What is the value of R for the following sequence

2, 10, 50, 250, 1250…

The function f(n) that generates this sequence is 2 + 1/(n+1). This function takes an input of n and outputs the corresponding term in the sequence.

What is an arithmetic sequence?

A progression or sequence of numbers known as an arithmetic sequence maintains a consistent difference between each succeeding term and its predecessor. The common difference in that mathematical progression is the constant difference.

Here, we have

Given: the sequence 2 and one-half comma 2 and one-third comma 2 and one-fourth comma and continues.

Each term in the sequence is 2 plus a fraction where the denominator is one more than the term number. For example:

In the first term, n=1, and the fraction is 1/2

In the second term, n=2, and the fraction is 1/3

In general, the n-th term in the sequence will have the fraction 1/(n+1).

Hence, the function f(n) that generates this sequence is 2 + 1/(n+1). This function takes an input of n and outputs the corresponding term in the sequence.

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The function 2y = -4x is linear and the graph is given below.

A linear function can be defined as the function whose graph is a straight line. It can also be defined as the function with one or more variables and the exponent of the variable is 1.

Here the given function is 2y = -4x.

The function is linear since the exponent of the two variables x and y are 1.

This can be written as,

y = -4x / 2

y = -2x

This is a proportional function with slope -2.

Graph will be a decreasing function.

The points on the graph includes (-2, 4), (-1, 2), (0, 0), (1, -2) and (2, -4).

Hence the graph of the function is given below.

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A 0 degree and 90 degree can form a pair of complementary angles, but 90 degree and an angle that is not adjacent cannot form a pair of complementary angles.

In mathematics, complementary angles are two angles whose sum is 90 degrees. So, if one angle is 0 degrees, the other angle would have to be 90 degrees in order for them to be complementary. However, if one angle is 90 degrees, it cannot form a pair of complementary angles with an angle that is not adjacent (i.e. next to it).

To understand why, consider a straight line, which is equivalent to 180 degrees. If one angle is 90 degrees, it takes up half of the line, leaving the other half for the other angle.

If the other angle is not adjacent, it would have to be placed somewhere on the other half of the line, which is already taken up by the 90 degree angle. Therefore, the two angles cannot add up to 90 degrees and cannot be complementary.

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The constant of proportionality is 4.01.

The number of widgets that can be produced is 42,225.30.

Joint variation is when a variable depends on more than one variable.

The equation that represents joint variation is:

z = kwt

Where:

32280.5 = k x 14 x 575

k = 32280.5 / (14 x 575)

k = 4.01

Number of widgets that can be produced by 585 workers in 18 hours

w = 4.01 x 18 x 585

w = 42,225.30

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princess trail is 13200 foot long

thank you

The measure of the angle using the sine rule will be 50.28 degrees.

It's a form of a triangle with one 90-degree angle that follows Pythagoras' theorem and can be solved using the trigonometry function.

Trigonometric functions examine the interaction between the dimensions and angles of a triangular form.

The value of the angle 'x' is given by the sine of angle 'x'. Then the equation is given as,

sin x = 30 / 39

x = sin⁻¹(30 / 39)

x = 50.28°

The measure of the angle using the sine rule will be 50.28 degrees.

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The percentage of sales tariff is 8.06%. Then the second item cost before the tariff will be $124.

Any product's quantity is given as if it were a percentage of one hundred. The ratio can be written as 100. The word "%" means "100%" in English. The sign for it is the letter "%".

The percentage is given as,

[tex]\text{Percentage (P) }= \dfrac{ \text{Final value - Original value}}{\text{ Original value }} \times 100[/tex]

An item that costs $310 before the tariff carries a $25 sales tariff. The sales tax % is then provided as,

[tex]\rm P = \left ( \dfrac{25}{ 310} \right ) \times 100[/tex]

P = 0.0806 x 100

P = 8.06%

The sales tax on a second object is $10. Then the second object cost before the tariff is given as,

⇒ $10 / 0.0806

⇒ $124

The percentage of sales tariff is 8.06%. Then the second item cost before the tariff will be $124.

