given that the following two are geometric series are convergent: 1+x+x^2+x^3+...and 1-x+x^2-x^3+... determine the value(s) of x for which the sum of the two series is equal to 8​

Answers

Answer 1

Let S and T denote the two finite sums,

S = 1 + x + x ² + x ³ + … + x

T = 1 - x + x ² - x ³ + … + (-x) ᴺ

• If both S = 8 and T = 8 as N goes to infinity:

Then

xS = x + x ² + x ³ + x ⁴ + … + x ᴺ⁺¹

-xT = -x + x ² - x ³ + x ⁴ + … + (-x) ᴺ⁺¹

so that

S - xS = 1 - x ᴺ⁺¹   ==>   S = (1 - x ᴺ⁺¹)/(1 - x)

and similarly,

T = (1 - (-x) ᴺ⁺¹)/(1 + x)

For both sums, so long as |x| < 1, we have

lim [N → ∞] S = 1/(1 - x)

lim [N → ∞] T = 1/(1 + x)

Then if both sums converge to 8, this happens for

S : 1/(1 - x) = 8   ==>   x = 7/8

T : 1/(1 + x) = 8   ==>   x = -7/8

• If the sum S + T = 8 as N goes to infinity:

From the previous results, we have

1/(1 - x) + 1/(1 + x) = 8   ==>   x = ±√3/2


Related Questions

which exponential function has an initial value of 2

Answers

Answer:

Exponential growth formula. a represents the initial value.

Step-by-step explanation:

The value of "b" will determine the growth or decay behavior of the exponential function.

An exponential function can be represented in the form:

f(x) = a * b^x

where "a" is the initial value or the value of the function when x = 0, and "b" is the base of the exponential function.

If we want an exponential function with an initial value of 2, we can set "a" equal to 2:

f(x) = 2 * b^x

The value of "b" will determine the growth or decay behavior of the exponential function. Without further information or constraints on the value of "b," we cannot determine the specific exponential function. Additional information, such as the growth/decay factor or a specific point on the function, is needed to determine the value of "b" and fully define the exponential function.

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Which equation can be used to find 60 percent of 50

Answers

Answer:

x = 0.6 * 50

Step-by-step explanation:

x = 60% of 50

x = 60% * 50

x = 0.6 * 50

Answer:

60% of 50 = 60 / 100 × 50 = ⅗ × 50 = 150 / 5 = 30

________

x% of y = x / 100 × y = xy / 100

Taree fourts of the 64 books were math books. determine the percent of the books that were math books.​

Answers

Answer:  75% - these are books on mathematics.

Step-by-step explanation:

[tex]\dfrac{3}{4} \cdot 64 = 48\\[/tex]

64 - 100%

48  -  x%

[tex]\dfrac{64}{100} =\dfrac{48}{x}\\\\x=\dfrac{48 \cdot 100}{64} \\\\x=75 \%[/tex]

The length of a rectangle is 4 meters and the width is 4 meters. What is the perimeter of the rectangle? Do not include units in your answer.

Answers

16

The perimeter of a shape is the distance around its edges.

The formula for the perimeter of a rectangle is:
2 (l + w)
Or just basically add up all measurements of length and width IF given (the shape we have is a rectangle so the other 2 measurements must be 4 meters also)

Use the formula:
2 (4 + 4) = 16

No units are included, so 16 is your answer.

Hope this helps!

A tank contains 200 L of salt solution which contains 100 grams of salt. Pure water enters the tank ata rate of 4 L/min, but the thoroughly mixed solution leaves the tank at a rate of 2 L/min.Write andsolve an IVP to determiney, the number of grams of salt in the tank at timet.

Answers

Answer:

a. dy/dt = - 2y/(200 + 2t)  where y(0) = 100 g

b. y = 20000/(200 + t)

Step-by-step explanation:

a. Write an IVP to determine y, the number of grams of salt in the tank at time, t.

Let y be the mass of salt.

