Sequence of numberPositive integer set(or its limitationsubset）ForDefine FieldsAn ordered list of numbers.Each number in a sequence is called an item of the sequence.The number in the first place is called the first item (usually also called the first item) of the sequence, the number in the second place is called the second item of the sequence, and so on. The number in the nth place is called the nth item of the sequence, usually using anexpress.
Legend of Ancient GreecePythagoras(ApproxB.C570 - about 500 BC) school mathematicians often study mathematical problems on the beach. They draw dots or use small stones to represent numbers on the beach.For example, they have studied:
Since these numbers can be represented by a triangular lattice as shown in Figure 1, they call them triangular numbers.
Number of squares
Similarly,
go by the name ofsquareBecause these numbers can be expressed as squares.Therefore, a sequence of numbers arranged in a certain order is called a sequence of numbers.
concept
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Function interpretation
Function understanding of sequence:
① Sequence is a special function.Its particularity is mainly reflected in its definition domain andrangeOn.A sequence of numbers can be regarded as a function whose domain is a positive integer set N * or its finite subset {1, 2, 3,..., n}, in which {1, 2, 3,..., n} cannot be omitted.
② It is an important way of thinking to understand the sequence of numbers from the point of view of function. Generally, there are three ways to express a function, and the sequence of numbers is no exception. There are usually three ways to express a function: a. tabulation;b。Image method;c.analytic method。Among them, the analytic method includes giving sequence by general term formula and giving sequence by recursive formula.
The item in the number sequence must be a number, which can be either a real number or acomplex。
Use symbol {an}It refers to a sequence of numbers, just "borrowing"aggregateThere are essential differences between them:oneThe elements in a set are different, while the items in a sequence can be the same.twoThe elements in the set are unordered, while the items in the sequence must be arranged in a certain order, that is, they must be ordered.
classification
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(1) Finite series and infinite series:
The sequence with limited terms is“finite sequence ”（finite sequence）；The number sequence with infinite terms is“infinite sequence ”（infinite sequence）。
(2) For positive sequence: (positive sequence refers to the sequence in which all items are positive)
1) Starting from the second item, the sequence in which each item is larger than its previous item is calledincreasing sequence ；For example: 1, 2, 3, 4, 5, 6, 7;
2) Starting from the second item, the sequence in which each item is smaller than its previous item is calleddecreasing sequence ；For example: 8, 7, 6, 5, 4, 3, 2, 1;
3) From the second item, some items are larger than their previous items, and some items are smaller than their previous itemsWobble sequence；
(1) General term formula: the relation between the Nth term an of the sequence and the ordinal number n of the term can be expressed by a formula an=F (n), this formula is called the general formula of this sequence, such as
。General term formula of sequenceThe characteristics of are: 1) The general term formula of some sequences can have different forms, that is, it is not unique;2) Some sequences have no general formula (for example, prime numbers are arranged from small to large into a column 2, 3, 5, 7, 11,...).
(2) Recursive formula: if the number sequence {an}The relationship between the nth term of the number sequence and its previous term or terms can be expressed by a formula, so this formula is called theRecursive formula。Features of recurrence formula of sequence: 1) The recurrence formula of some sequence can have different forms, that is, it is not unique.2) Some sequences have no recurrence formula, that is, there is a recurrence formula but not necessarily a general formula.
Arithmetical sequence
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definition
Generally, if a sequence starts from the second term, the difference between each term and its previous term is equal to the sameconstantThis sequence is called an arithmetic sequence, and this constant is called a common difference. The tolerance is usually expressed by the letter d,Sum of the first n itemsIt is represented by Sn.The arithmetic sequence can be abbreviated as AP.（Arithmetic Progression）[1]。
General formula
an=aone+(n-1)d
Where, a when n=1one=S1；When n ≥ 2 an=Sn-Sn-1。
an=The derivation process of kn+b (k, b are constants): an=dn+aone-D Make d=k, aone-D=b gives an=kn+b。
Mean term of equidifference
The arithmetic sequence composed of three numbers a, A and b can be called the simplest arithmetic sequence.At this time, A is called the arithmetic mean of a and b.Related: A=(a+b) ÷ 2.
