(2) The idea of classified discussion: the summation formula of equal proportion series should be divided and summed; When the time is known, it should also be classified;
③ Holistic thinking: When solving the sequence problem, we should pay attention to getting rid of the rigid thinking mode solved by formulas and use integers.
Body and mind solutions.
(4) When solving the related application problems of series, we should carefully analyze and abstract the actual problems into mathematical problems, and then use the knowledge and methods of series to solve them. Solving this kind of application problem is a comprehensive application of mathematical ability, and it is by no means a simple imitation and application. Pay special attention to the items of geometric series related to years.
First, the basic concept:
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. General formula of series 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.
14, the sum of any consecutive m terms in the series Sm, S2m-Sm, S3m-S2m, S4m-S3m, ... arithmetic progression still forms the arithmetic progression.
15, arithmetic progression, if m+n=p+q, then
16, geometric series, if m+n=p+q, then
The sum of any continuous m terms of Sm, S2m-Sm, S3m-S2m, S4m-S3m series, ... 17 and geometric series form geometric series.
18, the sequence of sum and difference of two arithmetic progression is still arithmetic progression.
19, the product, quotient and reciprocal sequence of the sum of two geometric series.
,,, or geometric series.
20. The series of arithmetic progression's arbitrary equidistant term is still arithmetic progression.
2 1, any equidistant term series of geometric series is still geometric series.
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. For arithmetic progression, then (c>0) is a geometric series.
25. (bn>0) is a geometric series, so (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, find the maximum and minimum value of a series of methods:
① an+ 1-an = ... For example, an= -2n2+29n-3.
② (An>0) as a =
③ an=f(n) Study function f(n)