The net flow into the tank dy/dt = mass flow in - mass flow out

Since only water flows into the tank, the mass flow in = 0 g/min

Let m be the mass of salt in the tank at time t. Since the volume of the tank is 200 L and water flows in at a rate of 4 L/min and out at a rate of 2 L/min, the net rate of increase of the volume of the tank is rate in - rate out = 4 L/min - 2 L/min = 2L/min. So, in time, t, the volume of the water in the tank increases by  2t. So, the volume of the tank in time, t is V = 200 + 2t.

So, the concentration of salt in the tank at time t is mass/volume = m/(200 + 2t).

Since the well mixed solution leaves at a rate of 2 L/min, the mass flow out is concentration × volume flow out =  y/(200 + 2t) × 2 = 2y/(200 + 2t)

The net flow into the tank dy/dt = mass flow in - mass flow out

dy/dt = 0 - 2y/(200 + 2t)

dy/dt = - 2y/(200 + 2t)

Since the initial mass of salt in the tank is 100 g, y(0) = 100 g

So, the initial value problem IVP is

dy/dt = - 2y/(200 + 2t)  where y(0) = 100 g

b. Solve an IVP to determine, the number of grams of salt in the tank at time, t.

Solving the IVP, we have

dy/dt = - 2y/(200 + 2t)  where y(0) = 100 g

Separating the variables, we have

dy/y = - 2dt/(200 + 2t)

Integrating both sides, we have

∫dy/y = - ∫2dt/(200 + 2t)

㏑y = - ㏑(200 + t) + ㏑C

㏑y + ㏑(200 + t) = ㏑C

㏑[y(200 + t)] = ㏑C

y(200 + t)] = C

y = C/(200 + t) since y(0) = 100, we have

100 = C/(200 + 0)

100 = C/200

C = 100 × 200

C = 20000

So, y = C/(200 + t)

y = 20000/(200 + t)

NEED HELP ASAP

Find the area of the irregular figure.
12 in.
6 in.
1
A = [? ]in.2
4 in.
13 in
4 in
5 in.

Answers

Answer:

Step-by-step explanation:

Which inequality is shown in the graph?

Answers

Answer:

A.

Step-by-step explanation:

the equation of the parabola shown in the graph is y=x²-5, then the inequality is y≥x²-5 (inside the parabola).

Santos flipped a coin 300 times. The coin landed heads up 125 times. Find the ratio of heads to total number of coin flips. Express a simplified ratio

Answers

Answer:

5:12

Step-by-step explanation:

125:300 simplified = 5:12

I hope this helps

10% of 360 is how much more than 5% of 360

Answers

Answer: 18


Explanation:

Firstly, you have to divide 360 by 10%, which is 36. An easy method to do this part is to remove the 0 at the back of 360. Then, calculate 5% of 360. Take the 10% of 360 (36) and divide it by two. You will get the answer 18. Finally, substructure 18 from 36, and you will get your answer, 18.


Hope this helps!

10% of 360 is 18 more than 5% of 360.

What is the percentage?

The percentage is defined as ratio expressed as a fraction of 100.

What are Arithmetic operations?

Arithmetic operations can also be specified by the subtract, divide, and multiply built-in functions.

Given data as :

10% of 360

5% of 360

Firstly, we have to determine 10% of 360,

⇒ 10% of 360

⇒ (10/100)360

⇒ (0.10)360

So, 10% of 360 is 36.

⇒ 5% of 360

⇒ (5/100)360

⇒ (0.05)360

So, 5% of 360 is 18.

Since 10% of 360 is more than 5% of 360

So, substract 18 from 36, and

⇒ 36 - 18

18

Hence, 10% of 360 is 18 more than 5% of 360.

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Exaluate: (4/9)^1/2.

Answers

Answer:

2/3

Step-by-step explanation:

We know that (a/b) ^ (1/2) is a^ (1/2)  / b^ 1/2

( 4/9) ^ 1/2

4^1/2

-----------

9 ^ 1/2

2

----

3

What happens to the mean of the data set shown below if the number 20 is added to the data set?




