Analysis: In this question, the changes are as follows: ① the number of operations, that is, the number of times to take the ball; ② Number of balls taken out; ③ The number of balls left in the box after each ball is taken out; ④ Number of balls put back each time ⑤ Number of balls put into the box each time; ⑥ Number of balls in the box after each operation. Each quantity varies with the number of operations. For this reason, the situation of each operation is listed in a table, and the rules of data can be found on the data in the table:
Running time 123... 10
The number of balls taken out is 1 2 3 … 10.
The number of balls left in the box is 0 27 ... A.
The number of balls put back is 4 8 12...b
Increase the number of balls in the box by 3 69...c
Total number of balls 4 10 19 … D
In the above table, if we can find the data of A, B, C and D, then the problem will be solved. It is easy to get the result that b is 4N and c is 3N from the table. So the result of the required d is obvious: the number of balls after each change is: 1, 1+3=4, 10= 1+3+6, 1+3+6 = 65438+. That is, d is 166.
Note: When solving this kind of problems, you should list the results of each process in a table, then observe the changes of data, find the rules from the changed data and draw conclusions.
Example 2: A party has 65,438+00 friends. If everyone only shakes hands with others once, how many times will 65,438+00 people shake hands? What if n friends?
Analysis: Students must understand: 1) Every two people shake hands; 2) The result of shaking hands between A and B and the result of shaking hands between B and A can only be regarded as one result. 3) Let 10 be A 1, A2, A3, A4, A5, A6, A7, A8, A9, A 10. Then A 1 shook hands with 9 other people for 9 times; A2 shook hands with the other 8 people 8 times; A3 shook hands with the other seven people seven times; ............. ……A9 and A 10 shake hands 1 time. So the handshake times are 9+8+7+6+5+4+3+2+ 1=45 (times).
Note: When solving this kind of problems, all the results should be given one by one according to certain rules, so as to sort out all the results.
The second category: numerical problems.
Example 3: Look at the numbers in the next column in turn. What is its arrangement law? Please write down the next three numbers. Can you name the numbers 100th, 2004th and 10000th?
① 2,-2,2,-2,2,-2,……
② - 1,3,-5,7,-9, 1 1,……
③ - ,,- ,,- ……
Analysis:
It is easy to find that this number is composed of positive and negative numbers with an absolute value equal to 2, that is, the odd number is 2 and the even number is -2. So the next three numbers are 2, -2, 2. No. 1000 is -2, No.2004 is -2 andNo. 10000 is -2.
(2) In addition to the sign change, it is easy to find that this number is odd; The symbol is negative and positive alternately; (Odd numbers are negative and even numbers are positive. Therefore, the symbol can be determined by (-1)N as the coefficient of each number. Odd numbers are often represented by (2N- 1), and the nth number of this series can be represented by (-1)N(2N- 1). The next three numbers in the original series are:-13, 15 and -65435. 100 is 199, 2004 is 4007, and 10000 is 19999.
(3) It is easy to find that the symbol characteristics of this series are the same as the second term, which can be expressed by (-1) n. And each score can be regarded as the reciprocal of an even number, which means that the nth number of this series can be expressed as (-1) n, so the last three numbers are,-,. No. 100, No.2004,No. 10000.
Note: It is not difficult for students to find the numerical law in this example. They only need to know the representation methods of a series of special sequences such as odd-numbered sequences and even-numbered sequences. Of course, the expression of symbols also needs to be mastered.
Example 4: What laws will you find by studying the following formula?
1×3+ 1=4=22
2×4+ 1=9=32
3×5+ 1= 16=42
4×6+ 1=25=52
Please use the formula to express the law of discovery: ▁ ▁ ▁ ▁.
Does this formula apply to all integers?
Analysis: Look for "1"in the first formula; Look for "2" in the second formula; ……; Look for "n" in the nth formula. At the same time, look for the numbers related to "1" and "2", ... and "n" are in the corresponding formulas. If the positions "1", "2", ... and "n" are found to be fixed, the "n" in the nth formula is located in "1", "2", ... and the corresponding first data in the question "N+ 1" can be The third data is the constant of 1, the fourth data is the result of (N+ 1)2, and the final conclusion is clear (N+ 1)2. Therefore, the law of discovery is expressed as:
N(N+2)+ 1 = N2+2N+ 1 =(N+ 1)2 .
Example 5: Observe the following types:
13+23=9=( 1+2)2
13+23+33=36=( 1+2+3)2
13+23+33+43=( 1+2+3+4)2
……
13+23+33+43+……+993+ 1003=?