Sn=aone+atwo+athree+·····+an=aone+(aone+d)+(aone+2d)+······+[aone+(n-1)d] ①
Sn=an+an-1+an-2+······+aone=an+(an-d)+(an-2d)+······+[an-(n-1)d] ②
2S from ①+②n=(aone+an)+(aone+an)+······+(aone+an）(n)=n (aone+an)
∴Sn=n(aone+an)÷2。
The sum of the first n terms of the arithmetic sequence is equal to half of the product of the sum of the first and last two terms and the number of terms:
Sn=n(aone+an)÷2=naone+n(n-1)d÷2
Sn=dntwo÷2+n(aone-d÷2)
Also available
aone=2sn÷n-an
an=2sn÷n-aone
Interestingly, S2n-1=(2n-1)an，S2n+1=(2n+1)an+1
nature
(1) Any two items am，anThe relationship of is: an=am+(n-m) d, which can be regarded as the generalized general term formula of arithmetic sequence.
(2) From the definition of arithmetic sequenceGeneral formula, the first n terms and formula can also deduce: aone+an=atwo+an-1=athree+an-2=…=ak+an-k+1，k∈N*。
(3) If m, n, p, q ∈ N*, and m+n=p+q, then there is am+an=ap+aq。
(4) For any k ∈ N*, with Sk，S2k-Sk，S3k-S2k，…，Snk-S(n-1)k... into an arithmetical series.
application
In daily life, people often use the differential number sequence. For example, when grading the size of various products, when the maximum size is not much different from the minimum size, it is often graded according to the differential number sequence.If it is an arithmetic sequence with an=m，am=n. Then am+n=0。Its application in mathematics can be exemplified by quickly calculating the integer of 6 from 23 to 132multipleHow many are there? There is more than one algorithm. Here we introduce how to calculate the first item a of the arithmetical sequenceone=24 (24 is 4 times of 6), equal difference d=6;So an=24+6 (n-1)<=132 gives n=19.
Proportional sequence
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definition
Generally, if a sequence starts from the second term and the ratio of each term to its previous term is equal to the same constant, this sequence is calledProportional sequence（geometric sequence）。This constant is called proportional sequenceCommon ratio（common ratio）The common ratio is usually expressed by the letter q.
The sequence of proportional numbers can be abbreviated as GP.（Geometric Progression）。
Proportional mean term
If a number G is inserted between a and b, so that a, G, and b are in equal proportion, then G is called theProportional mean term。
It matters:
；
。
Note: There are two terms in the equality ratio of two non-zero real numbers with the same sign, and they are each otherInverse number, so
It is a necessary and insufficient condition for a, G and b to form an equal proportion sequence.
General formula
(The first item is
, the common ratio is q);
(n≥2)。
Sum of the first n items
When q ≠ 1, the formula of the sum of the first n terms of the proportional sequence is:
；
When q=1, the formula of the sum of the first n terms of the proportional sequence is:
；
The relationship between the first n terms and the general term:
；
（n≥2）。
nature
(1) If m, n, p, q ∈ N *, and m+n=p+q, then
；
(2) In the sequence of proportional numbers, the sum of each k item in turn remains an proportional number sequence.
(3) From the definition of proportional sequence, general term formula, first n terms and formula, we can deduce:
(4) The middle term of equality ratio: q, r, p form an equality ratio sequence, then
，
Is
The middle term of the equation.
remember
, there is
。
In addition, one item isPositive numberThe same base logarithm is taken from the items of the proportional sequence ofArithmetical sequence；On the contrary, take any positive number C as the base, and use the terms of an arithmetic sequence to doindexConstructive power
Is an equal ratio sequence.In this sense, we say that a positive proportional sequence and an arithmetical sequence are "isomorphic".
(5) Sum of the first n terms of the proportional sequence
；
(6) Any two items
The relationship of is
；
(7) In a proportional sequence, the first term
And the common ratio q is not zero.
application
Proportional sequence is also often used in life.For example, the bank has a way to pay interest---Compound interest。That is, the interest and principal of the previous period are added together to calculate the principal, and then the interest of the next period is calculated, which is commonly known asat compound interest。Formula for calculating the sum of principal and interest based on compound interest: sum of principal and interest=principal*(1+interest rate) ^ deposit period.
Equal sum sequence
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“Equal sum sequence”: In a sequence, if the sum of each term and its subsequent term is the same constant, then this sequence is called the equal sum sequence, and this constant is called the common sum of the sequence.
For a sequence of numbers, if the sum of any continuous k (k ≥ 2) terms is equal, we will call this sequence of numbers equal sum sequence. Its nature is: it must be a cyclic sequence.