A. The mean increases by 0.75.
B. The mean does not change.
C. The mean increases by 12.
D. The mean increases by 0.25.

Answers

Answer:

This shows that the mean increases by 2.95

Step-by-step explanation:

Assume the given data is 2, 5, 6, 8

Mean = 2 +5+6+8/4

Mean = 21/4

Mean = 5.25

If number 20 is added, the data becomes 2, 5, 6, 8, 20

New mean = 2 +5+6+8+20/4

New mean = 41/5

New mean = 8.2

Taking the difference in the mean:

Difference = 8.2 - 5.25

Difference = 2.95

This shows that the mean increases by 2.95

NB: The data used was assumed since we are not given any data in question

Answer:

The answer is A

Step-by-step explanation:

what is the average speed for the interval t=1 hour to t=3 hours

Answers

Step-by-step explanation:

2 hours

3+1 / 2 = 4/2 = 2 hours speed average

Help please!!!!!A student needs to select 3 books from 3 different mathematics, 3 different physics and 1 history book. what is the probability that one of them is mathematics and the other 2 are either physics or history books ? A. 3/15 B.9/25 C. 15/35 D. 18/35​

Answers

Answer: Choice D) 18/35

===========================================

Explanation:

There are 3 ways to select the single math book and 4*3/2 = 12/2 = 6 ways to pick the two other books that are either physics or history (order doesn't matter). This is effectively because we have 3+1 = 4 books that are either physics or history, and we're using the nCr combination formula.

Overall, there are 3*6 = 18 ways to select the three books such that one is math, and the other two are either physics or history.

-------------------

There are 3+3+1 = 7 books total. Since we're selecting 3 of them, we use the nCr formula again and you should get 35.

Or you could note how (7*6*5)/(3*2*1) = 210/6 = 35

This says there are 35 ways to select any three books where we can tell the difference between any subject (ie we can tell the difference between the math books for instance).

-------------------

We found there are 18 ways to get what we want out of 35 ways to do the three selections. Therefore, the answer as a fraction is 18/35

Use the diagram to determine the height of the tree.

Answers

where's the diagram?

What is the answer to this? Is it d?

Answers

Answer:

Step-by-step explanation:

yeah of course..your answer is true.. it's part d and there is no solution for that system...w and v both of them remove in solution of system and we don't have any unknown variable for solving.

9514 1404 393

Answer:

  D. no solutions

Step-by-step explanation:

The first equation can be simplified to standard form:

  0.5(8w +2v) = 3

  4w +v = 3 . . . . . . . eliminate parentheses

__

The second equation can also be simplified to a comparable standard form:

  8w = 2 -v +4w

  4w +v = 2 . . . . . . add v -4w

__

Comparing these two equations, we find the variable expressions to be the same, but the constants to be different. If any set of variable values were to satisfy one of these equations, it could not satisfy the other equation. There are no solutions to the system.

The tiles below represent the polynomial 2x^2 + 7x+6.

What is the factorization of 2x^2 + 7x + 6?

A. (x+3)(x + 2)
B. (2x + 2)(x+3)
C. (2x+3)(x + 2)
D. (x+3)(x+6)

Answers

Answer:

C (x+2)(2x+3)

Step-by-step explanation:

You can count in the picture that in the top part there are two x's (2x) and three ones (3). So the top would be (2x+3)

On the left side there is one x (x) and two ones (2). It would be (x+2).

So combine them together, it would be (x+2)(2x+3).

C.

Answer:

answer is (2x+3)(x+2)

I hope this will help you

I need help with this​

Answers

Answer:

sorry but I don't know the answer

Answer:

B. 0.30

Step-by-step explanation:

30/100 = 0.30

he ride a bike for 15 miles oer hour how many miles did he ride

Answers

Yes it’s 15, Thats cool man

The number of typing errors made by a typist has a Poisson distribution with an average of three errors per page. If more than three errors appear on a given page, the typist must retype the whole page. What is the probability that a randomly selected page does not need to be retyped

Answers

Answer:

0.6472 = 64.72% probability that a randomly selected page does not need to be retyped.

