Please use the dislocation subtraction method to sum! （1）Cn=（2n+1）*2^n （2）Cn=（2n-1）*(1/2)^n

Please use the dislocation subtraction method to sum! （1）Cn=（2n+1）*2^n （2）Cn=（2n-1）*(1/2)^n

Cn=(2n+1)*2^n
Sn=3*2+5*4+7*8+...+(2n+1)*2^n
2Sn= 3*4+5*8+7*16+...+(2n-1)*2^n+(2n+1)*2^(n+1)
Subtraction of two formulas
-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
The second question is the same. Try it yourself
Summation of equal ratio sequence
In the equal ratio sequence an, S3 = 3.5, S6 = 31.5, find an
s6=a1+a2+a3+a4+a5+a6=a1+a2+a3+q^3(a1+a2+a3)31.5=3.5+3.5q^33.5q^3=28q^3=8q=2s3=a1+a2+a3=a1+a1q+a1q^2=a1(1+q+q^2)=a1(1+2+4)=7a17a1=3.5a1=0.5=1/2an=a1q^n-1=1/2*2^(n-1)=2^(n-2)
N = 2. If q = 1, then s6-s3 = A4 + A5 + A6 = 3an, the solution of an and Q substituting into the check is not in line with the meaning of the problem. So we can use S6 / S3 = (1-q6 power) / (1-q3 power) = 1 + Q3 power = 9 to get q = 2, and then substitute S3 to get A1 = 1 / 2. I wish you progress.
Let the common ratio be Q
S3=A1+A2+A3=3.5
S6=A1+A2+A3+A4+A5+A6=A1+A2+A3+(A1+A2+A3)q^3=(A1+A2+A3)(1+q^3)=31.5
From the above two formulas, we get 1 + Q ^ 3 = 9, q = 2
S3=A1+A2+A3=A1(1+q+q^2)=7A1=3.5,A1=1/2
An=1/2*2^(n-1)=2^(n-2)
An example of dislocation subtraction,
Bnan = (2 ^ n) × (2n-1) 2 ^ n is the nth power of 2. Sn = 1 × 2 + 3 × (2 & sup2;) + 5 × (2 & sup3;) +(2^n)×（2n-1) ①2Sn=1×（2²)+3×（2³)+5×（2^4)………… +(2n-3)×（2^n)+(2^(n+1))×（2n-1) ②...
On summation of equal ratio sequence
The second power of 10 + 10 + the third power of 10 + the fourth power of 10 + the fifth power of 10
Can we use a simple formula? The more detailed the better!
Sn=a1(1-q^n)/(1-q) =(a1-an*q)/(1-q) (q≠1)
q=10,a1=10
sn=10(1-10^n)/(-9)
The second power of 10 + 10 + the third power of 10 + the fourth power of 10 + the fifth power of 10
=(10^n-1)(10-1)=(1/9)(10^n-1)
Sn=a1(1-q^n)/(1-q) =(a1-an*q)/(1-q) (q≠1)
SN is the sum of sum, q is the common ratio, here is 10, A1 is the first term, here is also 10, n is the number of terms
…… What about the latter?
10+10^2+10^3+…… +10^10
The first item is 10, and the common ratio is 10 to seek the first 11 items
10[1-10^(11-1)]/1-10
The first term is A1, and the common ratio is Q to find the first n terms
a1[1-q^(n-1)]/1-q
First term A1 = 10, common ratio: q = 10,
Sn＝a1（1-q^n）/(1-q)=10*（1-10^n）/(1-10)=[10^(n+1)-10]/9
You first express the general term formula, which is the nth power of 10
The sum of the first n terms of the equal ratio sequence Sn = A1 (1-Q ^ n) / (1-Q) or Sn = (a1-an * q) / (1-Q) (Q ≠ 1) Sn = n * A1 (q = 1)
Where the common ratio is q = 10, and the first term A1 = 10 is substituted into the existing one
Urgent, everybody help! The derivation process of the formula of the side area of the cylinder
Formula: bottom circumference × height
Reason: the expanded drawing of the side area of a cylinder is a rectangle, the area of the rectangle = length × width, where the length refers to the circumference of the bottom surface of the cylinder, and the width refers to the height of the cylinder, so we use the circumference of the bottom surface × height to calculate the side area of the cylinder
S = 2x PI RL expand the side to get a rectangle
2X is the circumference of the bottom circle, i.e. the length of the rectangle, and l is the length of the generatrix, i.e. the width of the rectangle
If the side of the cylinder is a rectangle, the length is the circumference of the bottom circle of the cylinder, and the width is the height of the cylinder, then s = C × h, C = 2 π R (R is the radius of the bottom circle), so s = 2 π RH
After expansion, it is a rectangle, the length is the circumference of the bottom, and the width is the height of the cylinder
2∏r*h
As shown in the figure, the center angle of the circle is 30 ° and the radii are 1, 3, 5, 7 The area of shadow part is S1, S2, S3 , then S50 = 0___ π was retained
From the meaning of the question, we can get the general term formula: SN = 8 π 12 × (2n-1), that is, Sn = 2 π 3 × (2n-1), ∧ S50 = 23 π (100-1) = 66 π, so the answer is: 66 π
Given that X and y are opposite to each other, M is the number whose absolute value is equal to 2, n is the smallest natural number, P is neither positive nor negative, find the value of M (x + y) + PN. Write the solution clearly
Simple! (x + y) = 0, m (x + y) = 0, n = 0, PN = 0, so the result is 0
Why can't the base be 0 in the division of the same base power
As long as it is in division, 0 can not be used as the base, because it is meaningless. Dividing by 0 is equivalent to asking how many times the numerator is 0, and the answer must be infinite
As shown in the figure, the semicircle areas on the three sides of a right angle triangle from small to large are recorded as S1, S2 and S3 in turn, then the relationship among S1, S2 and S3 is______ .
