2 1

34

…

[Mathematics] Function

1

three

four

seven

1 1

18

29

47

76

123

…

Golden feature and twin Fibonacci-Lucas sequence

Another common property of Fibonacci-Lucas sequence: the absolute value of the difference between the square of the middle term and the product of the previous two terms is a constant value,

Fibonacci series: |1*1-1* 2 | = | 2 * 2-1* 3 | = | 3 * 3-2 * 5 | = | 5 * 3 * 8 | = | 8 *

Lucas sequence: | 3 * 3-1* 4 | = | 4 * 3 * 7 | = … = 5.

F [1, 4] series: | 4 * 4-1* 5 | =11.

F [2 2,5] series: |5*5-2*7|= 1 1.

F [2 2,7] series: |7*7-2*9|=3 1

Fibonacci series has the minimum value of 1, that is, the ratio of the front and rear terms is close to the golden section ratio, which is the fastest. We call it the golden feature, and the golden feature sequence of 1 is only Fibonacci sequence, which is the only sequence. The golden feature of Lucas sequence is 5, which is also the only child sequence. The first two series with only coprime are Fibonacci series and Lucas series.

The golden characteristics of F [1, 4] and f [2,5] are both 1 1, which are twin sequences. F [2,7] also has a twin sequence: F [3,8]. The other two coprime Fibonacci-Lucas sequences are twin sequences, which are called twin Fibonacci-Lucas sequences.

Generalized Fibonacci sequence

The golden characteristic of Fibonacci sequence is 1, which reminds us of Pell sequence: 1, 2,5,12,29, …, and | 2 * 2-1* 5 | = | 5 * 2 */kloc-.

The recurrence rules of Pell sequence Pn are: P 1= 1, P2=2 = p (n-2)+p (n- 1).

Accordingly, we can derive the third term from the first two terms: f(n) = f(n- 1) * p+f(n-2) * q, which is called generalized Fibonacci sequence.

When p= 1 and q= 1, we get Fibonacci-Lucas sequence.

When p= 1 and q=2, we get the number of Pell-Pythagoras strings (the set of series related to a right triangle with an integer side length).

When p=- 1 and q=2, we get arithmetic progression. When f 1= 1 and f2=2, we get that the natural sequence 1, 2, 3, 4 ... is characterized by the difference between the square of each number and the product of the two numbers before and after it is 1 (the difference of arithmetic progression is called natural feature).

Fibonacci sequence p = 1 in a broad sense has similar golden characteristics, pythagorean characteristics and natural characteristics.

When f 1= 1, f2=2, p=2 and q= 1, we get the geometric series 1, 2,4,8, 16. ...

Related mathematics

permutation and combination

There is a flight of stairs with 10 steps, and it is stipulated that each step can only span one or two steps. How many different ways are there to climb 10 steps?

This is a Fibonacci sequence: there is a way to climb the first step; There are two ways to climb two steps; There are three ways to climb three steps; There are five ways to climb these four steps. ...

1, 2, 3, 5, 8, 13 ... So there are 89 ways to climb the tenth level.

Similarly, a unified coin was thrown 10 times. How many possible situations are there for head discontinuity?

The answer is (1/√ 5) * {[(1+√ 5)/2] (10+2)-(1-√ 5)/2) (10+2.

Find the general formula A (1) = 1, a (n+1) =1/a (n).

Through mathematical induction, we can get: a(n)=F(n+ 1)/F(n). Substitute the general term of Fibonacci sequence and simplify it to get the result.

Rabbit reproduction problem

Fibonacci series is also called "rabbit series" because mathematician Leonardo Fibonacci introduced it by taking rabbit breeding as an example.

Generally speaking, rabbits can reproduce two months after birth, and a pair of rabbits can give birth to a pair of rabbits every month. If all rabbits don't die, how many pairs of rabbits can you breed in a year?

We might as well take a pair of newborn rabbits to analyze:

In the first month, the rabbits were infertile, so they were still a couple.

Two months later, two pairs of rabbits were born.

Three months later, the old rabbit gave birth to another pair. There are three pairs of rabbits because they can't reproduce.