Step-by-step explanation:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

In which

x is the number of sucesses

e = 2.71828 is the Euler number

[tex]\mu[/tex] is the mean in the given interval.

Poisson distribution with an average of three errors per page

This means that [tex]\mu = 3[/tex]

What is the probability that a randomly selected page does not need to be retyped?

Probability of at most 3 errors, so:

[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)[/tex]

In which

[tex]P(X = x) = \frac{e^{-\mu}*\mu^{x}}{(x)!}[/tex]

[tex]P(X = 0) = \frac{e^{-3}*3^{0}}{(0)!} = 0.0498[/tex]

[tex]P(X = 1) = \frac{e^{-3}*3^{1}}{(1)!} = 0.1494[/tex]

[tex]P(X = 2) = \frac{e^{-3}*3^{2}}{(2)!} = 0.2240[/tex]

[tex]P(X = 3) = \frac{e^{-3}*3^{3}}{(3)!} = 0.2240[/tex]

Then

[tex]P(X \leq 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 0.0498 + 0.1494 + 0.2240 + 0.2240 = 0.6472[/tex]

0.6472 = 64.72% probability that a randomly selected page does not need to be retyped.

Fond the square root of 169 by division method​

Answers

Answer:

13

Step-by-step explanation:

square root of 169 is 13

The answer it’s 13 because that’s the square root of 169

Martinez' General Store is having a 45% off sale on men's clothing. John paid $70.95 for a jacket that was on sale. What was the original price of the jacket?

Answers

Answer:

$129

Step-by-step explanation:

We can use this equation:

x - 0.45x = 70.95

0.55x = 70.95

x = $129

9 bushes in 81 minutes , how much time taken per bush

Answers

Answer:

9

Step-by-step explanation:

9 bushes in 81 minutes.

Take 81, divide by 9, 81÷9 equals 9. 9 minutes are taken per bush.

The area of rectangle is 105cm².If it length is 21 cm,what is its length and perimeter.​

Answers

Answer:

length of other side is 5cm and the perimeter is 52

Step-by-step explanation:

The area is side x times side y.

Knowing that the area A is 105cm2 and side x is 21cm

A= x*y

105cm2= 21 y /21

Arranging for y we get

y= 105/21

y= 5 cm

The perimeter is all sides added up

P= 21+21+5+5=52cm

Answer:

length 5cm and perimeter 52 cm

Step-by-step explanation:

length=area/breadth

=105/21

=5 cm

perimeter=2(length+breadth)

=2(5+21)

=52 cm

So for this problem, I almost got it however my rounding is off causing my answers to be wrong. Can someone please help me with the two that are wrong. Thank you for your help!

Answers

Answer:

it 94x26.2 i think it right if not sorry :/

Step-by-step explanation:

Help me please! Parabola question

Answers

The parabola opens upward with the y- intercept being at y=7 (0,7) and the x- intercepts on x= -2,-4 / (-2,0), (-4,0). The axis of symmetry is on x=-3, and the vertex is on (-3,-1).

5. Sarah and Aarnav had a 3-layer wedding cake made for their wedding. Each layer was 6 inches in height. The lower layer had a diameter of 12 inches, while the middle layer had a radius of 5 inches. The top layer had a diameter of 8 inches.