Analysis: It is not difficult to find various characteristics from the given three conditions: the cubic sum of several continuous natural numbers starting from 1 is equal to the square of the sum of these numbers. Students can easily find that the nth formula is as follows:
13+23+33+……+N3 =( 1+2+3+……+N)2 .
Therefore,13+23+33+43+…+993+1003 = (1+2+3+4+…+99+100) 2 = 50502.
(Through incomplete induction, it is not difficult to prove the conclusion of N type. Limited to space, it is not proved here. )
The third category: geometric figure types.
Example 6: Draw a picture with a matchstick as shown in the picture:
(1) Fill in the following table:
Graphic number ① ② ③ ④ ⑤
Number of matchsticks
(2) How many matches does the nth number need?
Analysis: when solving such problems, the method is clear; Is to turn the graphic problem into a digital problem, and then find out the law from the characteristics of numbers to answer.
Obviously, there are three matchsticks in the first figure; There are 9 matchsticks in the second figure; The third number is 18 matchsticks; There are 30 matchsticks in the fourth figure; ……
and 3 = 1×3； 9=3×3=( 1+2)×3; 18=6×3=( 1+2+3)×3; 30= 10×3=( 1+2+3+4)×3……
So the number of matchsticks in the nth graph is: (1+2+3+...+n) × 3. So it is not difficult to fill in every data in the form.
There are some questions like this for your reference:
1. When a line segment is marked with dots, there are three line segments in * * *. If you mark a point again, there are six line segments in * * * at this time ... and so on, how many line segments are there in * * * in the nth graph?
2. Draw a line segment from the vertex of the triangle to its opposite side. At this point, there are three triangles (as shown in Figure 2); If a line segment is drawn to the opposite side, there are six triangles in the figure (as shown in Figure 3); ..... and so on, then how many triangles are there in n * *?
Note: (1) When calculating the number of graphs, if we can grasp: first single, then two compound, then compound again and again ... and so on, we can count all the corresponding conclusions, which are not easy to repeat and omit.
(2) Only by knowing the laws and general expressions of some special sequences can we solve such problems more easily. Table below:
Natural sequence 123...n
Even sequence 2 4 6...2n
Odd sequence 135...2n- 1
Square of natural number 1.49...N2
The sum of the first n natural numbers is 1.
( 1) 1+2
(3) 1+2+3
(6) …… 1+2+3+……+N
()
The sum of the first n odd numbers is 1
( 1) 1+3
(4) 1+3+5
(9) …… 1+3+5+……+(2N- 1)
(N2)
The sum of the first n even numbers is 2.
(2) 2+4
(6) 2+4+6
( 12) …… 2+4+6+……+2N
N(N+ 1)
In order to further consolidate this knowledge, the following exercises are for your reference:
1) What laws will you find by observing the following?
3×5= 15=42- 1
5×7=35=62- 1
……
1 1× 13= 143= 122- 1
Use a formula containing only one letter to express the rule you guessed.
2) Observe the following types:
A 1=5× 1-3=2
A2=5×2-3=7
A3=5×3-3= 12
A4=5×4-3= 17
……
(1) According to the above law, guess and calculate AN=
(2) when N= 100, A 100=
Do you like Lamian Noodles? The master of the ramen noodle restaurant uses a rough noodle to pinch the two ends together and stretch them, then knead them, stretch them again and repeat them several times, and then pull this rough noodle into many fine noodles, as shown in the figure. Knead and stretch it several times like this, can you pull out 128 fine noodles?
4) As shown in the figure, the sides of the square are all 1, which are stacked according to the rules in the figure. If they are called the first floor, the second floor, the third floor, ... and the nth floor from top to bottom, please fill in the form:
Number of layers of small cube arrangement N 12345...n
The number of cubes at the lowest level is1.36 ...
Mathematical problems can be divided into two categories, one is the application of mathematical laws, and the other is the discovery of mathematical laws. The problem of applying mathematical laws refers to the problem that students need to solve by applying the mathematical laws they have learned before. The problem of discovering mathematical laws refers to a problem that has nothing to do with the mathematical laws that students have learned before. Students are required to find out the laws from known things before they can solve them. Most of the math problems that students do belong to the first category.
The discovery of mathematical law problems can enhance students' innovative consciousness and improve their innovative ability. Therefore, in recent years, people began to pay more attention to this kind of mathematical problems. Especially in the last two years, the senior high school entrance examination in most cities in China has such questions. The idea of discovering mathematical laws and solving problems can not only improve students' test scores, but also help to cultivate innovative talents.
First, we should be good at grasping the principal contradiction.