-

By analogy, the following table can be listed:

Number of past months

1

2

three

four

five

six

seven

eight

nine

10

1 1

12

Logarithm of offspring

1

1

1

2

three

five

eight

13

2 1

34

55

Eighty-nine

Logarithm of adult rabbit

1

1

2

three

five

eight

13

2 1

34

55

Eighty-nine

144

Population logarithm

1

1

2

three

five

eight

13

2 1

34

55

Eighty-nine

144

233

Logarithm of young rabbits = Logarithm of adult rabbits in last month

Logarithm of adult rabbits = logarithms of adult rabbits last month+logarithms of young rabbits last month.

Logarithm of population = Logarithm of adult rabbits this month+Logarithm of young rabbits this month.

It can be seen that the logarithm of young people, the logarithm of adults and the logarithm of population all constitute a series. This series has a very obvious feature, that is, the sum of the two adjacent items in front constitutes the latter item.

This series was written by Italian mathematician Fibonacci in. Besides the property of a(n+2)=an+a(n+ 1), the general formula of this series can also be proved as an = (1/√ 5) * {[(1+√ 5)/.

Sequence and matrix

Fibonacci sequence 1, 1, 2, 3, 5, 8, 13,. has the following definitions.

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

F( 1)= 1

F(2)= 1

For the following matrix multiplication

F(n+ 1) = 1 1 F(n)

Female (male) 10 female (male-1)

Its operation is to multiply the matrix 1 1 on the right by the matrix F(n):

10 Fahrenheit (n- 1)

F(n+ 1)=F(n)+F(n- 1)

F(n)=F(n)

It can be seen that the multiplication of this matrix completely conforms to the definition of Fibonacci sequence.

Let the matrix A= 1 1 iterate n times, and we can get: f (n+1) = a (n) * f (1) = a (n) *1.

1 0 F(n) F(0) 0

This is the definition of matrix multiplication of Fibonacci sequence.

Another algorithm of matrix multiplication, A n (n is an even number) = A (n/2) * A (n/2), so that we can realize matrix multiplication with logarithmic complexity through the idea of dichotomy.

Therefore, the answer can be obtained recursively.

Another solution of sequence value;

f(n)=[(sqrt(5)+ 1)/2)^ n]

Where [x] represents the integer closest to x.

Fibonacci arc

Fibonacci arc, also called Fibonacci fan line. First of all, this trend line is drawn based on two endpoints, such as the lowest point reversing to two points on the highest point line. Then draw an invisible (invisible) vertical line through the second point. Then, draw the third trend line from the first point: 38.2%, 50%, 6 1.8% invisible vertical lines intersect.

Fibonacci arc is the horizontal price of potential support points and resistance points. Fibonacci arc and Fibonacci fan line are often drawn at the same time in the chart. Support points and resistance points are obtained from the intersection of these lines.

It should be noted that the intersection of the arc and the price curve will change according to the numerical range of the chart, because the arc is a part of the circle, and their formation is always the same.

He died in 1 170, and his native place was Pisa. He is called "Leonardo of Pisa". 1202, he wrote the book Liber Abacci. He was the first European to study the mathematical theories of India and Arabia. His father was hired as a diplomatic consul by a business group in Pisa and was stationed in today's Algeria, so Da Vinci was able to study mathematics under the guidance of an Arab teacher. He also studied mathematics in Egypt, Syria, Greece, Sicily and Provence.

The application of Fibonacci sequence in stock market

Time cycle theory is one of the fundamental reasons for the rise and fall of stock prices, which can explain the mystery of the rise and fall of most markets. In the theory of time period, we can not only find the variable inventory with a fixed number of time periods, but also study the relationship between bands. However, no matter how to find variable inventory, Fibonacci series is one of the bases of all kinds of important analysis. This paper will briefly explain Fibonacci series and its relationship with the market.

Tools/raw materials

Steps/methods

Fibonacci sequence was discovered by Italian mathematician Fibonacci in13rd century. A series of numbers in a sequence are usually called magic numbers and odd numbers. The specific series are: 1, 1, 2, 3, 5, 8, 13, 2 1, 34, 55, 89, 144, 233, etc. Starting with the third number in the series, each number is equal to the first two numbers. The quotient of two adjacent terms in Fibonacci series is close to the golden section number of 0.6 18, and the numbers related to this number, such as 0. 19 1, 0.382, 0.5, 0.809, etc., constitute important numbers in the stock market for calculating the time and space of the market.

