1, definition and representation of sequence:
2. Items and number of items in the series:
3, finite sequence and infinite sequence:
4, increasing (decreasing), swing, cycle order:
5. The general formula of sequence {an} an:
6. The first n terms of the sequence and the formula Sn:
7. Structure of arithmetic progression, Tolerance D and arithmetic progression:
8. The structure of geometric series, Bi Gong Q and geometric series;
Second, the basic formula:
9. the relationship between the general term an and the first n terms and Sn of a general sequence: an=
10, the general formula of arithmetic progression: an = a 1+(n-1) Dan = AK+(n-k) d (where a1is the first term and AK is the known k term), when d≠0.
1 1, the first n terms of arithmetic progression and its formula: Sn= Sn= Sn=
When d≠0, Sn is a quadratic form about n, and the constant term is 0; When d=0 (a 1≠0), Sn=na 1 is a proportional formula about n.
12, the general formula of geometric series: an = a1qn-1an = akqn-k.
(where a 1 is the first term, ak is the known k term, and an≠0).
13, the first n terms of geometric series and their formulas: when q= 1, Sn=n a 1 (this is a direct ratio formula about n);
When q≠ 1, Sn= Sn=
Third, the conclusion about arithmetic and geometric series.
Arithmetic progression {an} formed by the sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 14 is still arithmetic progression.
15, arithmetic progression {an}, if m+n=p+q, then
16, geometric series {an}, if m+n=p+q, then
Geometric progression {an} formed by the sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 17 is still geometric progression.
18, the sum and difference of two arithmetic progression {an} and {bn} series {an+bn} is still arithmetic progression.
19, a sequence consisting of the product, quotient and reciprocal of two geometric series {an} and {bn}
{an bn},,, or geometric series.
20. arithmetic progression {an} Any equidistant series is still arithmetic progression.
2 1, the series of any equidistant term of geometric progression {an} is still geometric progression.
22. How to make three numbers equal: A-D, A, A+D; How to make four numbers equal: A-3D, A-D, A+D, A+3D?
23. How to make three numbers equal: A/Q, A, AQ;
Wrong method of four numbers being equal: a/q3, a/q, aq, aq3 (Why? )
24.{an} is arithmetic progression, then (c>0) is a geometric series.
25 、{ bn }(bn & gt； 0) is a geometric series, then {logcbn} (c >; 0 and c 1) are arithmetic progression.
26. In the arithmetic series:
(1) If the number of items is, then
(2) If the quantity is,
27. In geometric series:
(1) If the number of items is, then
(2) If the number is 0,
Four, the common methods of sequence summation: formula method, split item elimination method, dislocation subtraction, reverse addition, etc. The key is to find the general item structure of the sequence.
28. Find the sum of series by grouping method: for example, an=2n+3n.
29. Sum by dislocation subtraction: for example, an=(2n- 1)2n.
30. Sum by split term method: for example, an= 1/n(n+ 1).
3 1, sum by addition in reverse order: for example, an=
32. The method of finding the maximum and minimum term of series {an}:
① an+ 1-an = ... For example, an= -2n2+29n-3.
② (An>0) as a =
③ an=f(n) Study the increase and decrease of function f(n), such as an=
33. In arithmetic progression, the problem about the maximum value of Sn is often solved by the adjacent term sign change method:
(1) When >: 0, d < When 0, the number of items m meets the maximum value.
(2) When
We should pay attention to the application of the transformation idea when solving the maximum problem of the sequence with absolute value.
Six, the plane vector
1. Basic concepts:
Definition of vector, modulus of vector, zero vector, unit vector, opposite vector, * * * line vector, equal vector.
2. Algebraic operations of addition and subtraction:
( 1) .
(2) if a b= (). And B = (), AB = ().
Geometric representation of vector addition and subtraction: parallelogram rule and triangle rule.
Take the vector =, = as the adjacent side to make a parallelogram ABCD, then the vectors of the two diagonals are =+,=-and =-
And there are ||||-|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| | | || | | | | | | | | | | | | | | | | | | | | | | | | | |
Vector addition has the following laws:+=+(commutative law); +( +c)=(+)+c (law of association);
+0= +(- )=0.
