The basic formula of high school series: 1, and the relationship between the general term an and the first n terms and Sn of the general term series: an=
2. arithmetic progression's general formula: 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, an is a linear equation about n; When d=0, an is constant.
3. arithmetic progression's first n terms and formula: Sn=
Serial number =
Serial number =
What time? 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.
4. 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)
5. The first n terms and formulas of geometric series: When q= 1, Sn=n a 1 (this is a direct ratio formula about n);
When q? At 1, Sn=
Serial number =
Summary of knowledge points of high school mathematics series II: conclusions about arithmetic and geometric series in high school mathematics
The sequence Sm, S2m-Sm, S3m-S2m, S4m-S3m formed by the sum of any continuous m terms of 1 and arithmetic progression {an} is still arithmetic progression.
2. In arithmetic progression {an}, if m+n=p+q, then
3. In the geometric series {an}, if m+n=p+q, then
4. The sequences Sm, S2m-Sm, S3m-S2m, S4m-S3m formed by the sum of any continuous m terms of the geometric series {an} are still geometric series.
5. The sum and difference of two arithmetic progression {an} and {bn} series {an+bn} and {an-bn} are still arithmetic progression.
6. The product, quotient and reciprocal series of two geometric series {an} and {bn}.
safe
bn}、。
、
Or geometric series.
7. arithmetic progression {an} with arbitrary equidistant terms is still arithmetic progression.
8. geometric progression {an} with arbitrary equidistant terms is still geometric progression.
9. How to make three numbers into arithmetic progression: a-d, A, A+D; How to make four numbers equal: A-3D, A-D, A+D, A+3D?
10, how to make three numbers into geometric series: a/q, a, AQ;
Wrong method of four numbers being equal: a/q3, a/q, aq, aq3 (Why? )
1 1, {an} is arithmetic progression, then
(c>0) is a geometric series. 12 、{ bn }(bn & gt； 0) is a geometric series, then {logcbn} (c >; 0 and c
1) is arithmetic progression. 13. In arithmetic series
Medium: (1) If the number of items is
, then
(2) If the quantity is
Then,
,
14. It grows geometrically.
Medium: (1) If the number of items is
, then
(2) If the quantity is
Then,
Basic methods and skills of summation of mathematical series in senior high school 1. Find the sum of mathematical series by formula method;
① Sum formula of arithmetic sequence;
② Sum formula of equal ratio series, especially stating: When using sum formula of equal ratio series, we must check the relationship between its common ratio and 1, and discuss it in categories if necessary.
③ Common formulas:
,
,
Such as (1) proportional series.
Deqian
Term sum Sn=2n- 1, then
= _ _ _ _ (A:
); (2) The computer converts the information into binary numbers for processing. What's binary? Every binary 1? , for example
Represents a binary number, which is converted into decimal form.
, and then binary.
Converted into decimal number is _ _ _ _ _ (A:
) 2. Sum method of grouped series: When the formula method is difficult to sum directly, it is often used. Harmony? Medium? Similar items? Merge them together first, and then sum them with formulas. For example:
(a:
) 3. Find the sum of series by reverse addition: If the sum of two terms with the same distance from the beginning to the end is related to the general term of the series, you can often consider using reverse addition to give full play to its * * * function and sum (this is also before arithmetic progression).
And the derivation method of the formula). For example, ① verification:
; ② Known
, then
= _ _ _ _ (A:
4. Find the sum of series by dislocation subtraction: If the general term of series consists of a general term of arithmetic progression and a general term of geometric progression, then dislocation subtraction is often used (this is also before geometric progression).
And the derivation method of the formula). For example, suppose (1).
For geometric series,
, known
,
(1) Found the sequence.
The first sum of the common ratio; ② Find the sequence.
The general formula of. (A: ①
,
; ②
); (2) Setting function
, series
Satisfy:
, ① verification: sequence
Is a geometric series; ② Order
, find the function.
Order again
Take a derivative in ...
, and compare them.
and
The size of. (A: ① Omit; ②
, when?
When,
=
; while
When,
& lt
; while
When,
& gt
)
5. Elimination method of split term of series sum: If the general term of series can? Split into two differences? In the form of, and the adjacent items are related after splitting, the splitting item elimination method is often used to sum. Common decomposition project forms are:
①
; ②
; ③
,
; ④
; ⑤
; ⑥
. Such as (1) sum:
(a:
); (2) series connection
Yes,
, and Sn=9, then n = _ _ _ _
(A: 99);
6. General term transformation method for finding the sum of series: first, deform the general term to find out its internal characteristics, and then sum it by grouping summation method. such as
① Find the sequence 1? 4,2? 5,3? 6,? ,
,? front
peaceful and auspicious
= (A:
); ② Sum:
(a:
)
There are several common methods to find the general term formula of series in high school mathematics: 1. If the topic is known or judged by simple reasoning to be geometric series or arithmetic progression, directly use its general term formula.
