If the sequence {an}N-th item a ofnThe relationship with n can be expressed by a formula, which is called sequenceGeneral formula（general formulas）。The general terms of some sequences can be expressed by two or more formulas.Without general formulaseriesIt also exists, such as allPrime numberA sequence of numbers.[1]
A list of numbers in a certain order is calledseriesEach number in the sequence is called the item of the sequence[2], each item is called item 1 (or first item), item 2,Item n.The sequence of numbers can also be seen as a function whose domain is the natural number set N (or its finite subset {1,2,3,..., n}). When the independent variable takes values from small to large, it corresponds to a list of function values.
General form of sequenceIt can be written as aone，atwo，...，an，...，Where an is the nth term of the sequence, which can also be abbreviated as {an}.
If the sequence {an}N-th item a ofnThe relationship with n can be expressed by a formula, which is called sequenceGeneral formula。
Example 1 Known sequence {an}The general formula of is
Find the second and fourth items in this sequence.
Solution:
Example 2 The first four items given by
Write a general formula of it.
Solution: Because
therefore
nature
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one
If the general term formula of a sequence of numbers is known, then just use 1,2,3Instead of n in the formula, we can find the terms of this sequence.
two
Not every infinite number sequence has a general formula, for example, all prime number sequences have no general formula.
three
Given the first n terms of a sequence, the general term formula is not unique.
four
The general terms of some sequences can be expressed by two or more formulas.
General term formula of arithmetic sequence
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If the arithmetic sequence {an}, the tolerance is d, then an=aone+(n-1) d, which is the arithmetic sequence {an}The general formula of.
Note: 1) Because an=nd+(aone-d) Therefore, the image of the arithmetic sequence is some scattered points on the same line whose abscissa is the natural sequence, and the geometric meaning of the tolerance d is the slope of the line.
2) Arithmetic sequence {an}The general formula of can also be determined by the following formula: ① an=am+(n-m)d，②am+n=(mam-nan)/(m-n)
3) Arithmetic sequence {an}The tolerance d of can be determined by the formula d=(an-am）/(n-m) determined.
Example 3 How many natural numbers can be divided by 7 within 100?
Solve the natural number within 100 that can be divided by 7 to construct an arithmetic sequence, where aone=7, d=7. By
Available
Because n ∈ N, there are 14 natural numbers within 100 that can be divided by 7.
General term formula of proportional sequence
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If the sequence of proportional numbers {an}The first item of is aone, common ratio is q, then sequence anThe general formula of is an=aoneqn-1.
Note: 1) Because an=aoneqn-1Therefore, when q>0 and q ≠ 1, the image of the proportional sequence is some scattered points on the same exponential function whose abscissa is a natural number.
2) Proportional sequence {an}The general formula ofn=amqn-mFormula determination.
Example 4 Known proportional sequence {an}Medium, aone=1，atwo=2. Write the general formula.
Solution: obviously, its general formula is an=2n-1.
