Show that each equation is true. ( 1 - 3 )

Find the exact value of x. ( 4 - 6 )

a. The general formula for the sum of the first n terms is Sₙ = 2(1 - e⁻ⁿ)/(1 - e⁻¹)

b. The sum of a series is defined as the limit of the sequence of partial sums is converges.

c. The true statement are "The series converges. and The series is a geometric series." (option B and C).

a. To find the general formula for the sum of the first n terms, we need to add the first n terms of the series. Thus, we have:

Sₙ = 2e⁰ + 2e⁻¹ + 2e⁻² + ... + 2e⁻ⁿ

We can see that this is a geometric series with a first term a=2 and a common ratio r=e⁻¹. Therefore, we can use the formula for the sum of a geometric series to find the general formula for Sₙ:

Sₙ = a(1 - rⁿ)/(1 - r)

Substituting a=2 and r=e⁻¹, we get:

Sₙ = 2(1 - e⁻ⁿ)/(1 - e⁻¹)

the limit of the sequence of partial sums is zero, the series converges.

b. To determine whether the series converges or diverges, we can take the limit of the sequence of partial sums as n approaches infinity. Thus, we have:

∑n=0∞2e−n=limn→∞ S_n= limn→∞ 2(1 - e⁻ⁿ)/(1 - e⁻¹)

To evaluate this limit, we can use L'Hopital's rule, which states that if we have an indeterminate form of the type 0/0 or infinity/infinity, we can take the derivative of the numerator and the denominator until the form is no longer indeterminate. Applying L'Hopital's rule, we get:

lim n→∞ 2(1 - e⁻ⁿ)/(1 - e⁻¹) = limn→∞ 2e⁻ⁿ/(e⁻¹) = 0

c. We can now identify whether the series is a telescoping series, a geometric series, or a p-series.

The series is a geometric series, as we already saw in part (a).

Therefore, the correct answers are (B) and (C).

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The volume of each milkshake cup is given by

small cup = 50.24 square inches

medium cup =127.56 square inches

Large cup =226.08 square inches

Shape of the milkshake cups is cylindrical.

Formula used to calculate volume of each cup = πr²h

Where 'r' is the radius of cups.

And h is the height of the cups

Radius of small milkshake cup = 2in.

height of small cup = 4in.

Volume of the small cup = π × 2² × 4

                                         = 50.24 square inches

Radius of medium milkshake cup = 2.5in.

height of medium cup = 6.5in.

Volume of the medium cup = π × 2.5² × 6.5

                                         = 127.56 square inches

Diameter of large cup = 6in

Radius of large milkshake cup = 3in.

height of large cup = 8in.

Volume of the large cup = π × 3² × 8

                                         = 226.08 square inches

Therefore, the volume of each milkshake cup using given radius and height are as follow,

volume of small cup = 50.24 square inches

volume of medium cup =127.56 square inches

volume of Large cup =226.08 square inches

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The simplified expression is (z -3)/z - 5. Option C

From the information given, we have to simply the expressions by solving the quadratic equation;

From the information given, we have that;

z² -5zx + 6

Find the pair factors and substitute the values

z² - 2z - 3z + 6

Group in pairs, we have;

z(z - 2) - 3(z - 2)

For the denominator, we have;

z² - 7x + 10

Find the factors

z² - 5z - 2z + 10

Group in pairs, we have;

z(z - 5) - 2(z - 5)

Then, we have the simplified form;

(z -3)/z - 5

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If the small "coffee-shop" owner is trying to determine the preference of her customers by conducting a survey among a group of customers who visit the shop during a specific time frame, then this group would be a sample rather than a population.

A "Sample" is defined as a subset of the population that is selected to represent the entire population. In this case, the owner is trying to determine the preference of her customers by surveying a specific group of customers.

So, this group of customers is a sample of the population of all customers who visit the coffee shop.

To obtain meaningful results from the survey, the sample should be selected randomly and should be representative of the population. The results obtained from the sample could then be used to make inferences about the preferences of the entire population of customers who visit the coffee shop.

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The given question is incomplete, the complete question is

A small coffee shop owner wants to know which brand of coffee her customers prefer.

Is the data considered a "population" or "sample", Explain?

The standard form of each of the numbers is given as follows:

a) 2.35 x 10^-3 = 0.00235.

b) 2.35 x 10³ = 2350.

A number in scientific notation is given by the notation presented as follows:

[tex]a \times 10^b[/tex]

With the base being [tex]a \in [1, 10)[/tex], meaning that it can assume values from 1 to 10, with an open interval at 10 meaning that for 10 the number is written as 10 = 1 x 10¹, meaning that the base is 1, justifying the open interval at 10.

For item a, to obtain the standard form of the number, we must move the decimal points three places to the left, hence:

2.35 x 10^-3 = 0.00235.