Example: If a 1= 1 and an+ 1=an+2(n 1) in the series {an}, find the general term formula an of this series.
Solution: It consists of an+ 1=an+2(n 1) and arithmetic progression whose known derivative sequence {an} is a 1= 1 and d=2. So an=2n- 1. This kind of problem is mainly judged by the definition of equal proportion and arithmetic progression, which is a relatively simple basic minor problem.
Second, the sum of the first n terms of a known sequence is calculated by a formula.
S 1 (n= 1)
Sn-Sn- 1 (n2)
Example: The sum of the first n terms of the known sequence {an} is Sn=n2-9n, and the k-th term satisfies 5.
(A) 9 (B) 8 (C) 7 (D) 6
Solution: ∫ an = sn-sn-1= 2n-10,5 & lt2k-/kloc-0 < 8 ? K=8 (b)
When solving this kind of problems, we should pay attention to the situation of n= 1.
Thirdly, when the relationship between an and Sn is known, the relationship between Sn and n is usually obtained by transformation, and then the general term formula is obtained by the above method (2).
Example: It is known that the first n terms and Sn of the sequence {an} satisfy an=SnSn- 1(n2) and a 1=-, so the general term formula of the sequence {an} can be obtained.
Solution: ∫an = SnSn- 1(N2), while an=Sn-Sn- 1, SnSn- 1=Sn-Sn- 1, and both sides are divided by SnSn- 1. {-} is a arithmetic progression, the first term is-and the tolerance is-1. -= -，Sn= -，
Reuse the method of (2): When n2, an=Sn-Sn- 1=-, n= 1, this formula is not applicable, so,
- (n= 1)
-(nitrogen)
Fourth, find the general formula through accumulation and accumulation.
For the recursive formulas of an and an+ 1 and an- 1 given in the question, the general general formulas are all quadrature by accumulation.
Example: Let the series {an} be a positive series with the first term 1, and satisfy the general formula (n+1) an+12-nan2+an+1an = 0 to find the series {an}.
Solution: ∫ (n+1) an+12-nan2+an+1an = 0, which can be decomposed into [(n+1) an+1-nan] (an
You :{ an} is a positive series, and the first term is 1. an+ 1+an？ 0,? -=-,resulting in:-=-,-=-,-=-,? ,-=-,these n- 1 expressions are multiplied to get:? -=-,
∫a 1 = 1，？ An=-(n2), ∫n = 1 also holds. an=-(n？ N*)
Fifth, use the method of constructing sequence to find the general term formula.
If the formula given in the topic is recursive, but it is not easy to find the general formula through accumulation, accumulation and iteration, we can consider constructing the formula containing an (or Sn) by deformation, making it an equal proportion or arithmetic progression, so as to find the relationship between an (or Sn) and n, which is a hot spot in the college entrance examination in recent years, so it is both important and difficult.
Example: In the known sequence {an}, a 1=2, an+ 1 = (- 1) (an+2), n= 1, 2,3,
(1) Formula (2) for finding the general term of {an} is omitted.
Solution: An+ 1 = (- 1) (an+2) gives an+ 1-= (- 1) (an-).
? {an-} is a geometric series with the first term a 1-and the common ratio is-1.
An-= (- 1) n- 1 (2-) is obtained from a 1=2, so an = (- 1) n- 1 (2-)+-
Another example: in the sequence {an}, a 1=2, an+ 1=4an-3n+ 1(n? N*), it is proved that the sequence {an-n} is a geometric series.
Proof: This question proves that an+1-(n+1) = q (an-n) (q is a non-zero constant).
From an+ 1=4an-3n+ 1, it can be transformed into an+1-(n+1) = 4 (an-n), while ∵ a1-/kloc-.
Therefore, the sequence {an-n} is a geometric series with the first term 1 and the common ratio of 4.
If this problem is replaced by the general term formula of an, it can still be transformed into the general term formula of an by finding the general term formula of {an-n}.
Another example is: Let the first term {an} a 1? (0, 1), an=-, n = 2, 3, 4 (1) Find the general term formula of {an}. (2) Omission
Solution: from an=-, n = 2,3,4, to1-an =-(1-an-1), and 1-a 1? 0, so {1-an} is a geometric series with the first term of 1-a 1 and the common ratio of-,which is an =1-(1-a1) (-) n.