Definition of 1. sequence
A series of numbers arranged in a certain order is called a series, and each number in the series is called an item of the series.
(1) As can be seen from the definition of series, the numbers of series are arranged in a certain order. If the numbers that make up a series are the same but in different order, then they are not the same series. For example, the series 1, 2,3,4,5 is different from the series 5,4,3,2, 1.
(2) The definition of series does not stipulate that the numbers in the series must be different. Therefore, multiple identical numbers can appear in the same series, such as:-1, 1, 2, 3, 4, … to form a series:-1,-1.
(4) The term of a series is different from its number. The term of a series refers to a certain number in this series, which is a function value, equivalent to f(n), while the term of a number refers to the position serial number of this number in the series, which is the value of an independent variable, equivalent to n in f(n).
(5) Order is very important for a series. There are several identical figures. Because of their different arrangement order, the series is also different. Obviously, there is an essential difference between a series and a group of numbers. For example, if the five numbers 2, 3, 4, 5 and 6 are arranged in different order, different series will be obtained, while {2, 3, 4, 5 and 6}.
2. Series classification
(1) According to the number of items in the series, the series can be divided into finite series and infinite series. When writing a series, the last item should be written for a finite series, for example, the series 1, 3, 5, 7, 9, …, 2n- 1 indicates a finite series. If the sequence is written as 1,
(2) According to the relationship between items or the increase or decrease of series, it can be divided into the following categories: increasing series, decreasing series, swinging series and constant series.
3. General term formula of sequence
A sequence is a series of numbers arranged in a certain order, and its essential attribute is to determine the law of this number, which is usually expressed by formula f(n).
Although these two general formulas are different in form, they represent the same series, just as not every functional relationship can be expressed by analytical formula, and not every series can write its general formula. Although some series have general formulas, they may not be formally established. They only know the finite term in front of a series, and there is no other explanation. The series cannot be determined, and the general formula does not exist. For example, the series 1, 2, 3, 4, ...
The items written after the formula are different. Therefore, the induction of general term formula depends not only on its first few terms, but also on the synthesis law of series, and more observation and analysis are needed to truly find the internal law of series. There is no general way to write its general term formula from the first few terms of a series.
In order to understand the general formula of series, emphasize the following points again:
The general term formula of (1) series is actually that the domain is positive integer set N* or its finite set {1, 2, ..., n}.
(2) If we know the general term formula of the series, then we can use 1, 2, 3, … instead of N in the formula in turn to find out the terms of this series; At the same time, we can also use the general term formula of series to judge whether a number is an item in the series, and if so, what item it is.
(3) Just as all functional relationships do not necessarily have analytical formulas, not all series have general formulas.
If the approximation is less than 2, the sequence shall be accurate to 1, 0. 1, 0.0 1, 0.00 1, 0.000 1, ... 1, 1.
(4) Some general formulas of series are not necessarily formal, such as:
(5) Some series only give the first few terms, but do not give their composition rules, so the general term formula of series derived from the first few terms is not.
4. Series of images
For series 4, 5, 6, 7, 8, 9, 10, the corresponding relationship between the serial number of each item and this item is as follows:
Serial number: 1234567
Item code: 456789 10
In other words, the above can be regarded as a mapping from one set of serial numbers to another. Therefore, from the point of view of mapping and function, a sequence can be regarded as a positive integer set N* (or its finite set {1, 2,3, ..., n}), and a corresponding list of function values when the values are from small independent variables to large ones. Here,
Because the term of the series is a function value and the serial number is an independent variable, the general term formula of the series is the corresponding function and analytical formula.
Sequence is a special function, which can be expressed intuitively by images.
The sequence is represented by an image. With the serial number as the abscissa and the corresponding item as the ordinate, you can draw a picture to represent an order. When drawing, for convenience, the unit length taken on the two coordinate axes of the plane rectangular coordinate system can be different. From the image representation of the sequence, we can directly see the change of the sequence, but it is not accurate.
Compared with function, sequence is a special function, which is a group of positive integers or a group of finite continuous positive integers headed by 1, and its image is infinite or finite isolated points.
5. Recursive series
A pile of steel pipes is stacked in seven layers, and the number of steel pipes in each layer forms a sequence from top to bottom: 4, 5, 6, 7, 8, 9, 10.
The order ① can also be given by the following method: the number of steel pipes in the first floor from top to bottom is 4, and the number of steel pipes in each floor below is more than that in the previous floor 1 root.
Synchronous exercise questions
1. In the known sequence {an}, an=n2+n, then a3 is equal to ().
A.3B.9
C. 12D.20
Answer: c
2. In the following series, the one that is both an increasing series and an infinite series is ().
A. 1， 12， 13， 14，…
B.- 1,-2,-3,-4,…
C.- 1,- 12,- 14,- 18,…
D. 1，2，3，…，n
Analysis: choose C. For A, an= 1n, n∈N* is an infinitely decreasing sequence; For b, an=-n, n∈N* is also an infinite decreasing sequence; D is a finite sequence; For c, an=-( 12)n- 1, which is an infinitely increasing sequence.