For item b, to obtain the standard form of the number, we must move the decimal points three places to the right, hence:

2.35 x 10³ = 2350.

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Answer:

Step-by-step explanation:

Sector area is a portion of the circle's area.

formula is π r² × x/360°

where x is the central angle.

For example if the radius is 9 and the central angle is 120°.

π · 9² · 120/360

3.14 · 81 · 1/3 = 84.78 units²

or in terms of pi = [tex]\frac{1}{3}[/tex] (81π)

In the equation , it take 2 hours for the experienced crew to paint the apartment.

What is equation?

The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.

Here let us take x as time taken by new crew of painters apartment and y as time taken by experienced crew of painters apartment .

Now a new crew of painters takes two times as long to paint a small apartment as the experienced crew  then,

x = 2y ----> 1

both crews can paint the apartment in 6 hours then,

=> x+y = 6

Now put  equation 1  into x+y=6 then,

=> 2y+y=6

=> 3y = 6

=> y = 6/3 = 2 hours.

Hence it take 2 hours for the experienced crew to paint the apartment.

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The quotient is [tex]4x^2[/tex] and the remainder is [tex]3x^2 - x + 5[/tex].

Long division is a mathematical method used to divide one number by another. It is typically used for division of larger numbers, and is taught as a basic arithmetic skill in elementary and middle school.

In long division, the dividend (the number being divided) is written on the top, and the divisor (the number dividing the dividend) is written on the left. The quotient (the answer to the division problem) is written on the top of the line below the dividend, with the remainder (the left-over amount after division) written on the right.

The process involves a series of steps, including dividing, multiplying, and subtracting. The steps are repeated until the dividend is fully divided, or until a specified level of precision is reached.

Long division is also used to divide polynomials, by using the coefficients of the terms in the polynomial instead of the actual numbers. This process is similar to long division of numbers, but involves additional steps to ensure that the division is performed correctly.

Now We want to divide the polynomial [tex]4x^4 - 5x^3 + 2x^2 - x + 5[/tex] by the polynomial [tex]x^2 + x + 1[/tex] using long division.

[tex]x^2 + x + 1 |[/tex] ([tex]4x^4 - 5x^3 + 2x^2 - x + 5[/tex] )

First, we divide the highest degree term of the dividend, [tex]4x^4[/tex], by the highest degree term of the divisor, [tex]x^2[/tex], to get [tex]4x^2[/tex] :

[tex]x^2 + x + 1 | 4x^4 - 5x^3 + 2x^2 - x + 5\\-4x^4 - 4x^3 - 4x^2\\-----------------------------\\x^3 + 2x^2[/tex]

Next, we multiply the divisor,[tex]x^2 + x + 1[/tex], by the quotient, [tex]4x^2[/tex], to get [tex]4x^4 + 4x^3 + 4x^2[/tex], and subtract it from the dividend:

[tex]x^2 + x + 1 | 4x^4 - 5x^3 + 2x^2 - x + 5\\-4x^4 - 4x^3 - 4x^2\\----------\\x^3 + 2x^2\\- 4x^2 - 4x - 4\\-------\\3x^2 - x + 5[/tex]

We now have a new polynomial, 3[tex]x^2 - x + 5[/tex], as the remainder. Since the degree of the remainder is less than the degree of the divisor, the division process is complete.

Therefore, the quotient is [tex]4x^2[/tex] and the remainder is [tex]3x^2 - x + 5[/tex]. We can write the original polynomial as:

[tex]4x^4 - 5x^3 + 2x^2 - x + 5 = (x^2 + x + 1)(4x^2) + (3x^2 - x + 5)[/tex]

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Step-by-step explanation:

we can substitute book B as B

we can substitute book A as A

using the information above we can create an eqaution

we know B is $15.00 more than 3 times the cost of A

B = 3A + 15.00

we know the sum of the books is $75.00

A + B = 75.00

substitute B with eqaution 1 we will get

A + 3A + 15.00 = 75.00

4A + 15.00 = 75.00

- 15.00 -15.00

4A= 60.00

A = 15.00

which means book A is $15.00

The two possible solutions of the given equation are  a = √21.6 and a = -√21.6.

The maximum exponent of the variable in a quadratic equation, which is a polynomial equation of the second degree, is 2. The equation has two unique real solutions if the discriminant is positive. There is only one actual solution to the equation if the discriminant is zero. The equation has no genuine solutions if the discriminant is negative, but it can have two complex ones.

The given equation is 27 = a (a/0.8).

Multiply both sides of the equation by 0.8 thus we have:

21.6 = a²

Now, taking the square root on both sides we have:

a = √21.6 and a = -√21.6.

Hence, the two possible solutions of the given equation are  a = √21.6 and a = -√21.6.

