Mathematics Fibonacci series problems

The answer is wrong, it should be 233 pairs. The analysis is as follows: We might as well take a newly-born pair of rabbits for analysis: In the first month, the rabbits have no reproductive ability, so they are still a pair; Two months later, a pair of rabbits was born, and there were two pairs; Three months later, the old rabbit gave birth to another pair, because the little rabbit was not yet able to reproduce, so there were three pairs in total; And so on, to list the following table:

Months elapsed: --1---2---3---4---5---6---7---8---9---10---11---12 --13 (after one year) Rabbit logarithms: --1---1---2---3---5---8--13--21--34--55--89--144 --233 Alternatively, using the general formula f(n)=(1 5)*, using the Fibonacci sequence, and letting n=13, we get:

One year later, there were a total of pairs of rabbits in the fence f(13)=233

Obviously, there are 2 pairs after the first month, 3 pairs after the second month, and 5 pairs after the third month ......satisfying the Fibonacci sequence;

The Fibonacci sequence is defined as a1=a2=1, a(n)=a(n-1)+a(n-2)(n 3);

Assuming a(n)=a(n-1)+a(n-2)(n 3) from a(n)=c n, then c n=c (n-1)+c (n-2) gives n= (n-1) + n-2)(n 3), i.e., 2= +1, =(1 5) 2;Since a(n)=c1 1 n+c2 2 n, and a1=a2=1, then when n=1 and n=2 are taken, c1 1+c2 2=a(1)=1, c1 1 2+c2 2 2=a(2)=1, c1 = 5 5, c2=- 5 5, so a(n) = 5 5;According to the question, there is a correspondence between the number of months and n, then in the twelfth month, n=14, substituting a(n) gives a(14)=377.

This is a high school number series problem with 10 items: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 144It just adds up to no tricks.

The Fibonacci sequence, also known as the splitting sequence, was introduced by the mathematician Leonardo Fibonacci (Leonardoda Fibonacci) with the example of rabbit breeding, so it is also called the "rabbit sequence", which refers to such a sequence 、...Mathematically, the Fibonacci sequence is defined recursively as follows: f(1)=1, f(2)=1, f(n)=f(n-1)+f(n-2)(n>=2,n n*) In the fields of modern physics, quasicrystalline structure, chemistry, etc., the Fibonacci sequence has direct applications, for this reason, the American Mathematical Society has published a mathematical journal under the name of "Fibonacci Quarterly Series" since 1963 to publish research results in this area.

The Fibrache sequence, also known as the ** split sequence, refers to such a sequence: 1 1 2 3 5 8 13 21....

There are two ways to implement the Fibrache sequence, one is in the form of array subscripts, arr[i]=arr[i-1]+arr[i-2]; arr[0]=1;

arr[1]=0;

**：#include

int main()

for(i=0;i<12;i++)

return 0;

In the second method, the principle of exchange number is used, f3=f1+f2; f1=f2,f2=f3

**：#include

int fib(int num)

else }

return f3;

int main()

Ten million 4'21"

**Split ratio (Chinese-to-foreign ratio).

Ten million 4'37"

Why do they all say that seeing is not necessarily believing?

Ten thousand 2'18"

Can the human brain calculate faster than a calculator?

Ten thousand 2'18"

Fibonacci sequence.

The Fibonacci sequence, also known as the splitting sequence, was introduced by the mathematician Leonardo Fibonacci (Leonardoda Fibonacci) with the example of rabbit breeding, so it is also called the "rabbit sequence", which refers to such a sequence 、...Mathematically, the Fibonacci sequence is defined recursively as follows: f(1)=1, f(2)=1, f(n)=f(n-1)+f(n-2)(n>=3,n n*) In the fields of modern physics, quasicrystalline structure, chemistry, etc., the Fibonacci sequence has direct applications, for which the American Mathematical Society has published a mathematical journal named "Fibonacci Quarterly Series" since 1963 to publish research results in this area.

The Chinese name is the Fibonacci sequence.

Foreign name fibonacci sequence, also known as **split sequence, rabbit sequence.

Expression f[n]=f[n-1]+f[n-2](n>=3,f[1]=1,f[2]=1).

Presented by Leonardo Fibonacci.

The latter number is the sum of the first two numbers. The denominator of the multiplicity fraction is always greater than 1, so the value is always less than 1

The numerator always takes the previous denominator, except for the first time when the numerator denominator is 1, the value is equal to 1 2, and the subsequent values are greater than 1 2

And every time the complex fraction is calculated, the denominator in the complex fraction denominator is always the same, and the numerator is always the sum of the previous numerator and the denominator.