Fibonacci numbers can be found in the laws of various phenomena from the whole universe to small molecular atoms, from time to space, from nature to human society, politics, economy and military affairs. Notre Dame de Paris, Eiffel Tower, Egyptian pyramids and other world-famous buildings can all find the shadow of 0.6 18 from them. The themes of famous paintings, photography, sculpture and other works are all in 0.6 18. The announcer stood on the stage at 0.6 18, making the sweetest and most beautiful sound. The navel is 0.6 18 of the length of the human body, and the knee is 0.6 18 of the length from the sole of the foot to the navel. The use of 0.6 18 is also ubiquitous in the war, from the manufacture of weapons to the deployment of troops to the use of war time. It is said that Napoleon the Great was defeated by the golden section.

Fibonacci numbers frequently appear in financial market analysis methods. For example, in wave theory, a bull market can be represented by the rising wave of 1, or by five lower-level wavelets, and can be further subdivided into 2 1 or 89 wavelets; In the spatial analysis system, the height of the rebound market is usually 0.382, 0.5 and 0.18 of the previous downward trend; The callback market is usually 0.382, 0.5, 0.6 18 of the upward trend ahead.

Fibonacci sequence has two important meanings in practical operation:

The first practical significance lies in the sequence itself. The first ten figures in this series play an important role in the time relationship of daily market. When the market is in an important critical change time area, these figures can determine the specific change time. When using Fibonacci series, we can calculate the future market from an important stage in the market, and the probability of changing direction when the market arrives is greater.

Figure 1 comprehensive index (1a0001) Daily chart of July 29, 2009-65438+February 3 1

As shown in figure 1, the time relationship of the comprehensive index (1a000 1) from 3478 on August 4th, 2009 to 2639 on September 6th, 2009 is 2 1 trading day. The period from the low point of 2639 on September 1 day to the high point of 3068 on September 18, 2009 is 13 trading days, the low point of 27 12 on September 29, 2009 is 2 1 trading days, and it will be in 2009. It takes 55 trading days to reach the second highest point of the year 1 1 on October 24th, 2009.

Fig. 2 Daily weekly chart of July 10 to February 1 comprehensive index (1A00 1) in 2009.

As shown in Figure 2, the comprehensive index (1A00 1) runs for five weeks from the high point of 3478 on August 4th, 2009 to 2639 on September 4th, 2009. The time from the low point of 2639 on September 4th, 2009 to the high point of 336 1 1 on October 27th, 2009 is 13 weeks.

The application of Fibonacci sequence in stock market

The application of Fibonacci sequence in stock market

The second practical significance is that the derivative graph of this series is the theoretical basis for calculating the future market change time in the vertical period of the market. The derivative sequences of this series are: 1.236, 1.309, 1.5, 1.6 18,1.809,2,2.236. A series of derivative sequences are separated from gold by 0.66.

When using magic sequences, there are mainly six important time calculation methods:

First of all, through the complete time of the falling band, the running time of the rising band in the future market is calculated.

Second, through the complete rising band time, calculate the running time of the falling band in the future market.

These two proportional relationships are just like the relationship between action and reaction that we often see in our lives. It is the same reason that the height of ping-pong ball's vertical landing determines the height of ping-pong ball's rebound after landing.

Thirdly, the final running time of the rising zone is calculated by the time from the low point to the high point of the first sub-band in the rising zone.

Fourthly, the final running time of the descending band is calculated by the time from the high point to the low point of the first sub-band in the descending band.

These two proportional relationships are like the relationship between driving force and inertia that we often see in our lives. When the bows and chords of ancient bows and arrows are pulled apart, the distance the future arrows fly forward directly determines.

Fifthly, the final running time of the future rising band is calculated from the time of two adjacent low points of the first sub-band in the rising band.

Sixth, through the time of two adjacent high points of the first sub-band in the descending band, the final running time of the descending band is calculated.

These two proportional relations are as important as the influence of building foundation width on future height. With the same material, the wider the foundation, the higher the future height.

five

Among the six important time calculation methods, the most important is the parameters actually used in the calculation process. Using different parameters will get different answers. Almost all the important parameters are related to Fibonacci sequence. Due to space reasons, I will bury a foreshadowing here first, and I will elaborate the calculation method for investors in future articles.