Let the diameters of the three semicircles be: D1, D2, D3, S1 = 12 × π × (D12) 2 = d128 π, S2 = 12 × π × (D22) 2 = d228 π, S3 = 12 × π × (D32) 2 = d328 π. From the Pythagorean theorem, it can be concluded that: D12 + D22 = D32, ｜ S1 + S2 = π 8 (D12 + D22) = d328 π = S3
Five! Must be better! Not in the textbook!
For a project, it takes 9 days for Party A to complete it alone, 12 days for Party B to complete it alone, and 15 days for Party C to complete it alone. If Party A and Party C do it three days first, Party A leaves for some reason, and Party B takes over Party A's work. How many days will it take to complete the project
1. A pool, there are a, B, C three water pipes, a, B is the inlet pipe, C is the drainage pipe, single open a pipe 16 minutes can fill the pool, single open B pipe 10 minutes can fill the water, single open C pipe 20 minutes can complete the whole pool of water, now open a, B two
Four minutes later, close tube a and open tube C. how many minutes will it take to fill the pool?
2. Team a, team B and team C have 10, 12 and 16 people respectively to complete a project. Team a will complete the project within 40 days, while team B will complete the project within 40 days
Team C does it alone in 30 days, team C does it alone in 24 days, now three teams... Start
1. A pool, there are a, B, C three water pipes, a, B is the inlet pipe, C is the drainage pipe, single open a pipe 16 minutes can fill the pool, single open B pipe 10 minutes can fill the water, single open C pipe 20 minutes can complete the whole pool of water, now open a, B two
Four minutes later, close tube a and open tube C. how many minutes will it take to fill the pool?
2. Team a, team B and team C have 10, 12 and 16 people respectively to complete a project. Team a will complete the project within 40 days, while team B will complete the project within 40 days
Team a will complete the project in 30 days, team C will complete the project in 24 days. Now three teams cooperate to complete the project. Team a has five participants, and team B has five participants
There are three members in team C. Although all the members in team C participate, they leave 18 days ahead of time. If the work efficiency of each member in each team is not high enough
It's the same. How many days did the project last?
3. A total of 50 tons of water was stored in pool a and B. after 5 tons of water was used in pool a and 8 tons of water was injected into pool B, the water in pool a was 3 tons less than that in pool B
How many tons of water are there in each pool?
4. A reservoir has two inlet pipes (A and b) and one drainage pipe (c). If pipe a is opened for 3 hours, the reservoir can be filled
Tube B can inject half pool water in 2 hours, and tube C can be opened separately. Tube a can discharge half pool water in 3 hours
B and C pipes are also opened. How many hours will it take to fill a pool of water?
5. For a project, Party A will do it alone for 20 days and Party B will do it alone for 15 days. Now Party A and Party B will cooperate for several days, and the rest will be completed
Part of the work is done by Party B alone, sharing 12 days. How many days does Party A do?
6. For a project, team a completed it in 10 hours, team B completed it in 15 hours, and team C completed it in 20 hours. Team a started to cooperate with each other. In the middle of the project, team a had another task, which was completed by team B and team C. It took 6 hours from the beginning to the completion of the project?
7. A worker originally planned to produce a batch of parts in 13 hours. Later, because he produced 10 more parts per hour, he not only completed the task in 12 hours, but also produced 60 more parts than the original plan. How many parts did he produce in the original plan? Put it away