What is dislocation subtraction

The method of dislocation subtraction is a method to solve the problem of summation. In the type of problem, it can be used only when the coefficient before a and the index of a are equal
S=a+2a2+3a3+…… +(n-2)an-2+(n-1)an-1+nan （1）
Multiply a on both sides of (1) to obtain equation (2) as follows:
aS= a2+2a3+3a4+…… +(n-2)an-1+(n-1)an+nan+1 （2）
Using (1) - (2), equation (3) is obtained as follows:
（1-a）S=a+(2-1)a2+(3-2)a3+…… +(n-n+1)an-nan+1 （3）
（1-a）S=a+a2+a3+…… +an-1+an-nan+1
S=a+a2+a3+…… +An-1 + an use the sum formula of this
（1-a）S=a+a2+a3+…… +an-1+an-nan+1
Finally, divide both sides of the equation by (1-A) at the same time to get the general formula of S

How to subtract by dislocation subtraction
I know the principle. My problem is how to subtract the first form from the second form. I don't understand the simplified part when subtracting. Excuse me!

Dislocation subtraction method is a common summation method of sequence, which is applied to the form of multiplication of equal ratio sequence and equal difference sequence. For example, an = bncn, where BN is equal difference sequence and cn is equal ratio sequence; SN is listed separately, and then all formulas are multiplied by the common ratio of equal ratio sequence, that is, KSN at the same time; then the two formulas are subtracted by one bit
catalog
brief introduction
give an example
Solving problems by dislocation subtraction
Edit an introduction to this paragraph
Dislocation subtraction is more commonly used in the general term of sequence, which is the product of an arithmetic sequence and an arithmetic sequence, such as an = (2n-1) * 2 ^ (n-1), in which the 2N-1 part can be understood as arithmetic sequence, and the 2 ^ (n-1) part can be understood as arithmetic sequence
Examples of editing this paragraph
For example: sum Sn = 1 + 3x + 5x ^ 2 + 7x ^ 3 + +(2n-1) * x ^ (n-1) (x ≠ 0) when x = 1, Sn = 1 + 3 + 5 + +(2n-1) = n ^ 2; when x is not equal to 1, Sn = 1 + 3x + 5x ^ 2 + 7x ^ 3 + +(2n-1)*x^(n-1)；　　∴xSn=x+3x^2+5x^3+7x^4+… +(2n-1) * x ^ n; by subtracting the two formulas, we get (1-x) Sn = 1 + 2x [1 + X + x ^ 2 + x ^ 3 + +X ^ (n-1)] - (2n-1) * x ^ n; Sn = (2n-1) * x ^ (n + 1) - (2n + 1) * x ^ n + (1 + x) / (1-x) ^ 2
To solve the problem by subtraction
The dislocation subtraction method is a kind of method to solve the problem of summation. In the type of problem, it can be used only when the coefficient in front of a and the index of a are equal. This is an example (format problem, the number and N after a are exponential): S = a + 2A2 + 3a3 + +(n-2) An-2 + (n-1) an-1 + Nan (1) multiplies a on the left and right sides of (1) to obtain the following equation (2): as = A2 + 2A3 + 3A4 + +(n-2) an-1 + (n-1) an + Nan + 1 (2) using (1) - (2), we obtain equation (3) as follows: (1-A) s = a + (2-1) A2 + (3-2) A3 + +(n-n+1)an-nan+1 （3）　　（1-a）S=a+a2+a3+…… +an-1+an-nan+1 　　S=a+a2+a3+…… +An-1 + an use this summation formula. (1-A) s = a + A2 + a3 + +Finally, divide both sides of the equation by (1-A) to get the general formula of S. example: sum Sn = 3x + 5x ^ 2 + 7x ^ 3 + When x = 1, Sn = 1 + 3 + 5 + ... + (2n-1) = n ^ 2;; when x is not equal to 1, Sn = 3x + 5x ^ 2 + 7x ^ 3 + ... + (2n-1) · x to the power of n-1, so xsn = x + 3x ^ 2 + 5x ^ 3 + 7x to the fourth power (1-x) Sn = 1 + 2x (1-x) Sn = 1 + 2x (1 + x) Sn = 1 + 2x (1 + x) Sn = 1 + 2x (1 + X + X + x ^ 2 + x ^ 2 + x ^ 2 + x ^ 2 + x ^ 2 + X ^ 2 + 2 + 3 +... + X'n-2 (N-2-1) · X's N-power of (2n-1) · X's n-square (1-x) Sn = 1 + 2x (1 (1-x) Sn = 1 + 2x (1) Sn = 1 + 2x (1 + x) Sn = 1 + 1 + 2x (1) Sn = 1 + 1 + 2x (1) Sn = 1 + 2x (1 + 2x (1 + 2 + 2 + 2 + 2 + 2 (n = 3 * 2 + 2 + 5 * 5 * 4 * 4 * 4 * 4 * 4 + 4 * 4 + 4 * 4 + 4 + 4 + 4 + 4 + 4 + 4 + 4 2 ^ (n + 1) - Sn = 6 + 2 * 4 + 2 * 8 + 2 * 16 +... + 2 * 2 ^ n - (2n + 1) * 2 ^ (n + 1) = 6 + 2 * (4 + 8 + 16 +... + 2 ^ n) - (2n + 1) * 2 ^ (n + 1) = 6 + 2 ^ (n + 2) - 8 - (2n + 1) * 2 ^ (n + 1) (sum of equal ratio sequence) = (1-2n) * 2 ^ (n + 1) - 2, so Sn = (2n-1) * 2 ^ (n + 1) + 2 dislocation subtraction is used in the sum formula of equal ratio sequence 1 / 2Sn = 1 / 4 + 1 / 8 +. + 1 / 2 ^ n + 1 / 2 ^ (n + 1)

