How to verify that a sequence is a geometric or arithmetic sequence?

Teaching objectives

1. To enable students to understand the induction method and understand the principles and essence of mathematical induction.

2. To master the two methods of mathematical induction forensic questions Steps; be able to use "mathematical induction" to prove simple propositions related to natural numbers.

3. Cultivate students' ability to observe, analyze, and demonstrate, further develop students' abstract thinking ability and innovation ability, and allow students to Experience the process of building knowledge and experience the mathematical ideas of analogy.

4. Strive to create a pleasant classroom situation, put students in an atmosphere of positive thinking and bold questioning, and improve students' interest in learning and classroom efficiency.

5. Through the exploration of example problems, you can experience a method of studying mathematical problems (guess first and then prove), stimulate students' enthusiasm for learning, and help students initially form the consciousness and scientific spirit of doing mathematics.

Recognition of the significance of teaching key induction method and analysis of the generation process of mathematical induction method

Understanding of the recursive thought in mathematical induction method of teaching difficulties

Teaching method analogy inspired inquiry teaching method< /p>

Teaching methods Multimedia-assisted classroom teaching

Teaching procedures

The first stage: input stage - creating learning situations and providing learning content

1. Create problem situations and activate students' thinking

(1) Examples of incomplete induction:

There is a joke in "Ying Xie Lu" compiled by Liu Yuanqing of the Ming Dynasty: The rich man's son Learn to write. In this joke, the rich man's son came to the conclusion that "four means four horizontal lines, five means five horizontal lines..." and used the "inductive method". However, the conclusion deduced by this induction is obviously wrong.

(2) Comparative quotation of complete induction method:

There was a master who wanted to test his two apprentices to see who was smarter. He gave each of them a basket of peanuts to peel. See if each peanut kernel is wrapped in pink and see who can give the answer first. The first apprentice spent a lot of effort to peel all the peanuts; the second apprentice only picked a few plump ones, a few shriveled ones, and a few Some are cooked, some are uncooked, some are three-kernel, some are one-kernel, and some are two-kernel, but the total is no more than a handful of peanuts. Obviously, the second apprentice gave the answer first, and he is smarter than the eldest apprentice.

In practical life and production, the inductive method is also widely used. For example, meteorologists and hydrologists use the inductive method to make meteorological predictions and hydrological forecasts based on accumulated historical data. These inductive methods are Complete induction cannot be used.

2. Review old mathematical knowledge and trace inductive consciousness

(From life to mathematics, review previously learned mathematical knowledge with students to further experience inductive consciousness , and at the same time let students feel that we have already been exposed to induction in our previous studies.)

(1) Example of incomplete induction: given the first four terms of the arithmetic sequence, write the general term of the sequence Formula.

(2) Example of complete induction method: Prove the circumferential angle theorem in three cases: the center of the circle is inside, outside, and on one side of the circumferential angle.

3. With the help of mathematical historical materials, promote Students' thinking

(On the basis of life examples and learned mathematical knowledge, students are guided to read mathematical historical materials, which can allow students to experience the inductive method from multiple aspects and angles and feel the universality of using the inductive method. At the same time Guide students to think: using incomplete induction in mathematics often leads to wrong conclusions, whether it is us or everyone in mathematics. So, is there a better induction method?)

Question 1 Known = (n∈N),

(1) Find ; ; ; .

(2) What conclusion can you get from this? Is this conclusion correct?

(Cultivation of students’ awareness of bold conjecture and mathematical generalization ability. Generalization ability is the core of thinking ability. Rubinstein pointed out: Thinking is completed in generalization. Psychology believes that "transfer is generalization", The breakthrough I am looking for in the transfer of knowledge, skills, thinking methods and mathematical principles is the students’ generalization process.)

Question 2

Fermat was a famous French mathematician in the 17th century. He once believed that when n∈N, they must all be prime numbers. This is what he got after verifying n=0,1,2,3,4 Later, the great Swiss scientist Euler in the 18th century proved that =4 294 967 297 = 6 700 417 × 641, thus denying Fermat's conjecture. Unexpectedly, this conclusion does not hold when n=5.

Question 3, when n∈N, are all prime numbers?

Verification: f(0)=41,f(1)=43,f(2)=47 ,f(3)=53,f(4)=61,f(5)=71,f(6)=83,f(7)=97,f(8)=113,f(9)=131, f(10)=151,…,f(39)=1 601. But f(40)=1 681= , which is a composite number.

The second stage: the interaction stage of old and new knowledge - old and new The role of knowledge and building a new knowledge structure

4. Search for life examples to stimulate interest in learning

(Based on the first stage, starting from life examples, analyze and summarize the principles with students, Revealing the process of recursion. Confucius said: "Those who know well are not as good as those who are good at it, and those who are good at it are not as good as those who are happy." The personality psychological tendency of interest is generally accompanied by good emotional experience.)

Example :Play domino video

Key: (1) The first card is knocked down; (2) If a certain card falls, the next card must fall. So, we can Draw a conclusion: all dominoes will fall.

Search: Give a few more life examples: overturning bicycles, aligning in line for morning exercises, etc.

