Sequence is an important part of high school mathematics, which is closely related to numbers, formulas, functions, equations, inequalities, etc. It is a compulsory part of the college entrance examination every year. At the same time, there are many mathematical ideas and methods (such as function thought, equation thought, classification discussion, transformation thought, inductive conjecture and so on). ) of the synthetic sequence. When dealing with the problem of sequence synthesis, if we can use these mathematical ideas and methods flexibly, we will get twice the result with half the effort.

I. Functional thought

Sequence is a special function. The general term formula and the sum formula of the first n terms of a sequence can be regarded as a function of n, and can also be regarded as an equation or equation. In particular, arithmetic progression's general term formula can be regarded as a linear function of n, and its summation formula can be regarded as a quadratic function with zero constant term. Therefore, many series problems can be analyzed and solved by the idea of functional equation.

Example 1. Given the general formula of a series, which term does the series start from and the value of each term increases gradually? From which item, the value of each item is positive? Is there an item with the same value as the first item in the sequence?

Analysis: According to the conditions, the points of the series are all on the image of the function. For example, the right picture can be obtained according to the properties of quadratic function. This series begins with the fifth item, and the value of each item increases gradually. From the ninth item, all values are positive, and the ninth item is the same as the first item.

Example 2. The known series is arithmetic progression. If,,, find it.

Solution: So it is arithmetic progression, and its general term is a linear function. Let, then point out,,, in its image,

Therefore, the solution is:.

Comment: It is a linear function of n, and its image is a discrete point on a straight line. This problem is solved by establishing linear function with undetermined coefficient method.

Example 3. Let the sum of the first n terms of arithmetic progression be known, ...

(1) Find the range of tolerance d; (2) Point out which of the following ... is the biggest, and explain the reasons.

Analysis: For (1), the inequality set about d can be established by. For (2), the quadratic function of n is considered and transformed into the problem of finding the maximum value of the quadratic function.

Solution: (1) By understanding,

(2)

, is a quadratic function about n, and the equation of image symmetry axis is:

So when an integer is the largest, it is the largest.

Commentary: For arithmetic progression, it is a quadratic function of n, and its constant term is zero. It can be written in a form, and its image must pass through the origin. For this problem, because, the abscissa and X axis of another intersection point of the image.

, meet, so the axis of symmetry is, therefore, to maximize the judgment, the above thinking process is relatively simple.

Example 4. The first term of arithmetic progression is 2, and the sum of the former 10 terms is 15, which is the maximum.

Analysis: tolerance d can be obtained from the known content. The key to solve this problem is to understand the expression ""accurately and comprehensively: it is a subsequence of a sequence, in which 2, 4, 8, ... constitute a geometric series, which is the sum of the first n terms of this subsequence. Recognizing the above three points, the problem is not only easier to solve, but also different solutions of the maximum can be obtained from different angles.

Solution: Let the tolerance of arithmetic progression be d, which is given by the following formula:

, solution

There are three solutions to the maximum value.

Solution 1:

pass by

Make, solve

Thirdly, get the solution.

That is, in the series:

,

Therefore, at that time, the value of was the largest, and its maximum value was:

Solution 2:

General term of sequence

Order, obtain,

From this, you can get

Therefore, the maximum value of is 4.

∴

Solution 3:

Therefore, if there are natural numbers,

Therefore, the value of and is the largest.

Solution takes time and has a maximum value:

Review: The above three methods of finding the maximum value all use the function idea. The first solution is to find the maximum sum of the first n terms of the subsequence through monotonicity and positive and negative values of the sequence. The second solution is to study subsequences directly. The third solution is to find the maximum value through the monotonicity of learning, and the third solution can be simplified to the monotonicity of learning function.

Second, the idea of equation.

The general term formula of sequence and the sum formula of the first n terms are closely related to the five basic quantities, and "knowing three to find two" is the most basic operation. Therefore, the viewpoint of equation is the basic mathematical thought and method to solve this kind of problem.

Example 5. Let it be a positive sequence, the sum of the first few terms is, for all natural numbers, the arithmetic mean of 2 is equal to the arithmetic mean of 2, and the general term formula is obtained.

