The method of directly using the definition of arithmetic progression or geometric progression to find the general term is called the definition method, which is suitable for the problem of known series types.
Example 1. Arithmetic progression is an increasing series, and the sum of the first n terms is 0, which becomes a geometric series. General formula for finding sequence.
Solution: Let the tolerance of series be
... in geometric series, ...
namely ......
∵ ,∴ ……………………①
∵
∴ …………②
From ① ②: ...,
∴
Comments: When using the definition method to find the general term of series, we should pay attention to the definition, try to find the first term and tolerance (common ratio) first, and then write the general term.
Second, the accumulation method
To find the general term of a series in the form of an-an- 1=f(n) (f(n) is an arithmetic or geometric series or other summable series), we can use the accumulation method, that is, let n = 2,3, … n- 1 get the expression of n- 1 to get the general term.
Example 2. In the known sequence {an}, there is a 1= 1 for any natural number n, so find.
Solution: From the known,
,……,
, ,
The above formula is cumulative, use-=
= ,
Comments: The accumulation method is to obtain n- 1 expressions by repeatedly using recursive relations, thus obtaining the general term. This method is finally converted into the sum of the first n- 1 terms of {f(n)}, and attention should be paid to the skills of summation.
Thirdly, iterative method.
To find the general term of a series with the shape (where it is a constant), we can use recursive relation to solve it iteratively.
Example 3. Given sequence {an} satisfies a 1= 1, an+ 1 =+1.
Solution: an = 3an-1+1= 3 (3an-2+1)+1= 32an-2+31= … = 3n-65438+.
Comments: Because there are a lot of data when using iterative method to solve problems, we should be careful in calculation to avoid calculation errors and lead to a dead end.
Fourth, the formula method
If the relationship between the sum of the preceding items of a series and is known, the general term of the series can be solved by a formula.
Example 4. The sum of the antecedents of the known sequence satisfies. Find the general term formula of sequence;
Solution: by
When? Yes.
……,
After verification, it also conforms to the above formula, so
Comments: When solving with formulas, we should pay attention to the discussion of N classification, but if we can write them together, we must merge them.
Five, cumulative multiplication
For the general term of a series, the general term can be multiplied by n=2, 3, … n- 1 to get n- 1 formulas.
Example 5. In the known sequence, the relationship between the sum of antecedents is to find the general term formula.
Solution: by
Subtract two expressions to get:,
,
Multiply the above n- 1 equations to get:
Comments: Multiplication is to multiply n- 1 formulas repeatedly by recursive relation to find the general term. This method is finally converted into the product of the n- 1 term of {f(n)}, so we should pay attention to the technique of product.
Sixth, the discussion method of dividing N parity
In some series problems, it is sometimes necessary to classify and discuss the parity of n in order to deal with the problems.
Example 6. Given the sequence {an}, a 1= 1, anan+ 1=2, find the general term formula.
Solution: When anan+ 1=2 is divided by an+ 1an+2=2, we get =, then a 1, a3, a5, …a2n- 1, … and a2, a4, a6, …a2n. (2) When n is an even number, the sum of.
Comments: Another way to classify the parity of n is that if the topic contains it, dividing n into parity can naturally lead to discussion. Classified discussion is equivalent to additional conditions, turning uncertainty into certainty. Finally, pay attention to the merger when you can write together. This is a new hot spot of college entrance examination in recent years, such as the 2 1 question of Jiangxi college entrance examination in 2005.
Seven. conversion method
The method of trying to turn unconventional problems into familiar sequence problems to find general formulas is called reduction method, and it is also an idea that we must have when solving any mathematical problems.
Example 7. Known sequence satisfies
Qiuan
Solution: When
Divide both sides by the same number,
That's for sure,
∴ arithmetic progression, the first item is 5, and the tolerance is 4.
Comments: With the help of arithmetic progression's general formula, this question is a typical reduction method. Common reduction methods include logarithmic reduction and undetermined coefficient reduction. , and it is a common method in the college entrance examination, summarized as geometric series or arithmetic progression.
Eight, "induction-guess-proof" method
When it is difficult to solve or deform directly, first find the first few terms of the sequence, guess the general term, and then prove it by mathematical induction, that is, induction-guess-proof method.
Example 8. If the sequence is satisfied: calculate the values of a2, a3 and a4, and then summarize the formula of an to prove your conclusion.
Solution: ∫ A2 = 2a1+3× 2 = 2×1+3× 2,
a3 = 2(2× 1+3×2)+3×2 1 = 22× 1+2×3×2 1，
a4 = 2(22× 1+2×3×2 1)+3×22 = 23× 1+3×3×22；
Guess an = 2n-1+(n-1) × 3× 2n-2 = 2n-2 (3n-1);
Prove by mathematical induction:
1 When n= 1, A 1 = 2- 1× = 1, the conclusion is correct;
2 If n=k, AK = 2k-2 (3k- 1) is correct.
When n=k+ 1,
= The conclusion is correct;
From 1 and 2, it is known that n∈N* exists.
Comments: When using the "induction-guess-proof" method, you should guess carefully and don't guess wrong, otherwise all previous efforts will be in vain; When proving by mathematical induction, we should pay attention to the completeness of the format and must use inductive hypothesis.
Nine, undetermined coefficient method (construction method)
In order to find the general term of a recursive formula, such as (P and Q are constants), we can turn it into a well-known series by the method of undetermined coefficients, which is equivalent to method of substitution.
Example 9. The known sequence {an} satisfies a 1= 1, an+ 1 = +2.
Solution: If, then,
For geometric series,
,
Comments: To find the general term of a recursive series (p, q is constant), we can construct a new series an+ 1+ =p(an+) by iterative method or undetermined coefficient method, and we can also find it by induction-guessing method, which is also a kind of problem that has been tested a lot in the college entrance examination in recent years.
Example 10. The known sequence satisfies.
Solution: divide the two sides together, get and transform into.
Okay, then. Orders,
Yes Conditions can be translated into,
The sequence is the first term, which is the geometric series of tolerance.
Because, so =
Get =。
Comments: When the recursive formula is (both P and Q are constants), it can be divided by the same, and then it can be classified as (both P and Q are constants).
Example 1 1. Known sequences satisfy the need to find one.
Solution: Suppose. ....
After expansion, we must .....
Get from ...
Conditions can be transformed into
The problem of obtaining sequence ... as the first semester ... because the geometric progression of tolerance is transformed into the problem of finding the general term of sequence by accumulation method, and the solution is obtained.
Comments: When the recursive formula is (p, q are constants), it can be set, and its undetermined constants S and T are calculated, thus simplifying to the above-mentioned known problems.