In recent years, the problem of finding the general term formula by giving the analytical formula of sequence (including recursive relationship and non-recursive relationship) has appeared in the college entrance examination. It is difficult for students to solve such problems. This paper introduces the basic methods and skills of finding the general formula of this kind of problem with examples for teaching reference.

1, superposition method

There is an analytical formula in the form of an+ 1=an+f(n), f (1)+f (2)+...+f (n) is solvable, and an can be obtained by multimodal addition.

Example 1. In the sequence {an}, a 1 =- 1, an+ 1= an+2n, find an(n≥2).

Solution: According to the conditions, A2 = A 1+2× 1, A3 = A2+2× 2 ……, An = An- 1+N (n? 0? 1- 1), and the above n- 1 expression is supplemented and simplified: an? 0? 1? 0? 1 = a 1+n(n- 1)= n？ 0? 1? 0? 12-n- 1。

2. Iterative multiplication

Is the form of the sequence an=f(n)? 6? 1an- 1, and f( 1)? 6? The product (2)...f (n) of 1f is solvable, and an can be obtained by multimodal multiplication.

Example 2. In the sequence {an}, ≥2), find.

Solution: Through the condition an- 1,

This n- 1 formula is simplified by multiplication:

.

3, undetermined coefficient method

The linear recurrence relation of sequence has a shape such as and b is a constant), and an can be obtained by undetermined coefficient method.

Example 3. In the sequence {an}, find.

Solution: Add the undetermined number to both sides to get +(- 1)/3), so that the series {is a geometric series with a common ratio of 3.

∴an =

4. Factorization method

When the sequence relationship is complex, we can consider decomposing factors and contracts into simpler forms, and then get an by other methods.

Example 4. The known sequence satisfies (n∈) and the condition is ≥2).

Solution: From:

For n∈, obtained by undetermined coefficient method:

∴

5. Difference method

The sequence has a similar relation (non-recursive relation), which can be obtained by other elementary methods after calculating the difference.

Example 5. Let it be a sequence of positive numbers, the sum of the preceding items is, and the arithmetic mean of all natural numbers and 2 is equal to the arithmetic mean of 2:

(1) Write the first three items of the series;

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

The questioner's intention is to find the first three items of the series by asking questions (1), and then guess the general formula; (2) The conjecture is proved to be correct by mathematical induction. In fact, it is simpler to find the general term formula by difference method.

Solution: (1) abbreviated

(2) It is also restricted by conditions.

Namely ①

②

①-② Yes,

that is

factoring

For ∑> 0, ∴

∴ is arithmetic progression with an error of 4,

6. Reciprocity method

The sequence has a similar shape relationship and can be obtained by multiplying the two sides of the equation.

Example 6. Set the order to meet the demand.

Solution: transform the original condition into two sides multiplied by the same.

∵

∴

7. The method of composing arithmetic and geometric series by compound sequence.

Sequence has the relationship between form and form. We can turn the compound sequence into arithmetic progression or geometric progression, and then get it by other elementary methods.

Example 7. In the series, find

Solution: According to conditions

∴

∴ Then use multimodal addition to obtain:

8. Circular method

There is a similar relationship between sequences. If the complex series cannot form arithmetic and geometric series, it can sometimes be found by considering the cyclic relationship.

Example 8. In this series,

Solution: According to conditions

that is

That is, once every six items. 1998=6×333,

∴

9. Opening method

For some series, you can ask for it first and then ask for it.

Example 9. There are two series, each of which is a positive integer. For any natural number, all become arithmetic progression, and all become geometric series.

Solution: The conditions are as follows:

Derived from formula ②: ③.

④

Substitute ③ and ④ into ① to get:

Distorted).

∵ >0,∴ - .

∴: This is arithmetic progression. because

Therefore.