Proof of the Equal Difference Series, how to prove the Equal Difference Series

sn-s(n-1)=2sn 2 2sn-12sn 2-sn-2sns(n-1)+s(n-1)=2sn 2sn-2sns(n-1)+s(n-1)=0 divided by sns(n-1).

1/s(n-1)-2+1/sn=0

1/sn-1/s(n-1)=2

The number column is the first of 1 and the tolerance is 2.

1/sn=1+2(n-1)=2n-1 sn=1/(2n-1)1/s(n-1)=1+2(n-2)=2n-3 s(n-1)=1/(2n-3)

an=1/(2n-1)-1/(2n-3)

When n>=2, an=sn-sn-1, substituted and sorted to obtain sn-1-sn-2snsn-1=0, and divided by snsn-1 on both sides at the same time, (1 sn)-1 (sn-1)=2

A number column is an equal difference series.

s1=a1=1, then according to the general formula of the equal difference series, 1 sn=1+2(n-1)=2n-1 can be obtained

Then sn=1 (2n-1) can be obtained by substituting the medium formula into the problem again.

an=2 (2n-1)(3-2n), where when n=1, a1=1

The two most commonly used methods are:

1. Prove with definitions, that is, prove an-an-1=m (constant) 2. Sell the property of the same difference series, that is, prove 2an=an-1+an+1 Other methods: 1. Prove that there is always an equal difference in the middle term, that is, 2an=a(n-1)+a(n+1).

2. The first n terms and conform to sn=an 2+bn

Here's how to prove the difference series:

Let the difference series an=a1+(n-1)d the maximum number plus the minimum number divided by two, i.e., [a1+a1+(n-1)d] 2=a1+(n-1)d 2, the mean is sn n=[na1+n(n-1)d 2] n=a1+(n-1)d 2 to prove that the three numbers abc are equal difference series, then c-b=b-a,c 2(a+b)-b 2(c+a) the first = (c-b)(ac+bc+ab), b 2(c+a)-a 2(b+c)=（b-a）（ac+bc+ab）。

Since C-B=B-A, then (C-B)(Ac+BC+AB)=(B-A)(AC+BC+AB), that is, C 2(A+B)-B 2(C+A)=B 2(C+A)-A 2(B+C), so A 2(B+C), B 2(C+A), C 2(A+B) are equal difference series, and the equal difference series refers to the number series from the second term onwards, the difference between each term and its previous term is equal to a constant, which is often represented by a and p. This constant is called the tolerance of the equal difference series, and the tolerance is often denoted by the letter d. The application of the difference series in daily life, the difference series is commonly used, such as when the size of various products is divided into levels, when the maximum size and the minimum size are not much different, they are often graded according to the difference series.

Sequence Definition:

A sequence is a function that defines a domain from a set of positive integers. Each number in the sequence is called the term of the series, the number in the first place is called the first term of the series, the number of acres in the second place is called the second term of the series, and so on, the number in the nth position is called the nth term of the series, which is usually represented by an. Famous sequences include Fibonacci sequences, trigonometric functions, Cattelan numbers, Yang Hui triangles, etc.

A sequence is a special kind of function. Its particularity is mainly reflected in its definition domain and value range. A sequence can be thought of as a set of positive integers defined as n* or a finite subset of its (1,2,3,..., n), where (1, 2, 3,..., n) cannot be omitted.

In general, there are three ways to represent functions, and sequences are no exception, and there are usually three ways to express them, namely the list method, the image dislike method, and the analytical method. The analytic method includes giving a series of numbers with a general formula and giving a series of numbers with a recursive formula.

The two most commonly used methods are:

1. Prove with definitions, that is, prove an-an-1=m (constant) 2. Sell the property of the same difference series, that is, prove 2an=an-1+an+1 Other methods: 1. Prove that there is always an equal difference in the middle term, that is, 2an=a(n-1)+a(n+1).

2. The first n terms and conform to sn=an 2+bn

1. Prove by definition, i.e., prove an-an-1=m (constant).

2. Prove the properties of the difference series, that is, prove 2an=an-1+an+1.

3. Prove that there is always an equal difference in the middle term of the brigade, that is, 2an=a(n-1)+a(n+1).

4. The first n terms are in line with sn=an2+bn.

A series of equal differences refers to a series of numbers from the first or second term, and the difference between each term and its previous term is equal to the same constant, which is often expressed by a and p. This constant is called the tolerance of the equal difference series, and the tolerance is often denoted by the letter d.

Extension: Definition of Equal Difference Series An equal difference series is a series of numbers in which the difference between two adjacent items in an exponential column is the same, and this constant is called the difference of the equal difference series.

The four ways to prove an equal difference series are as follows:

Prove by definition, i.e. prove an-an-1=m (constant); Prove by the properties of the difference series, i.e., prove 2an=an-1+an+1;Prove that there is a constant equal difference term, i.e., 2an=a(n-1)+a(n+1); The first n terms and conform to sn=an2+bn.

Definition of Equal Difference Series:

A series of equal differences refers to a series of numbers from the second term onwards, in which the difference between each term and its previous term is equal to the same constant, which is often represented by a and p. This constant is called the tolerance of the equal difference series, and the tolerance is often denoted by the letter d.

For example: 1, 3, 5, 7, 9 ......2n-1。The general formula is:

an=a1+(n-1)*d。The first term a1 = 1, tolerance d = 2. The first n terms and the formula are:

sn=a1*n+[n*(n-1)*d]2 or sn=[n*(a1+an)]2. Note: The above n is a positive integer.

The basic properties of the difference series:

If the tolerance is d, the series obtained by adding one number to each item is still an equal difference series, and its tolerance is still d; If the tolerance number is vertically d, the series obtained by multiplying the same number by the constant k is still an equal difference series, and its tolerance is kd; If it is an equal difference series, then and (k and b are non-zero constants) are also equal difference series.

For any m and n, in the series of equal differences, there are: an = am + n m)dm, n n +), in particular, when m = 1, the general term formula of the equal difference series is obtained, which is more general than the general term formula of the equal difference series; In general, when m+n=p+qm,n,p,q n+), am+an=ap+aq.

A series of equal differences with a tolerance of d, from which the terms of equal distance are taken out to form a new series, which is still a series of equal differences, and its tolerance is kd (k is the difference of the number of terms taken out); The following table is composed of terms with a tolerance of m, and m n+) is composed of an equal difference series with a tolerance of md.

In the series of equal differences, from the second term onwards, each term (except the last term of the infinite series) is the middle term of the two terms before and after the major of the key of the same number; When the tolerance d is 0, the number in the difference series increases with the increase of the number of terms; When d 0, the number in the difference series decreases with the decrease in the number of terms; At d 0, the number in the equal-brightness difference sequence is equal to a constant.

Practical applications of contour series:

Finance: The equal difference series can be used to calculate fixed deposits, regular investment, equal principal and interest repayment, etc. Logistics: Differential series can be used to calculate the efficiency of container loading and unloading, and can also be used to plan route optimization.

Engineering: Equal difference series can be used to calculate the length of steel bars, the length of steel plates, etc. Geography: Equal difference series can be used to calculate changes in altitude, changes in the temperature of sea water, etc.

Medical field: Equal difference series can be used to calculate the dosage of drugs, the metabolism of drugs, etc. Education: Equal difference series can be used to calculate learning progress, changes in test scores, etc.