a) Calculate the total volume of cake to the nearest cubic inch. (3 marks)











b) The baker who made the cake needed to calculate the total surface area of the outside of the cake to the nearest square inch to find out how much icing to buy. The baker will not be icing the bottom of any of the layers of cake. However, the baker will ice the top and sides of each layer separately and then stack the three layers of cake. Calculate the surface area of the cake that the baker plans to ice to the nearest square inch. (3 marks)












c) The baker bought the buttercream icing from Crave Cupcakes. Using your surface area calculation from part b, calculate how many containers of icing the baker needs to buy in order to ice Sarah and Aarnav’s cake. Each container of icing contains 16 ounces of icing. If it takes 56 ounces of icing to cover 678 square inches of cake, calculate the total cost for the icing before tax if each 16-ounce (2 cup) container of icing costs $10. (3 marks)

Answers

Answer:

Total volume of cake = 1452 cubic inches

Step-by-step explanation:

Each layer of the cake has a cylindrical shape, so that:

volume of a cylinder = [tex]\pi[/tex][tex]r^{2}[/tex]h

Where r is the radius and h represents the height.

i. For the lower layer;

h = 6 inches and r = 6 inches

volume = [tex]\pi[/tex][tex]r^{2}[/tex]h

            = [tex]\frac{22}{7}[/tex] x [tex](6)^{2}[/tex] x 6

            = 678.86 cubic inches

ii. For the middle layer;

h = 6 inches and r = 5 inches

volume = [tex]\pi[/tex][tex]r^{2}[/tex]h

             = [tex]\frac{22}{7}[/tex]x [tex](5)^{2}[/tex] x 6

            = 471.43 cubic inches

iii. For the top layer;

h = 6 inches and r = 4 inches

volume = [tex]\pi[/tex][tex]r^{2}[/tex]h

             = [tex]\frac{22}{7}[/tex] x [tex](4)^{2}[/tex]x 6

             = 301.71 cubic inches

Total volume of cake = 678.86 + 471.43 + 301.71

                = 1452 cubic inches

can someone help me with this question

Answers

9514 1404 393

Answer:

local minima: at x=-1, x=3local minimum values: -2 and -1 (respectively)

Step-by-step explanation:

A local minimum is where the curve stops going down and starts going up. It is the bottom of any U-shaped spot. Here, those are identified with dots at the coordinates (-1, -2) and (3, -1).

(a) the x-values at which f has a local minimum are -1 and 3.

(b) the local minimum values of f are -2 and -1 at those x-values.

graph a circle with General form.x^2 +y^2+8x-12y+24=0

Answers

Answer:

jhshejwjabsgsgshshsnsjs

Answer:

Step-by-step explanation:

Put the equation into center-radius form.

x² + y² + 8x - 12y + 24 = 0

x² + y² + 8x - 12y = -24

(x²+8x) + (y²-12y) = -24

(x²+8x+4²) + (y²-12y+6²) = 4²+6²-24

(x+4)² + (y-6)² = 28

Center: (-4,6)

radius: √28

Suppose taxi fares from Logan Airport to downtown Boston is known to be normally distributed and a sample of seven taxi fares produces a mean fare of $16.51 and a 95% confidence interval of [$15.96, $17.04]. Which of the following statements is a valid interpretation of the confidence interval??

a. 95% of all taxi fares are between $14.77 and $16.23.
b. We are 95% confident that a randomly selected taxi fare will be between $14.77 and $16.23.
c. The mean amount of a taxi fare is $15.51, 95% of the time.
d. With 95% confidence, we can report that the average taxi fare between Logan Airport and downtown Boston will fall between $14.77 and $16.23.

Answers

Answer:

The answer is "Option d".

Step-by-step explanation:

Please find the correct and complete question in the attachment file.

The taxis were known to also be uniformly distributed through the Logan airport to the city of Boston, as well as a sample of seven taxi rates gives an average of $15.51 for [$14.77 and $16.23] for a 95% trust interval. Having 95 percent confidence, we could give a valid interpretation of the confidence level for the averages taxi rates among both Logan Airport and Boston Centre.

Counting numbers are to be formed using only the digits 1, 3, 7,5,4,8, and 2. Determine the number of different possibilities for two-digit numbers.​

Answers

Answer:

11699 numbers and 42 two-digit numbers

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