Dislocation subtraction (the first n terms of geometric series and formula derivation method)
Group summation method
Decomposition summation method
Superposition summation method
The key of series summation is to analyze the characteristics of its general term 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 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.