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The two possible solutions of the given equation are  a = √21.6 and a = -√21.6.

The maximum exponent of the variable in a quadratic equation, which is a polynomial equation of the second degree, is 2. The equation has two unique real solutions if the discriminant is positive. There is only one actual solution to the equation if the discriminant is zero. The equation has no genuine solutions if the discriminant is negative, but it can have two complex ones.

The given equation is 27 = a (a/0.8).

Multiply both sides of the equation by 0.8 thus we have:

21.6 = a²

Now, taking the square root on both sides we have:

a = √21.6 and a = -√21.6.

Hence, the two possible solutions of the given equation are  a = √21.6 and a = -√21.6.

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Answer:

see below

Step-by-step explanation:

A box plot—displays the five-number summary of a set of data.

The five-number summary is the

minimum, first quartile, median, third quartile, and maximum.

In a box plot, we draw a box from the first quartile to the third quartile.

A vertical line goes through the box at the median.

So the median of this set of data = 7

only C & D show a median of 7.

Then, Just like the median divides the data into half so that 50% of the measurement lies below the median and 50% lies above it, the quartile breaks down the data into quarters so that 25% of the measurements are less than the lower quartile, 50% are less than the median, and 75% are less than the upper quartile.

D is correct

Answer:

(2,5] or (7,∞)

Step-by-step explanation:

The number line where we plotted -1 1/6 and 17/6 is added as an attachment

From the question, we have the following parameters that can be used in our computation:

-1 1/6 and 17/6

To start with, we convert both numbers to the same form

i.e. decimal or fraction

When converted to fractions, we have

-7/6 and 17/6

This means that we can plot -7/6 at -7 and 17/6 at point 17 where the difference in each interval is 1/6

Using the above as a guide, we have the following:

The number line is attached

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we estimate the population mean m to be 75.

What is confidence interval?

A confidence interval (CI) for an unknown parameter in frequentist statistics is a range of estimations. The most popular confidence level is 95%, but other levels, such 90% or 99%, are occasionally used for computing confidence intervals. The fraction of related CIs over the long run that actually contain the parameter's true value is what is meant by the confidence level. The degree of confidence, sample size, and sample variability are all factors that might affect the width of the CI. A larger sample would result in a narrower confidence interval if all other factors remained constant. A wider confidence interval would also be required by a higher confidence level and would be produced by a sample with more variability.

Since we want a 95% confidence interval, α = 0.05/2 = 0.025 and we need to find the critical value from the t-distribution with (36-1) = 35 degrees of freedom. Using a t-table or calculator, we find that t0.025,35 = 2.032.

Now, plugging in the values we have:

CI = 75 ± (2.032 * (24/√36))

CI = 75 ± (2.032 * 4)

CI = 75 ± 8.128

So the 95% confidence interval for the population mean m is (66.872, 83.128). This means we are 95% confident that the true population mean falls within this interval.

As for the estimated population mean, we can simply take the sample mean, which is X = 75.

Therefore, we estimate the population mean m to be 75.

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The value of the car went down by an average of $1,725 each year over the six-year period.

Depreciation is the loss in value of an asset due to factors such as age, wear and tear, market conditions, and obsolescence. It is usually calculated on a linear basis, meaning that the same amount of depreciation is spread evenly over a period of time.

In this case, the car was purchased for $22,995, and six years later it was worth $12,645.

To calculate the linear depreciation of the car, we need to subtract the value at the end of the period from the value at the beginning, and divide by the number of years:

Depreciation = ($22,995 - $12,645) / 6

Depreciation = $1,725

This means that the car depreciated by an average of $1,725 each year. In other words, the value of the car went down by an average of $1,725 each year over the six-year period.

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Answer:

The common ratio of this geometric sequence is -1/2, so we have

a(7) = 16 × -1/2 = -8, a(8) = -8 × -1/2 = 4.

Dividing by -1/2 is the same as multiplying by -2. Working backwards, we have:

a(5) = -2 × 16 = -32

a(4) = -2 × -32 = 64

a(3) = -2 × 64 = -128

a(2) = -2 × -128 = 256

a(1) = -2 × 256 = -512

Answer:

the answer would be 30 because you would be adding 7 years by 23 and since you said years ago i think you meant by years old

you're welcome if this help :)

Step-by-step explanation:

If mum was 23 seven years ago, that means she is currently 23 + 7 = 30 years old.

In seven years, she will be 30 + 7 = 37 years old.

The equation of the line through the points is 7y = -2x - 15.

