I. Formula method
Example 1 Given that the sequence satisfies, find the general term formula of the sequence.
Solution: If the two sides are divided by, then, then, therefore, the series is based on the first term, and the arithmetic progression of the tolerance is obtained by arithmetic progression's general term formula, so the general term formula of the series is.
Comments: The key to solve this problem is to transform the recursive relation into a formula indicating that the series is arithmetic progression, and then directly use arithmetic progression's general formula to get it, and then get the general formula of the series.
Second, the accumulation method
Example 2 Find the general term formula of a given sequence if it is satisfied.
Solution: if you get it, you will get it.
So the general formula of the series is.
Comments: The key to solve this problem is to convert the recursive relation into, and then find out, that is, the general term formula of the sequence.
Example 3 Find the general term formula of a given sequence if it is satisfied.
Solution: if you get it, you will get it.
therefore
Comments: The key to solve this problem is to convert the recursive relation into, and then find out, that is, the general term formula of the sequence.
Given that the sequence satisfies, find the general term formula of the sequence.
Solution: get it from both sides.
So, therefore,
Therefore,
rule
Comments: The key to solve this problem is to convert the recursive relation into, then find out, that is, the general term formula of the series, and finally find out the general term formula of the series.
Third, cumulative multiplication
Example 5 Given that the series satisfies, find the general term formula of the series.
Solution: Because, so, then, so.
So the general formula for this sequence is
Comments: The key to solve this problem is to convert the recursive relation into, and then find out, that is, the general term formula of the sequence.
Example 6 The general term formula of known sequence is satisfied and solved.
Solution: Because ①
So ②
Use type ②-①
rule
therefore
So ③
By,, and then, know, and then, substitute ③.
Therefore, the general formula of is
Comments: The key to solve this problem is to transform the recursive relation into, then find out the appropriate expression, and finally find out the general term formula of the series.
Fourth, the undetermined coefficient method
Example 7 Given that the series satisfies, find the general term formula of the series.
Solution: Setting ④
Substituting into Equation ④, both sides of the equation are eliminated. Separate the two sides of formula 4, and you can substitute it into formula 4.
By and (5), then, then the series is a geometric series with the first term and 2 as the common ratio, then, therefore.
Comments: The key to solve this problem is to transform the recursive relation into, so that we can know that the series is geometric progression, then we can get the general term formula of the series, and finally we can get the general term formula of the series.
Example 8 Given that the series satisfies, find the general formula of the series.
Solution: Let ⑥.
Substitute 6 and you get.
Tidy up.
Sequence, and then, into the 6 type.
⑦
On the seventh day,
Yes, well,
So the series is a geometric series with the first term and 3 as the common ratio.
Comments: The key to solve this problem is to transform the recursive relation into, so that we can know that the series is geometric progression, then we can get the general term formula of the series, and finally we can get the general term formula of the series.
Example 9 Given that the series satisfies, find the general term formula of the series.
Solution: Hypothesis 8
Will be substituted into the formula and get.
, then
Both sides of the equation are eliminated, so,
Solve the equation, and then substitute it into the formula to get
⑨
Pet-name ruby by sum formula, get
So, therefore, the sequence is a geometric series with the first term and 2 as the common ratio, so, then.
Comments: The key to solve this problem is to transform the recursive relation into, so that we can know that the series is geometric progression, then we can get the general term formula of the series, and finally we can get the general term formula of the series.
Five, logarithmic transformation method
Example 10 The known sequence satisfies the general formula for finding the sequence.
Solution: Because, so. Take the ⑩ of the common logarithm on both sides of the formula.
set up
Substitute ⑩ into ⑩, get, cancel sorting on both sides, get, and then.
, so
Substitute into the formula and get
From and types,
OK,
Then,
So the series is a geometric series with the first term and 5 as the common ratio, so, therefore,
Then.
Comments: The key to solve this problem is to transform the recursive relation into by logarithmic transformation, so that we can know that the series is a geometric series, then we can get the general term formula of the series, and finally we can get the general term formula of the series.
Six, iterative method
Example 1 1 Given that the series satisfies, find the general term formula of the series.
Solution: Because, so.
So the general formula of this series is.
Comments: This question can also comprehensively use cumulative multiplication and logarithmic transformation to find the general term formula of series. That is to say, the two sides of the equation are obtained by taking the common logarithm, which can be inferred from the cumulative multiplication.
Seven, mathematical induction
Example 12 Given that the series satisfies, find the general term formula of the series.
Solution: From sum, we get.
From this, we can guess and prove this conclusion through mathematical induction.
(1) So the equation holds.
(2) Assuming that the equation was established at that time, that is, then at that time,
So the equation was also established at that time.
According to (1) and (2), this equation holds for either one.
Comments: The key to solve this problem is to find the first n terms of the series through the first term and recursive relationship, then guess the general term formula of the series, and finally prove it by mathematical induction.
VIII. Substitution method
Example 13 Given that the series satisfies, find the general term formula of the series.
Solution: So, order.
Therefore, instead
that is
Because, therefore,
Then, that is,
Can become,
Therefore, the geometric series, who is the first term, is considered fair. Therefore, it is, that is, it is
Comments: The key to solve this problem is to transform the given recursive relation into a form by changing the elements of to, so that we can know that the series is geometric progression, then get the general term formula of the series, and finally get the general term formula of the series.