3. The following statement is incorrect ()
A. according to the general formula, you can find any item in the sequence.
B. Any series has a general formula.
C. A series may have several different forms of general term formulas.
D. some series may have no projects.
Analysis: Choose B. Not all series have general formulas, such as 0, 1, 2, 1, 0, ...
Item 10 of series 4.23, 45, 67 and 89 is ()
A. 16 17B
C.202 1D.2223
Analysis: Choose C. From the meaning of the question, the general formula of the sequence is an=2n2n+ 1.
∴a 10 = 2× 102× 10+ 1 = 202 1。 So, C.
5. The recurrence formula of known nonzero sequence {an} is an=nn- 1? an- 1(n & gt； 1), then a4= ()
A.3a 1B.2a 1
C.4a 1D. 1
Analysis: choose C. assign values to n in the recursive formula in turn. When n=2, A2 = 2a1; When n=3, a3 = 32a2 = 3a1; When n=4, a4=43a3=4a 1.
two
1. Definition of inequality
In the objective world, the unequal relationship between quantity and quantity is universal. We use mathematical symbols to connect two numbers or algebraic expressions to express the inequality between them. Expressions containing these inequalities are called inequalities.
2. Compare the sizes of two real numbers
The size of two real numbers is defined by the operational properties of real numbers.
There is a-b > 0? ; a-b=0？ ; a-b & lt； 0? .
In addition, if b>0, there is > 1? ; = 1? ; & lt 1? .
Can be summarized as: differential method, commercial method, intermediate method and so on.
3. The nature of inequality
(1) symmetry: A >；; b？ ;
(2) Transitivity: a>b, b>c? ;
(3) additivity: a>b? a+cb+c，a & gtb，c & gtd？ a+c b+ d；
(4) Availability: a>b, c>0? Ac> BC; a & gtb & gt0，c & gtd & gt0? ;
(5) Multiplier: a>b>0? (n∈N，N≥2)；
(6) Prescription: a>b>0? (n∈N，n≥2)。
Review guide
1. "One skill" is the skill of difference method deformation: deformation is the key in difference method, and factorization or formula is often carried out.
2. "One Method" undetermined coefficient method: When calculating the range of algebraic expression, the known algebraic expression is used to represent the target expression, then the parameters are obtained by using the principle of polynomial equality, and finally the range of the target expression is obtained by using the properties of inequality.
3. "Two common attributes"
(1) Reciprocity: ① a >; b，ab & gt0? & lt； ②a & lt； 0
③a & gt； b & gt0,0; ④0
(2) If a>b>0, m>, then 0
① the nature of true fraction: (b-m > 0);
② the nature of false score: >; 0).
three
1. The values of x and y that satisfy the binary linear inequality (group) form an ordered number pair (x, y), which is called the solution of the binary linear inequality (group). The set of all such ordered number pairs (x, y) is called the solution set of the binary linear inequality (group).
2. Each solution (x, y) of binary linear inequality (group) corresponds to a point on the plane as the coordinate of the point, and the solution set of binary linear inequality (group) corresponds to a half plane (plane area) in the plane rectangular coordinate system.
3. The straight line L: ax+by+c = 0 (both A and B are not zero) divides the coordinate plane into two parts, one of which (half plane) corresponds to the binary linear inequality ax+by+c > 0 (or ≥0), and the other part corresponds to the binary linear inequality AX+BY+C.
4. Given the plane area, it is expressed by inequality (group). The method is: take any point outside all straight lines (such as the origin (0,0) of this question), substitute its coordinates into Ax+By+C, and judge whether it is positive or negative to determine the corresponding inequality.
5. The plane area represented by a binary linear inequality is a half plane divided by the corresponding straight line, which can be judged by substituting special points into the binary linear inequality test. When the straight line does not pass through the origin, the origin inspection is often selected, and when the straight line passes through the origin, (1, 0) or (0, 1) is often selected for substitution inspection. The plane region represented by binary linear inequalities is the plane region represented by its various inequalities. Delineation of lines and positioning of points.
6. The ordered number pair (x, y) consisting of the values of integers x and y satisfying the binary linear inequality (group) is called the solution of this binary linear inequality (group). All points corresponding to integer solutions are called integer points (also called lattice points), which are all in the plane region represented by this binary linear inequality (group).
7. When drawing the plane area represented by the binary linear inequality Ax+By+C≥0, the boundary should be drawn as a solid line, and the binary linear inequality AX+BY+C > 0, the boundary should be drawn as a dotted line.
8. if point P(x0, y0) and point P 1(x 1, y 1) are on the same side of the straight line l: ax+by+c = 0, then Ax0+By0+C and ax1+byl+c. If point P(x0, y0) and point P 1(x 1, y 1) are on both sides of the straight line L: ax+by+c = 0, then the signs of Ax0+By0+C and Ax 1+Byl+C are opposite.
9. The steps of abstracting binary linear inequalities (groups) from practical problems are:
(1) Set variables according to the meaning of the question;
(2) Analyze the variables in the problem, and list the inequalities between the constant and the variables X and Y according to various inequality relations;
(3) Combine inequalities with meaningful practical ranges of variables X and Y to form an inequality group.