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.
Split term summation method
example
1/ 1*4+ 1/4*7+ 1/7* 10......... 1/(3n-2)(3n+ 1)
How to solve this split term method that is not n(n+ 1)?
explain
1/(3n-2)(3n+ 1)
1/(3n-2)- 1/(3n+ 1)= 3/(3n-2)(3n+ 1)
As long as it is the summation of fractional series, the split term method can be used.
The method of splitting the term is to subtract the reciprocal of the smaller factor from the reciprocal of the larger factor in the denominator, and then compare it with the original general term formula to get the required constant.