If students can fully master the problem-solving methods and skills of high school mathematics series in the process of high school mathematics learning, it will be of great help to learn mathematics during college. In recent years, in the mathematics college entrance examination, the examination of the knowledge points of series has become a test center that the college entrance examination examiner pays more attention to, and even some high-scoring questions are directly related to the series. However, in the learning stage of senior high school mathematics, many students still lack the methods and skills to solve the series of senior high school mathematics problems, and some problems and contents have not been fully understood and absorbed, which often occurs in the process of solving problems. Therefore, exploring and studying the problem-solving methods and skills of different types of series can help students learn high school mathematics better.
Thoughts and skills of solving problems in the teaching of mathematical sequence test questions in senior high school
Discussion on the concept of 1. sequence
In the series of high school questions, some questions can bring the learned general formula or summation formula into direct answer. In the face of this type of test questions, there is no skill, just need to master the relevant series formula skillfully.
For example, in the geometric series {b} where all terms are positive numbers, the first term is b 1=3, and b 1+b2+b3=2 1, so what is b3+b4+b5?
Analysis: (1) This topic mainly examines the concept of positive series, the general formula of geometric series and the knowledge points of summation formula, and examines students' ability to master basic knowledge and basic operations of series.
(2) This test requires students to master the general formula and summation formula taught by the teacher in class.
(3) First of all, let's look for the comparison. Obviously, q is not equal to 1. Then we can list the equation about the common ratio according to the summation formula of geometric series, that is, 3 (1-Q3)/(1-q) = 21.
For this equation, we should first choose its operation mode, and ask students to skillfully convert high-order equations into low-order equations for operation in their usual practice.
2. Study on sequence properties.
In some test questions of series, some statements are often changed to test students' ability to understand and master the basic properties of series.
For example, arithmetic progression {xn} is known, where xl+x7=27. What is x2+x3+x5+x6?
Analysis: We have learned this formula in class: m+n=p+q In arithmetic progression and geometric progression, we can make full use of this feature to solve this problem, namely:
xl+x7= x2+x6= x3+x5=27，
Therefore, x2+x3+X5+X6 = (x2+X6)+(x3+X5) = 27+27 = 54.
This type of series test requires teachers to explain the nature of series in detail and deduce it carefully in classroom teaching. Let students really understand the source nature of the sequence.
3. Discussion on the formula of finding the general term
① Use the general term formula of arithmetic and geometric series to find the general term formula.
② Use the relationship an={S 1, n =1; Sn-Sn- 1, n≥2} formula for finding the general term
③ Using superposition and overlapping multiplication to find the general term formula.
④ Using mathematical induction to find the general term formula.
⑤ Use the construction method to find the general term formula.
4. Several methods to find the sum of the first n items
In recent years, in the mathematics college entrance examination questions, two knowledge points, the general formula of series and the sum of series, are tested every year. Therefore, in the classroom teaching of mathematics series in senior high school, teachers should explain the knowledge points of the sum formula of series in detail. The main methods to solve the sum of series are dislocation subtraction, grouping sum and merging sum. Here are three methods to solve the sum of series in detail.
(1) dislocation subtraction
Dislocation subtraction is mainly used to sum geometric series. In recent years, this method is often used to solve the sum of series in college entrance examination questions. This kind of problem solving method is mainly used to sum the first n terms of {arithmetic progression geometric series} series.
For example, it is known that {xn} is arithmetic progression, the sum of its first n terms is Sn, {yn} is geometric progression, x 1=y 1=2, x4+y4=27, S4-y4= 10, and (1.
Analysis: (1)xn=3n- 1, yn = 2n.
(2)Tn = 2xn+22xn- 1+23xn-2+…+2nx 1，
2Tn = 22xn+23xn- 1+…+2nx 2+2n+ 1x 1
It is calculated that TN =-2 (3n-1)+3× 22+3× 23+…+3× 2n+2n+1=12 (1-2n+1).
-2an+ 10bn- 12 =-2(3n- 1)+ 10×2n- 12 = 10×2n-6n- 10
Therefore, Tn+ 12=-2xn+ 10yn, n∈N*
Dislocation subtraction is mainly used in the calculation of sum test questions of series like an=bncn, which is geometric progression, arithmetic progression. The skills to solve this kind of problem are as follows: first, list the sum of the first n of arithmetic progression and geometric progression, that is, Sn, and then multiply both sides of Sn by the common ratio Q of geometric progression at the same time to get qSn；; Finally, there is an error, and then the formulas on both sides are subtracted.
(2) grouping summation method
In the high school series of questions, we often encounter some irregular series of questions. At first glance, they do not belong to arithmetic progression or geometric series. However, if we split this series, we can get what we know as arithmetic progression and geometric progression. When we encounter this type of series of questions, we can solve the problem by grouping and summarizing. First, we split the series, and then we can use arithmetic progression and geometric progression to calculate. Finally, we can combine them to get the answers to the questions.
(3) Combined summation method
In the series of college entrance examination questions, we often encounter some very special questions. At first, these questions seem irregular, but through merger and division, we can find out their special properties. This requires our teachers to cultivate students' ability to combine series, find out the law through combination, and finally successfully solve the summation problem of this kind of special series.
Concluding remarks
Sequence knowledge is the connection point of all kinds of mathematical knowledge. In mathematics examination, students' comprehensive knowledge of mathematics is often examined on the basis of sequence knowledge. In the process of learning series in senior high school, we must first master the basic concepts and properties of series, otherwise any problem-solving skills will be of no help.