Solution: according to the meaning of the question, we can get. Once again, finishing: once again, that is, arithmetic progression with the first term of 2 and the tolerance of 4.

Comments: In this example, the elimination idea of the equation is used to get the result.

This equation finds the recurrence relation of two adjacent terms in the sequence and solves the problem. It is worth noting that sometimes we can use recursive relations and eliminate elements to solve problems.

Example 6: Given that the tolerance of arithmetic progression is positive, find the sum of the first n items.

Solution: arithmetic progression knows:, so it is the two roots of the equation. If you solve it, you get:. Solve equation: again, so.

Comments: This topic uses this property to construct a clever solution of quadratic equation, and then uses the equation to find the value and tolerance of the first term, thus solving the problem. It can be seen that when solving the sequence problem, we can often find the idea of the equation and/or the shortcut to solve the problem. .

Third, the idea of classified discussion.

The so-called classified discussion means that when the given object of the problem cannot be studied uniformly, we need to study the studied object in different categories, synthesize various results, and finally get the solution to the problem.

Example 7: Find the sum of the first n items in arithmetic progression.

Solution: (1) When,;

(2) when,;

Comprehensive (1)(2) shows that.

Comments: This example embodies the idea of classified discussion from the relationship between points and sums, because the middle foot code must be a positive integer.

Example 8. It is known that {} is a geometric series whose common ratio is q, which is called arithmetic progression.

(i) find the value of q;

(ii) Let {} be a arithmetic progression, the first term is 2, the tolerance is Q, and the sum of the first n terms is Sn. When n≥2, compare the sizes of Sn and bn, and explain the reasons.

Solution: (1) by the question.

(ii) If

Danggu

if

while

So for

Example 9. (Jiangxi Volume) The first n terms and Sn of the known sequence {an} satisfy the general formula sn-sn-2 = 3 to find the sequence {an}.

Solution: Method 1: First consider the even items as follows:

………

Similarly, odd-numbered items are:

………

Synthetic available

Example 10. Let the common ratio of geometric series be the sum of the first n terms.

(i) The numerical range to be obtained;

(Ⅱ) Let the sum of the first n items in the record be, and try to compare the sum.

Solution: (i) Because it is a geometric series,

while

The above formula is equivalent to the inequality group: ①

Or ②

Q & gt 1; Solution ②, because n can be odd or even, we get-1

To sum up, the value range of q is

(ii) By

therefore

∫> 0 and-1 0

When or when.

When and ≠0, that is

When or =2, that is

Fourth, the transformation and transformation of ideas.

When dealing with mathematical problems, we often turn the problems we want to solve into a kind of problems we are familiar with.

Example 1 1. Given the first term of a series, the sum of the first n terms is, and the general term formula is.

Analysis and brief solution: when n≥2,,. Subtract two expressions to get.

,

.

It can be seen that the common ratio of geometric series is 2.

Say it again,

OK,

Then.

Therefore.

Divide both sides by the same number and you will get

(constant),

As you can see, the first term is arithmetic progression, and the tolerance is. therefore

,

Therefore.

Comment: In this case, through two transformations, the first transformation is classified as geometric series, and the second transformation is classified as arithmetic series. This question reduces the difficulty.

Example 12. Let it be a positive series whose first term is 1, and (n= 1, 2, 3…), find the general term.

Analysis and brief solution: known.

As a positive sequence, we know it, so we have it. (*)

Method 1 (iterative multiplication): multiply (*)

.

use

,

get

Method 2 (superposition method): by (*), remember, and then.

Use it.

Method 3 (Constant series): We can know from formula (*) that it is a constant series.

Then,

Yes

Comments: Some series are not easy to be directly converted into arithmetic or geometric series, but they can be converted into series with general terms by seeking special relations through reasoning. Clever use of the above examples

And solve.

Example 13. The general formula of a known series is to find the front sum of this series.

Solution:

Comments: This example uses the idea of reduction to disassemble each item (elimination item) in the series and skillfully find the sum before.

Example 14, verification:

Proof method 1: Lingyou

Evidence 2: Order

rule

;

Comments: Proof 1 is a grouping summation proof, and Proof 2 is a reverse addition summation proof.

