The limit value of sequence If xn = 1 / 2n power, and how to judge the convergence or divergence of the sequence, please give me another example, thank you!

A sequence xn is a convergent sequence, which simply means that there is a real number a, xn will be more and more connected to A. for example, xn = 1 / N, obviously xn is closer and closer to 0, so xn is a convergent sequence at this time
Let me give you another example of divergent sequence, such as xn = (- 1) ^ n, because when n is odd, xn = - 1, when n is even, xn = 1, so we can't find such a real number a
For your example
Xn = (1 / 2) ^ n, which is a convergent sequence, because it keeps approaching 0

How does the third-order determinant in linear algebra transform from matrix form to general operation formula
In my textbook of linear algebra, there is only the derivation of the solution equation of the second-order determinant, but not the derivation of the third-order determinant. The third-order determinant directly uses the diagonal rule to get the formula, which is not easy to understand. The general formula of the third-order determinant is derived by solving the equation, and why the diagonal rule is used is explained

In fact, there is no need to deduce the formula, diagonal rule is actually very easy to understand
When a determinant is transformed into a triangular determinant by multiplication, substitution, and multiplication, the value of the determinant will not change. At this time, you use the definition of determinant to calculate the value of the determinant, which is obviously the product of the diagonal elements. Because if you use elements other than the diagonal, the value of the resulting term is 0
If you don't want to deduce it directly, you can expand it by row or column
Let the determinant be as follows
a b c
d e f
g h i
=a(ei-hf)-b(di-gf)+c(dh-eg)

Proof of linear algebra matrix
There is a third order real symmetric matrix A, a square = 0, using diagonalization method to prove that a = 0

A is a real symmetric matrix, and there exists an invertible matrix P, such that P ^ (- 1) AP = diag (λ 1, λ 2, λ 3) a = pdiag (λ 1, λ 2, λ 3) P ^ (- 1) a ^ 2 = [pdiag (λ 1, λ 2, λ 3) P ^ (- 1)] = pdiag (λ 1 ^ 2, λ 2, λ 3 ^ 2) P ^ (- 1) = 0  λ 1 ^ 2 = λ 2 ^ 2 = λ 3 ^ 2

If a > 0, b > 0, and a is not equal to B, please explore the size relationship between 1 / A and 1 / B, and verify your conclusion with specific rational numbers

When AB is less than 1,
A is less than B, 1 / A is greater than 1 / b (a = 1 / 3, B = 1 / 2, 1 / a = 3, 1 / b = 2,)
2、 A is greater than B, 1 / A is less than 1 / b (a = 1 / 2, B = 1 / 3, 1 / a = 2, 1 / b = 3,)
When AB is greater than 1,
A is less than B, 1 / A is greater than 1 / b (a = 2, B = 3, 1 / a = 1 / 2, 1 / b = 1 / 3,)
Two a is greater than B, 1 / A is less than 1 / b (a = 3, B = 2, 1 / a = 1 / 3, 1 / b = 1 / 2,)
When a is greater than 1 and B is less than 1,
1 / A is greater than 1 / b (a = 1 / 2, B = 3, 1 / a = 2, 1 / b = 1 / 3,)
If a is less than B, then 1 / A is greater than 1 / B, and vice versa

Mother Lu raised 100 chickens and ducks. Later, she sold 1 / 5 of the chickens and bought back 8 ducks. At this time, the chickens and ducks were equal. How many chickens and ducks were there?

Suppose there were x chickens, then there were (100-x) ducks
X*(1-1/5)=(100-X)+8
4X/5=108-X
4X=540-5X
9X=540
X=60
Then there are 100-60 ducks = 40
There were 60 chickens and 40 ducks

Find the value range of a whose value range of function y = (X & # 178; + AX-2) / (X & # 178; - x = 1) is (- ∞, 2)
There are answers in teaching aids, but I don't understand one step, or I can work it out with other methods
Let (X & # 178; + AX-2) / (X & # 178; - x = 1) < 2
∵ X & # 178; - x = 1, the formula must be greater than 0
∴x²+ax-2＜2(x²-x=1)
This inequality holds for X ∈ R
The square of ⊿ = [- (a + 2)] - 4 × 1 × 4 ﹤ 0 [?
The solution is - 6 < a < 2
The value range of a is {a | - 6 ＜ a ＜ 2}

This is where you draw a picture
In the coordinate system, to make a quadratic function (parabola) constant greater than 0
It's this parabola that must always be above the x-axis. Right?
That is to say, the equation corresponding to the quadratic function cannot have roots, otherwise, if there are roots
So if the parabola goes through the x-axis, it can't always be above the x-axis
So if the equation has no solution and the coefficient of quadratic term is greater than 0, it must be greater than 0

The first equation is x-7y + 3x = 17
5X-6Y-Z=24
3X+7Y-2X=1
The second one: 3 / x equals 4 / y = 5 / Z
X + y + Z = 24
A car runs back and forth between ab. with a distance of 70kN. Because there is a hill with even slope in the journey, it takes 2 hours and 30 minutes for the car to go from a to B, and 2 hours and 18 minutes for the car to return from B to A. suppose that the car travels 30KN per hour on the plane and 20KN per hour uphill. The downhill speed is 40kN per hour. Uphill. How many kilometers are the downhill roads

Let X / 3 = Y / 4 = Z / 5 = k, then x = 3k, y = 4K, z = 5K, so x + y + Z = 12K = 24, so k = 2, so x = 6, y = 8, z = 10, so from a to B level

How can the general term formula of sequence 1.3.6.10.15.21 be (n ˇ 2 + n) / 2?

1=1
3=1+2
6=1+2+3
10=1+2+3+4
``````
The nth term is 1 + 2 + 3 + 4 + '+ n
So it's like a proposition

How to convert mesh and micron? How many microns should 200 mesh be equal to~
5000 mesh, 4000 mesh, 3500 mesh, 2200 mesh, 1250 mesh, 850 mesh ~ ~ how many microns are there respectively ~ ~ I'm in urgent need of your help now ~ ~ ` thank you~``

Micrometer is a unit of length. Mesh is the number of holes per square centimeter

The equation of first degree with one variable,
First question:
1）
The students of a middle school renovate the playground by themselves. It takes 7.5 hours to finish if the students of grade one are allowed to work alone. It takes 5 hours to finish if the students of grade two are allowed to finish the work alone. If the students of grade one and grade two are allowed to finish the work together for 1 hour, and then the students of grade two are allowed to finish the rest separately, how much time does it take?
Second question:
2）
It takes 80 hours for one person to do a batch of data collation. Now it is planned that one person will do it for 2 hours first, and then 5 people will do it for 8 hours to finish 3 / 4 of this work. How to arrange the specific number of people involved in data collation?

The workload of renovating the playground is 1. It will take x hours for junior two students to finish the rest
1/7.5+1/5+1/5*X=1
The solution is x = 10 / 3
1 + 10 / 3 = 13 / 3 hours
A: it takes 13 / 3 hours
Let's plan to have X people to do the work first
According to the meaning of the question: 80 = 2x + (x + 5) * 8
X = 4
So there are four people to do the work first

The limit value of sequence If xn = 1 / 2n power, and how to judge the convergence or divergence of the sequence, please give me another example, thank you!