5. Analogies to mathematical problems to stimulate thinking Waves

By analogy with the domino process, prove the general term formula of arithmetic sequence:

(1) When n=1, the equation holds; (2) Assume that when n=k, etc. The formula holds, that is, then = , that is, the equation also holds when n=k+1. Therefore, we can draw the conclusion: the general formula of the arithmetic sequence holds for any n∈.

(Breaking Luna's discovery learning theory believes that "guided discovery learning" emphasizes the process of knowledge generation and development. Here, through the analogy of the domino process, students are allowed to discover the prototype of mathematical induction, which is a kind of re-created discovery learning.)

6. Guide students to generalize and form scientific methods

The key steps to prove a proposition related to positive integers are as follows:

(1) Prove that when n takes the first value;

(2) Assume that the conclusion is correct when n=k (k∈ ,k≥), prove that the conclusion is also correct when n=k+1.

After completing these two steps, you can conclude that the proposition is correct for all positive integers n from the beginning.

This method of proof is called mathematical induction.

The third stage: operation Stage - Consolidate the cognitive structure and enrich the cognitive process

7. Including conjectures and proofs to cultivate research awareness

(This example requires students to conjecture first and then prove, which can not only consolidate the induction method and mathematical induction, it can also teach students how to do mathematics and cultivate their awareness and ability to independently study mathematical problems.)

The example problem is in the sequence { }, =1, (n∈ ), first Calculate the value of , , and then speculate on the formula of the general term, and finally prove your conclusion.

8. Basic feedback exercises to consolidate the application of the method

(Textbook examples and arithmetic sequence generalization The proof of the formula is almost the same, and it is not difficult to solve it by applying the proof steps of mathematical induction, so I use it as an exercise, which not only takes into account the students' ability level, but also does not dilute the focus of this lesson. The third question of the exercise happens to be an equal ratio The proof of the formula of the general term of a sequence is a contrast and supplement to the former. Through these two exercises, we can see how well students have mastered the steps of mathematical induction method proof problems.)

(1) (Page 63 Example 1) Using mathematical induction

Prove: 1+3+5+…+(2n-1)= .

(2) (Exercise 3 on page 64) The first term is, and the common ratio is the general formula of the geometric sequence of q Yes.

9. Teachers and students *** summarize together to complete the summary improvement

(1) The core content of this lesson is induction and mathematical induction;

(2) Induction is a method of reasoning from the specific to the general. It can be divided into two types: complete induction and incomplete induction. Complete induction is limited to a limited number of elements, while incomplete induction has The conclusion drawn is not necessarily reliable, and mathematical induction is a complete induction method;

(3) As a proof method, the basic idea of ??mathematical induction is recursive (recursive) thinking, and the key points can be used It can be summarized as: two steps and one conclusion, the basis of recursion is indispensable, the inductive hypothesis must be used, and the conclusion should not be forgotten;

(4) The mathematical thinking methods involved in this lesson are: recursion Inference thinking, analogy thinking, classification thinking, inductive thinking, dialectical materialism thinking.

10. Assign after-school homework to consolidate the foundation for extension

(1) Exercises on page 64 of the textbook Questions 1, 2; Question 2 of Exercise 2.1 on page 67.

(2) In the second step of the mathematical induction proof, to prove that the proposition is true when n=k+1, n must be used =k, the proposition is established. Here is an analysis question for students to discuss and think after class:

Use mathematical induction to prove: When (n∈ ), the second step adopts the following proof method:< /p>

Assume that the equation holds when n=k, that is, when n=k+1,

.

Do you think the above proof is correct? Why ?

Teaching design instructions

1. Mathematical induction is a proof method used to prove the correctness of propositions related to the natural number n. Its operation steps are simple and clear. The focus of teaching should not be the application of methods. I think the teaching process cannot be regarded as the instillation of methods and the practice of skills. For this reason, I envision strengthening the teaching of the generation process of mathematical induction, and embodying the generation of mathematical induction in the understanding of induction. In the analysis and understanding, the emergence of mathematical induction and the perfection of incomplete induction are combined. This not only allows students to see the background of the emergence of mathematical induction, but also pays attention to its functions from the beginning, laying a good foundation for its use. It not only strengthens the teaching of inductive thinking, but also is an important supplement to the teaching of deductive thinking in middle school mathematics. It is also a good opportunity to guide students to develop innovative abilities.

2. In terms of teaching methods , here the method of discussion and exploration between teachers and students under the guidance of teachers is used. The purpose is to strengthen students' participation in the teaching process. In order to make this participation have a certain degree of intelligence, teachers should do a good job in mobilizing, organizing, Guidance and guidance. Students' thinking participation often starts with questions. This lesson arranges a series of questions according to the order of thinking, allowing students to invest in thinking activities. The research content of this lesson is placed in the questions. Gradually unfold, guide students to use the knowledge and methods they have learned to solve problems, and obtain the update and expansion of the knowledge system.

3. Use mathematical induction to prove mathematical propositions related to positive integers, two steps Both are indispensable. Understand the idea of ??recursion in mathematical induction, especially the second step. To prove that the proposition n=k+1 is true, you must use the condition that the proposition is true when n=k. These contents will be placed below Completed in one lesson, this understanding not only enables us to correctly understand the principles and essence of mathematical induction, but also points out the direction of thinking for the design of the second step in the proof process.