The equation of a line that passes through two given points is calculated as follows;

y - y1 = m(x - x1)

Where;

The slope is calculated as;

m = (y2 - y1) / (x2 - x1)

m = (-3 + 1) / (3 + 4)

m = -2/7

The equation of the line is calculated as;

y - y1 = m(x - x1)

y - (-1) = (-2/7)(x - (-4))

y + 1 = (-2/7)(x + 4)

y + 1 = (-2/7)x - (8/7)

y = (-2/7)x - (15/7)

7y = -2x - 15

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Answer:

Step-by-step explanation:

The inequality that represents the situation "the temperature should be at least 40 degrees" is:

Temperature ≥ 40

This means that the temperature must be greater than or equal to 40 degrees.

The best estimate of the number of times that Lou is on time to class follows a binomial distribution and is equal to 24.

The binomial distribution is a probability distribution that describes the number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure.

The number of times that Lou is on time to class follows a binomial distribution, where the number of trials (classes) is n=32 and the probability of success (Lou being on time) is p = 75% = 0.75.

The expected number of times that Lou is on time to class can be calculated as:

E(X) = n × p = 32 × 0.75 = 24

Therefore, the best estimate of the number of times out of 32 that Lou is on time to class is 24, using the binomial distribution.

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Answer:

x = 2.4

Step-by-step explanation:

4x + 5 / x + 7
combine like terms

X + 4x = 5x

5 + 7= 12

12/ 5x

Divide

12 /5= 2.4

X= 2.4

Answer:

-1/2

Step-by-step explanation:

The information provided about the line is that the line passes through the points (2, 2) and (6, 0)

Therefore, the first step consists in computing the slope. The formula for the slope is:

[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]

Now, by plugging the corresponding numbers is , we get that the slope is:

[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{0-2}{6-2} =-\frac{1}{2}[/tex]

So then, we find that the slope is [tex]m=-\frac{1}{2}[/tex] and that the line passes through the point [tex](2,2)[/tex].

Conclusion: Based on the data provided, we conclude that the slope of the line is [tex]m=-\frac{1}{2}[/tex].

The calculated value of the surface area of the shape is 192 square cm

From the question, we have the following parameters that can be used in our computation:

The triangle (see attachment)

The surface area is calculated as

Area = 1/2 * Base * Height

Substitute the known values in the above equation, so, we have the following representation

Area = 1/2 * 24 * 16

Evaluate

Area = 192

Hence, the surface area is 192 sq cm

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Answer:

100

Step-by-step explanation:

We can estimate 485% as 500%. We know that 500% is equal to 5.

Therefore,

20 × 500%

= 20 × 5

= 100

3.96 cm is the value of the side x.

From the general formula of the sine rule:

[tex]\frac{sin\ a}{A} =\frac{sin\ b}{B} =\frac{sin\ c}{C}[/tex]

Where A, B, and C are the sides of the triangle and a, b, and c are the corresponding angles of the triangle.

In the given problem,

a= 90 degrees

A = 10 cm

b= 37 degrees

B = unknown

c =180 -90-37 = 53 (Sum of all inner angles of a triangle is 180 degrees)

C = x

Thus,

[tex]\frac{sin\ 90}{10} =\frac{sin\ 53}{x}[/tex]

x = 10 sin53°

x = 10*0.3959

x = 3.96

therefore, the value of the side x is 3.96 cm.

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3.96 cm is the value of the side x.

From the general formula of the sine rule:

[tex]\frac{sin\ a}{A} =\frac{sin\ b}{B} =\frac{sin\ c}{C}[/tex]

Where A, B, and C are the sides of the triangle and a, b, and c are the corresponding angles of the triangle.

In the given problem,

a= 90 degrees

A = 10 cm

b= 37 degrees

B = unknown

c =180 -90-37 = 53 (Sum of all inner angles of a triangle is 180 degrees)

C = x

Thus,

[tex]\frac{sin\ 90}{10} =\frac{sin\ 53}{x}[/tex]

x = 10 sin53°

x = 10*0.3959

x = 3.96

therefore, the value of the side x is 3.96 cm.

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Adam's income is 16.67% less than Pat's income.

Let's assume Adam's income is $100

Then Pat's income is 20% more than Adam's income, which means Pat's income is:

                 $100 + $20 = $120

Now, we need to find out how much percent Adam's income is less than Pat's income. We can use the following formula to calculate the percentage decrease:

Percentage decrease = (Decrease in value / Original value) x 100

Decrease in value is the difference between Pat's income and Adam's income, which is:

                            $120 - $100 = $20

The original value is Pat's income, which is $120

So, the percentage decrease in Adam's income compared to Pat's is:

(20 ÷ 120) × 100

= 0.16666 × 100

= 16.666%

= 16.67% (approx)

Therefore, Adam's income is 16.67% less than Pat's income.

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The number of employees for a certain company has been decreasing each year by 8%. If the company cumenty has 720 employees and this rate continues, find the number of employees in 10 years.

The number of employees in 10 years will be approximately

Round to the nearest whole number as needed)