If there is a corresponding relationship between the nth term an of a sequence and one or more other terms of the sequence, this relationship is called the recurrence formula of the sequence.for exampleFibonacci sequenceThe recurrence formula of is an=an-1+an-2
Recursive formula of arithmetic sequence: an=D (n-1)+a (d istoleranceA isFirst item）
Recursive formula of proportional sequence: bn=Q (n-1) * b (q isCommon ratioB is the first item)
The method of writing the sequence by recursive formula:
1. Write the first few items of the sequence according to the recurrence formula, and then put them into the calculation in turn;
2. If the last term is known, the given formula is usually arranged in the form of using the later term to represent the previous term.
Recursive Column
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Also called recursive column.The sequence of the following items can be deduced from the preceding items.It means that for all n>p, the form an=f(an-1，an-2，…，an-p）The sequence of the relation {an}, where f is a function.P is a fixed positive integer, aone，atwo，…，apIs a known number.P is called the order of the recurrence sequence. The above relationship is called the recurrence formula. Given aone，atwo，…，ap, you can get all a from itn。In the form of an+conean-1+ctwoan-2+…+cpan-p=0(cone，ctwo，…，cpIs a constant) is called a linear recurrence formula, and the corresponding sequence is called a linear recurrence column.The simplest recursive column is a first-order recursive column, that is, an=f(an-1）Sequence {a ofn}. It is also called iteration column.Arithmetical sequenceAndProportional sequenceAre linear iterative columns.[1]
Proportional sequence
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IsometricseriesIt refers to a sequence of numbers from the second term onwards, in which the ratio of each term to its previous term is equal to the same constant. It is often represented by G and P.This constant is called proportional sequenceCommon ratio, the common ratio is usually expressed by the letter q (q ≠ 0), and the sequence of proportional numbers aone≠ 0。Where {an}Each item in is not 0.Note: when q=1, anbyConstant column。
Proportional formula
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(1) Definition formula:
（2）General formula(The general formula of the proportional sequence is obtained by multiplying the definition):
(3) Summation formula:
The summation formula is described in words: Sn=the first term (1-the n-th power of the common ratio)/1-the common ratio (the common ratio ≠ 1) If the common ratio q=1, then each term in the sequence of proportional numbers is equal, and its general formula is
, any two
，
The relationship of is
；When using the sum of the first n terms of the proportional sequence, we must pay attention toDiscuss common ratioqIs it 1
(4) From the definition of proportional sequence, general term formula, first n terms and formula, we can deduce:
In addition, one item isPositive numberSame items of the proportional sequence ofbase numberForm aArithmetical sequence；On the contrary, take any positive number C as the base, and use the terms of an arithmetic sequence to doindexIf the power Can is constructed, it is an equal ratio sequence.In this sense, we say that a positive proportional sequence and an arithmetical sequence are "isomorphic".
Definition of the middle term of the equation: from the second term, each term（finite sequence (except for the last term of) is the middle term of the equal ratio of its previous term and its subsequent term.
Items and formulas of infinite recursive proportional sequence: the absolute value of common ratio is less than 1Infinite proportional sequenceWhen n increases infinitely, the limit is called the sum of the terms of this infinite proportional sequence.
Arithmetical sequence
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Starting from the second term, each term is equal to a finite sequence or an infinite sequence of numbers of the previous term plus the same number d.It is also called arithmetic sequence. This number d is called the tolerance of arithmetic sequence.
Arithmetic sequence starts from the second item, and each item is of the preceding and following itemsArithmetic mean。
If the tolerance of an arithmetical sequence is positive, it is an increasing sequence;If the tolerance of an arithmetic sequence is negative, it is a decreasing sequence;If the tolerance of an arithmetic sequence is equal to zero, it is a constant sequence.
For a sequence al，atwo，…，an,..., if the difference between two adjacent terms atwo-aone，athree-atwo，…，an+1-an,... constitute an arithmetic sequence with non-zero tolerance, which is called sequence {an}It is a second order arithmetic sequence.
The recursive method can be used to define the sequence of arithmetical numbers of each order: for the sequence {an}, if {an+1-an}If it is an arithmetical sequence of order r, then the sequence {an} is an arithmetical sequence of order r+1. An arithmetical sequence of order 2 or more is calledHigher order arithmetic sequence。
The general term formula of order r arithmetic sequence can be expressed by a polynomial of order r with respect to the number of terms n. On the contrary, the sequence of order r arithmetic sequence whose general term formula is a polynomial of order r with respect to the number of terms n.[2]