3. Product of real number and vector: The product of real number and vector is a vector.
( 1)| |=| | | |;
(2) When > 0, it is in the same direction; When < 0, the opposite; When =0, = 0.
(3) If = (), then = ().
Necessary and sufficient conditions for two vector lines;
The necessary and sufficient condition for the straight line between (1) vector b and non-zero vector * * * is that there is only one real number, so b =.
(2) If = () and b = (), then ‖ b 。
The basic theorem of plane vector;
If e 1 and e2 are two nonlinear vectors on the same plane, there is only one pair of real numbers for any vector on this plane, so = e 1+ e2. ..
4. The ratio of P-divided directed line segments:
Let P 1 and P2 be two points on a straight line, and point P is any point in the world different from P 1 and P2, then there is a real number that makes =, which is called the ratio of point P to directed line segment.
When point p is on the line segment, > 0; When point P is on the extension line of line segment or, < 0;
Formula of vernal equinox coordinates: if =；; The coordinates of are (), () and () respectively; Then (≦- 1), the midpoint coordinate formula:.
5. Quantity product of vectors:
(1). Vector angle:
Given that two nonzero vectors and b make =, =b, then ∠AOB= () is called the included angle between the vector and b.
(2). Quantity product of two vectors:
If two nonzero vectors and b are known and their included angle is, then b = |||| b | cos.
Where | b | cos is called the projection of vector b in the direction.
(3) Properties of the product of vector numbers:
If = () and b = (), then e = e = || cos (e is the unit vector);
⊥ b b = 0 (,b is a non-zero vector); | |= ;
cos = =。
(4) Vector product algorithm:
b = b()b =(b)=(b)； (+b) c= c+b c。
6. Main ideas and methods:
This chapter mainly sets up the viewpoint of number-shape transformation and combination, handles geometric problems with algebraic operation, especially the relative position relationship of vectors, correctly uses the basic theorems of * * * line vector and plane vector to calculate the modulus of vectors, the distance between two points and the included angle of vectors, and judges whether the two vectors are vertical or not. Because vectors are new tools, they are often combined with trigonometric functions, sequences, inequalities and solutions. And it is the intersection of knowledge.
Seven, solid geometry
The basic properties of 1. plane: If you master three axioms and inferences, you will explain the problems of * * * points, * * lines and * * * planes.
Able to draw by tilt measurement.
2. The positional relationship between two straight lines in space: the concepts of parallelism, intersection and nonplanarity;
Will find the angle formed by straight lines in different planes and the distance between straight lines in different planes; Generally, two straight lines are proved to be non-planar straight lines by reduction to absurdity.
3. Lines and planes
① positional relationship: parallel, straight line in the plane, and straight line intersects with the plane.
(2) The method and nature of judging the parallelism between a straight line and a plane, and the judgment theorem is the basis of proving the parallelism problem.
(3) What are the methods to prove that a straight line is perpendicular to a plane?
④ The angle formed by a straight line and a plane: the key is to find its projection on the plane, and the range is {00.900}.
⑤ Three Verticality Theorem and Its Inverse Theorem: This theorem should be examined in college entrance examination questions every year. The theorem of three perpendicular lines and its inverse theorem are mainly used to prove the vertical relationship and the measurement of spatial graphics, such as proving that straight lines in different planes are vertical, determining the plane angle of dihedral angle, and determining the perpendicular line from point to straight line.
4. Airplanes and Airplanes
(1) positional relationship: parallel, intersecting, (vertical is a special case of intersection)
(2) Master the proof method and nature of plane parallel to plane.
(3) Master the proof method and property theorem of the plane perpendicular to the plane. Especially, it is known that two planes are perpendicular, which can be proved by property theorem.
(4) Distance between two planes → Distance from point to surface →
(5) dihedral angle. Method and solution of dihedral angle plane intersection;
(1) definition method, generally using the symmetry of graphics; Generally, the oblique triangle should be solved in the calculation;
(2) The vertical line, diagonal line and projection method generally require that the vertical line of the plane is easy to find, and a right triangle should be solved in the calculation.
(3) The projective area method is generally used when two surfaces have only one common point and the intersection of the two surfaces is difficult to find?