This is completely consistent with the law of the Fibonacci sequence.

So what is the value of this simplest fraction of infinite consecution?

That is, what is the limit of the ratio of two consecutive Fibonacci numbers?

Let x=1 (1+1 (1+1 (1+..)

Apparently there is: x=1 (1+x).

That is: x 2 + x - 1 = 0

x=( 5-1) 2=rounding off negative values).

This is the division ratio, and it is also the limit of the ratio of two consecutive Fibonacci sequences.

This is what the landlord said: "getting closer and closer to the ** ratio".

The so-called "as n increases, the gap between the two numbers gets smaller and smaller", in fact, it is getting closer and closer to the limit.

So why is it that "any two numbers keep adding up"?

**The split ratio is actually a matter of the ratio of the outside:

The so-called east-to-foreign ratio is to divide the known line segment into two parts, so that one part is the proportional term between the whole line segment and the other part.

If you set the longer segment to x, the shorter segment is 1-x

So, x 2 = 1*(1-x) [where "1" represents the entire line segment].

That is: x 2 + x - 1 = 0, which is exactly the same as the equation above that solves the simplest infinite continuous fraction.

Note that the whole line segment here is represented by 1, which means that the ** division ratio has nothing to do with the actual length of the line segment.

In the same way, for the Fibonacci sequence, if the ratio of the two terms is examined.

Then, it doesn't matter which two numbers start to add.

Because it is always the ratio of the large number of the two numbers to the sum of the two numbers, this is exactly the same meaning as the ratio of the middle and foreign parts of the ** division.

Moreover, all ratios are always between and 1, except for the first ratio, which is not a ratio to "and".

If the two numbers at the beginning are not the same, then: m, n, m+n, m+2n, 2m+3n, 3m+5n, ,..

It can be seen that it is still according to the law of the Fibonacci sequence, of course, this is a general understanding, and the strict proof depends on the relevant information.

Think about it again, what if the first two numbers of the Fibonacci sequence were 1 and 2? It's different.

It's not the same, except for the first one, it's not the same.

If the two numbers starting are the same, then: m, m, 2m, 3m,..In fact, it is a Fibonacci sequence, but each number is only M times different, and it does not affect the value of the ratio of two consecutive items at all. And from the third term, the coefficients before a exactly form the Fibonacci sequence;

From the second term, the coefficients before b exactly form the Fibonacci sequence;

Thus, the formula for the general term of the Fibonacci sequence is:

The coefficient before the nth number a = (1 5)*

The coefficient before the nth number b = (1 5)*

So the nth number (n 3) is:

1/√5)**a+(1/√5)**b。

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

Item 1 + Item 2 = Item 3 1 + 1 = 2

Item 2 + Item 3 = Item 4 1 + 2 = 3

Item 3 + Item 4 = Item 5 2 + 3 = 5

Item n-2 + Item n-1 = Item n.

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

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946，……

The first two numbers add up to themselves, n+(n+1)=n+2

1,1,2,3,5,8,13...

In addition to the beginning of the 1,1

Any number is equal to the sum of the previous two numbers.

The initial values are x(1)=1 and x(2)=1. Then recursively by the following formula: x(n)=x(n-1)+x(n-2).

The initial values are x(1)=1 and x(2)=1. Then recursively by the following formula: x(n)=x(n-1)+x(n-2).

Name a few values: 1, 1, 2, 3, 5, 8, 13, 21, ......

The Fibonacci sequence was introduced by the mathematician Leonardo Fibonacci using rabbit breeding as an example, so it is also called the "rabbit sequence". Generally speaking, rabbits have the ability to reproduce two months after birth, and a pair of rabbits can give birth to a pair of baby rabbits every month. If all rabbits don't die, how many pairs of rabbits can be bred after a year?

We might as well take a newly-born pair of rabbits for analysis: In the first month, the rabbits have no reproductive ability, so they are still a pair; Two months later, a pair of rabbits was born, and there were two pairs; Three months later, the old rabbit gave birth to another pair, because the little rabbit was not yet able to reproduce, so there were three pairs in total; The following table can be listed by analogy: Number of months elapsed:

1---2---3---4---5---6---7---8---9---10---11---12 Rabbit logarithms: --1---1---2---3---5---8--13--21--34--55--89--144 The numbers 1,1,2,3,5,8 in the table form a sequence. The very obvious feature of this sequence is:

The sum of the first two adjacent terms constitutes the latter term. Proof of this feature: the number of big rabbits per month is the number of rabbits in the previous month, and the number of small rabbits per month is the number of big rabbits in the previous month, that is, the number of rabbits in the previous month, and the number of rabbits in the previous month, are added.