In a word, there are many mathematical methods such as elimination, construction, dislocation subtraction, reverse addition, decomposition elimination, decomposition and grouping summation.

Fifth, inductive conjecture, mathematical inductive thought

Example 15. Let a 1 = a≦, the first term of series {an}, and,

Remember, n = = 1, 2, 3, ... .

(i) Find a2 and A3;

(2) Judge whether the sequence {bn} is a geometric series and prove your conclusion;

(iii) Seek.

Solution: (i) A2 = A 1+= A+, A3 = A2 = a+;

(II)∫a4 = a3+= a+, so a5= a4= a+,

So B 1 = A 1-= A-, B2 = A3-= (a-), B3 = A5-= (a-),

Guess: {bn} is the geometric series of common ratio?

Proved as follows:

Because bn+1= a2n+1-= a2n-= (a2n-1-) = bn, (n ∈ n *)

So {bn} is a geometric series with the first term A-, and the common ratio is?

㈢。

Example 16. Fish is a renewable resource in its natural state. In order to make sustainable use of this resource, it is necessary to investigate the effects of its regeneration ability and fishing intensity on the total fish resources from a macro perspective. Xn is used to indicate the total number of a fish population at the beginning of the nth year, n∈N*, x 1>0 > 0. Without considering other factors, set the fish stock in the nth year.

(1) Find the relationship between xn+ 1 and xn;

(2) conjecture: if and only if x 1, a, b and c meet any conditions, the total number of fish at the beginning of each year will remain unchanged. (No proof required)

(2) Let A = 2 and B = 1. In order to ensure that Xn > 0 and n∈N* exists for any x1∈ (0,2), the fishing intensity of B is

What is the maximum allowable value? Prove your conclusion.

Solution (1) From the beginning of n to the beginning of n+ 1, the reproduction quantity of fish is axn, the catch quantity is bxn, and the death quantity is.

(ii) If the total fish population at the beginning of each year remains unchanged, xn is always equal to x 1, n∈N*, which is obtained by formula (*).

Because x1>; 0, so a>b.

Guess: If and only if a>b, and at the beginning of each year, the total number of fish remains the same.

(iii) if the value of b is such that xn >；; 0，n∈N*

From xn+ 1 = xn (3-b-xn), n ∈ n *, we know.

0 & ltxn & lt3-b, n ∈ n *, especially, 0.

And x1∈ (0,2), so

It is speculated that the maximum allowable value of b is 1.

The next proof is that when x1∈ (0,2) and b= 1, there are xn ∈ (0,2), n∈N*

① When n= 1, the conclusion is clearly established.

② Assuming that n=k, the conclusion holds, that is, xk ∈ (0,2),

Then when n=k+ 1, xk+ 1 = xk (2-xk? )& gt0.

And because xk+1= xk (2-xk) =-(xk-1) 2+1≤1

So xk+1∈ (0,2), so when n=k+ 1

According to ① and ②, for any n∈N*, there is xn ∈ (0,2).

To sum up, in order to ensure that any x 1 ∈ (0,2) has Xn >: 0, n∈N*, the maximum allowable value of fishing intensity b is1.

Example 17. Known series

(1) proof

(2) Find the general term formula an of the sequence.

Solution: (1) Method 1 is proved by mathematical induction:

1 When n= 1,

This proposition is correct.

If n=k, yes.

rule

but

and

This proposition is correct.

From 1, 2, we know that n∈N sometimes happens.

Method 2: Prove by mathematical induction:

1 When n= 1, ∴;

2 if n=k,

Order, monotonically increasing on [0,2], so by assuming

Have: namely

That is, it holds when n=k+ 1, so for everything,

(2) Let's find the general term of the sequence: So

,

And bn =- 1, so

Finally, mathematical thoughts and methods are the "soul" of mathematics, which is not completely abstract, but the objective content with mathematical knowledge as the carrier. It is the accumulation of people's experience in solving problems, the refinement and summary of problem-solving methods, and it is applicable, universal and instructive. Therefore, when reviewing series, we should pay attention to the infiltration of mathematical thinking methods, let students know its value and cultivate application consciousness.