Some topics look big and complicated, but in fact there are not many key contents. Make a detailed analysis of the topic, get rid of the rough, get rid of the false and keep the true, and extract the main and key contents, so that the difficulty of the topic will be greatly reduced and the problem will be solved.
In addition, in 2006, Shaoyang Junior High School Graduation Examination Paper Mathematics Test (Curriculum Reform Area), "The spiral in the figure consists of a series of isosceles right-angled triangles, the serial numbers are ①, ②, ③, ④, ⑤ ..., and the length of the hypotenuse of the nth isosceles right-angled triangle is _ _ _ _ _ _ _ _." It can also be solved according to this idea.
Second, we must grasp the variables in the topic.
Finding the topic of mathematical laws will involve one or several variables. The so-called finding the law, in most cases, refers to the changing law of variables. Therefore, grasping the variables is equivalent to grasping the key to solving the problem.
For example, as shown in the following figure, if the ground is paved with black and white square bricks of the same specification, the third figure will have black brick, and the first figure needs black brick (represented by inclusion algebra). (2006 Hainan Junior High School Entrance Examination Mathematics Subject Test (Curriculum Reform Area))
The key to this question is how many black tiles do you need for the first picture?
In these three pictures, the four black tiles in front are unchanged, while the black tiles in the back are changed. Their numbers are: 0×3 black tile in the first picture, 1×3 black tile in the second picture, 2×3 black tile in the third picture, and so on, (n- 1)×3 black tile in the nth picture. So the nth graph has 4+(n- 1)×3 black tiles.
In 2006, the unified entrance examination for senior high schools (technical secondary schools) in the experimental area of curriculum reform in Yunnan Province also had a similar topic: "Observe the arrangement law of small circles in figures (L) to (4) and continue to arrange them according to this law. Remember that the number of small circles in the nth graph is m, then m= (represented by an algebraic expression containing n). "
Third, be good at comparison.
Only by comparison can we tell. Through comparison, we can find the similarities and differences of things, and it is easier to find the changing law of things.
To find a regular topic, we usually give a series of quantities in a certain order, which requires us to find a universal law according to these known quantities. The laws revealed often contain the serial numbers of things. Therefore, it is easier to find the mystery by comparing variables with serial numbers.
For example, observe the following numbers: 0, 3, 8, 15, 24, ... try to write the number 100th according to this rule. "
To solve this problem, we can first find the general law, and then use this law to calculate the number 100. Let's compare the related quantities together:
Numbers given: 0, 3, 8, 15, 24, ...
Serial number: 1, 2, 3, 4, 5,.
It is easy to find that each term of the known number is equal to the square of its serial number minus 1. So the nth term is n2- 1, and the first term 100 is 1002- 1.
If the topic is complex or contains many variables. When solving a problem, we should not only consider the serial number of the known number, but also consider other factors.
For example, Rizhao City in 2005, the senior high school entrance examination questions "known the following equations:
① 13= 12;
② 13+23=32;
③ 13+23+33=62;
④ 13+23+33+43= 102 ;
…… ……
According to this law, the fifth equation is. "
This topic, in a given equation, the addend on the left is changing, the base of the addend is changing, and the sum on the right is also changing. So there are many factors to compare. As far as the left is concerned, from top to bottom, it is found that the addend increases in turn 1. Therefore, the fifth equation should have five addends; Comparing the radix of addend from left to right, it is found that they are all natural numbers. Therefore, the left side of the fifth equation is 13+23+33+43+53. Looking at the right side of the equation, the exponent has not changed, but the cardinality has changed. On the left side of the equation, the exponent has not changed, but the cardinality has changed. Comparing the bases on both sides of the equation, it is found that the base of sum is equal to the base of addend. So the radix on the right side of the fifth equation is (1+2+3+4+5), and the sum is 152.
Fourth, be good at finding the cycle of things.
Some topics contain the circular law of things. If we find the circular law of things, other problems will be solved.
For example, the math problem of the senior high school entrance examination in Yulin City in 2005: "Observe the arrangement of the following balls (where ● is a solid ball ○ is a hollow ball):
●○○●●○○○○○●○○●●○○○○○●○○●●○○○○○●……
From the 1 ball to the 2004 ball, * * * has a solid ball. "
These balls, from left to right, are arranged in a fixed order, and are circulated once every 10 balls, with a circulation interval of ● There are three solid balls in each loop. As long as we know how many cycles are included in 2004, it is easy to calculate the number of solid balls. Because 2004÷ 10=200 (remainder 4). So in 2004, there were 200 knots, and there were 4 balls left. There are 200×3=600 solid balls in 200 cycle segments, and there are 2 solid balls in the remaining 4 balls. So, a * * * has 602 solid balls.