Offset subtraction

Dislocation subtraction is a common method for summation of sequence. Here is an example +(n-2) An-2 + (n-1) an-1 + Nan (1) multiplies a on the left and right sides of (1) to obtain the following equation (2): as = A2 + 2

The steps of dislocation subtraction
In the dislocation subtraction method, the second step is to multiply Sn by the common ratio. Whose common ratio is the common ratio?

Let the equal ratio sequence {an}, the first term be A1, and the common ratio Q ≠ 1
Sn=a1+a2+a3+----+an
Sn=a1+a1q+a1q^2+-----+a1q^(n-1)--------------(1)
The two sides of the above formula are multiplied by Q
qSn=a1q+a1q^2+a1q^3-----+a1q^(n-1)+a1q^n-----(2)
(1) Formula - (2) is obtained
Sn-qSn=a1-a1q^n
Sn=a1(1-q^n)/(1-q)
The common ratio q is the common ratio of the sequence {an}

(1) Let {an} satisfy an + 1-an = 2, A1 = 2, and find the general formula of {an}. (2) let {an} satisfy a1 + 3a2 + 32a3 + +3n-1an = N3, a ∈ n *. Find the general term of sequence {an}

(1) ∵ an + 1-an = 2, A1 = 2, ∵ sequence {an} is arithmetic sequence, ∵ an = 2 + (n-1) 2 = 2n. (5 points) (2) ∵ a1 + 3a2 + 32a3 + 3n−1an＝n3，①∴a1+3a2+32a3+… +3n-2an-1 = n − 13, (n ≥ 2) ② ① - ②, we get: 3N − 1An = N3 − n − 13 = 13 (n ≥ 2)

Senior high school mathematics compulsory five series questions, using superposition and dislocation subtraction
Let {an} satisfy a & # 8321; = 2, an + 1-an = - 3.2 (n-1)
(1) Find the general term formula of the sequence {an}; (N and N + 1 are the corner marks, n-1 is the power)
(2) Let BN = Nan, find the first n terms and Sn of the sequence

Tn=3×20+7×2+… +(4n-1)•2n-1
2Tn=3×2+7×22+… +（4n-5）•2n-1+（4n-1）•2n
∴Tn=(4n-1)•2n-[3+4(2+22+… +2n-1)]
How do you get it out? Which one is bigger···

What you wrote above is: the formula of the second line minus the formula of the first line
The main reason for your big head is that you don't write regularly
The first line: 1, 2, 3 （n-1） ,n
The second line: 1 2 （n-2）,（n-1) ,n
In strict accordance with the above good position, the second line minus the number of the first line. The result is the one you wrote
In fact, this is the operation method of "dislocation. Subtraction. Method". I hope you can fully understand the answer to two questions like this

An equal ratio sequence problem of compulsory 5 in senior high school mathematics
If A5 = - 8, q = - 1 / 2, then the nth term of a =?
It's better to have a detailed problem-solving process, thank you!

an=am·q^(n-m)（m,n∈N*）
∴an=a5·q^(n-5)=-8·(-1/2)^(n-5)=-128·(-1/2)^(n-1)

The third and fourth exercises of 5 equal ratio Series in senior high school mathematics
It is known that {an} is an infinite equal ratio sequence, and the common ratio is Q. (1) remove the first k items from the sequence, and the remaining items form a new sequence. Is the new sequence an equal ratio sequence? If so, what are the first item and the common ratio? (2) take out all the odd items in the sequence, and form a new sequence. Is the new sequence an equal ratio sequence? If so, what are the first item and the common ratio, What are the first term and the common ratio? (3) in the sequence {an}, take out one term every 10 terms to form a new sequence. Is this new sequence an equal ratio sequence? If so, what is its common ratio? Can you make a guess according to the conclusion?

(1) If the first term is A1, the first term is AK + 1 and the common ratio is Q
(2) If the first term is A1, the first term after removing is A2, and the common ratio is Q2 (the square of Q)
(3) It's not an equal ratio series. There's no way to have a common ratio. For example, this series:
The 1024 in 2 4 8 16 32 64 128 256 512 (1024) 2048 4096 8192 can not be used as an equal ratio sequence
Guess, I can't think of any personal experience
Make a topic to think hard, come on!

The formula of five equal difference and equal ratio Series in senior high school mathematics

I have also taken the mathematics of compulsory 5. Here are all the formulas of arithmetic and ratio: the sum of the first n terms of arithmetic sequence formula an = a1 + (n-1) d is: SN = Na1 + n (n-1) d / 2 Sn = (a1 + an) n / 2 if M + n = P + Q, then there is am + an = AP + AQ if M + n = 2p, then am + an = 2AP (1) arithmetic sequence

What is dislocation subtraction