Fifth, we should grasp the hidden invariants in the topic.
Some topics have changed in form, but the essence has not changed. As long as we always pay attention to finding its invariants in the process of observing morphological changes, we can reveal the essential laws of things.
For example, in 2006, Wuhu City (experimental area of curriculum reform) graduated from junior high school. Please carefully observe the transformation law of equilateral triangle in the picture and write down the mathematical facts you found about the distance from one point to three sides in equilateral triangle.
In these three figures, the white triangle is an equilateral triangle with three black triangles embedded in it. From left to right, the upper two black triangles rotate clockwise, but their shapes have not changed, and of course the height of the black triangles has not changed. In the first picture from the left, the sum of the heights of the black triangles is the sum of the distances from one point to three sides in the equilateral triangle, and in the last picture, the sum of the heights of the three black triangles is the height of the equilateral triangle. Therefore, the sum of the distances from any point to three sides in an equilateral triangle is equal to its height.
Sixth, try to calculate
Looking for rules, of course, looking for mathematical rules. Mathematical laws are mostly analytical expressions of functions. Analytic expressions of functions often include mathematical operations. Therefore, to find the law, to a great extent, is to find a mathematical expression that can reflect the known quantity. Therefore, starting with calculation and trying to do some calculations is also a good way to solve the problem of finding the law.
For example, Hanchuan City observed the following kinds in the 2006 senior high school entrance examination paper Mathematics: 0, x, x2, 2x3, 3x4, 5x5, 8x6, ... Try to write the 10 formula according to this rule. "
This problem contains two variables, one is the index of each item, and the other is the coefficient of each item. It is not difficult to see that the index of each term is equal to its serial number minus 1, but the changing law of the coefficient is not so easy to find. However, if we take out the coefficient and try to do some simple calculations, it is not difficult to find the changing law of the coefficient.
Arrangement of coefficients: 0, 1, 1, 2, 3, 5, 8, ...
Observe the arrangement of coefficients from left to right, and find the sum of two adjacent terms in turn. You will find that this and the latter are exactly the same. That is to say, the sum of the coefficients of two adjacent terms in the original series is equal to the coefficient of the latter term. Using this rule, it is not difficult to deduce that the coefficient of the eighth term of the original sequence is 5+8= 13, the coefficient of the ninth term is 8+ 13=2 1, and the coefficient of the first term 10 is13+21.
So the original number 10 is 34x9.
All roads lead to Rome. There are many ways to solve the problem of finding a method. Here is just a brief summary of the "common" problem-solving ideas. Interested teachers can further study new ways to solve this kind of problems from the angles of solving equations, Lagrange interpolation theorem and solving resolution function.
(1)1,(2)1+5 = 6, (3)1+5+9 =16. What is the number n? Please write down the process.
The first number is 1.
The second number is 1+5 = 6.
The third number is 1+5+9 = 15.
The fourth number is 1+5+9+ 13 = 28.
From the above law, it can be found that each additional layer increases by 4 more than the previous layer.
The solution of the last added number of the nth number is 4 × (n- 1)+ 1 ∴ The sum of the numbers added continuously from 1 to the last number is (1+ the last number) ÷2n.
Then the first two formulas are combined to get the nth number [2+4? (n- 1)]⊙2n means n(2n- 1).
There is a column number: 1,1/2,2/1/3,2/2,3/1,1/4,2/3,3/.
(1) What is the first number after the number 1/5?
(2) If you count from the first number on the left to the right, what is the number of 1/9 in this column?
Solution: The sequence is: 1,1/2,2/1/3,2/2,3/1,1/4,2/3,3/. So the first number after the number is =. From the first number on the left to the right, 1 to 1 is 1, 1 is 2,3,4. To is a number of 8, so 1+2+3+4+5+6+7+8=36. So there are 37 numbers in this column.
3,10,29,66 What's the next number?
Solution: 3 =13+210 = 23+229 = 33+266 = 43+2. The next number is: 53+2= 127.
The number of (1)-1,2,-4, 8,-16, 32, ..., 10 is _ _ _ _ _ _
Each number can be written as
The frequency is 0, 1, 2, 3. ...
When the number of times is even, it is preceded by a negative sign.
So the number 10 is expressed as.
(2) 1, -3, 5, -7, …, 15 is _ _ _ _ _ _.
The absolute value of each number is expressed as,,, (n is a number).
If the number is even, it will be preceded by a negative sign.
So the absolute value of the number 15 is.