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Based on a system of equations, the cost of a soccer ball is $15.50.

A system of equations is two or more equations solved concurrently or at the same time.

A system of equations can also be described as simultaneous equations.

                            Shin Guards    Soccer Balls    Total Cost

First purchase             3                     2                  $48.97

Second purchase       5                     3                  $76.45

Let the cost of shin guards per unit = x

Let the cost of soccer balls per unit = y

Equation 1: 3x + 2y = 48.97

Equation 2: 5x + 3 y = 76.45

Multiply Equation 1 by 1.5:

Equation 3: 4.5x + 3y = 73.455

Subtract Equation 3 from Equation 2:

5x + 3 y = 76.45

-

4.5x + 3y = 73.455

0.5x = 2.995

x = 5.99

Substitute x = 5.99 in Equation 1:

3(5.99) + 2y = 48.97

17.97 + 2y = 48.97

2y = 31

y = 15.5

= $15.50

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Jeremy needs to buy soccer supplies for his team. He bought 3 packs of shin guards and 2 soccer balls for $48.97.

The number of rectangular prisms that can be made by 24 unit cubes, two having dimensions mentioned below.

1.  3 units by 4 units by 2 units.

2. 2 units by 3 units by 4 units

To measure and compare various things we use one term which is known as a unit. When all of the measuring units are the same, comparison is simple. Depending on their intended usage, different units can be categorized.

For example, for  length measurement we use units like:- meter, centimeter, inch, miles.

One possible right rectangular prism that can be made using 24 unit cubes has dimensions of 3 units by 4 units by 2 units.

This means it has 3 layers of 4 unit cubes each, with each layer having 2 rows of 3 unit cubes.

Another possible right rectangular prism that can be made using 24 unit cubes has dimensions of 2 units by 3 units by 4 units.

This means it has 2 layers of 3 unit cubes each, with each layer having 4 rows of 2 unit cubes.

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The unit cost per pound of trail mix, in dollars is $3.75.

How to calculate Unit price of an item?

To calculate the unit price of an item, you have to understand that unit price means price per unit. To find the unit price we divide the total cost by the amount of items we are given in the question.

Given :

15x - 4y = 0

Where ,y is total cost in dollars of the trail mix and x is  the number of pounds bought.

To calculate , unit cost of trial mix , we need to calculate y

Therefore , from equation 15x - 4y = 0

We can say that,

y =(15/4)x

For unit cost we need to substitute x = 1

Thus, y = (15/4)*1

or

y = (15/4)

or

y = 3.75

Hence, the unit cost per pound of trail mix, in dollars is $3.75.

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The correct derivative expressions of the given ones are -

[tex]$1.\;\;\int\limits^3_3 {f^{2} (x)} \, dx = 0[/tex]

[tex]$2.\;\int\limits^3_8 {g(x)-f(x)} \, dx =[/tex][tex]$\int\limits^3_8 {g(x)} \, dx - \int\limits^3_8 {f(x)} \, dx = -2 - 6 = -8[/tex]

[tex]$3.\;\;\int\limits^3_8 {4f(x)} \, dx - \int\limits^3_8 {2g(x)} \, dx = 24 - 4 = 20[/tex]

It is given that -

[tex]$\int\limits^3_8 {f(x)} \, dx = 6\\[/tex]

and

[tex]$\int\limits^3_8 {g(x)} \, dx[/tex]

The correct statements of the given are -

[tex]$1.\;\;\int\limits^3_3 {f^{2} (x)} \, dx = 0[/tex]

[tex]$2.\;\int\limits^3_8 {g(x)-f(x)} \, dx =[/tex][tex]$\int\limits^3_8 {g(x)} \, dx - \int\limits^3_8 {f(x)} \, dx = -2 - 6 = -8[/tex]

[tex]$3.\;\;\int\limits^3_8 {4f(x)} \, dx - \int\limits^3_8 {2g(x)} \, dx = 24 - 4 = 20[/tex]

Therefore, the correct derivative expressions of the given ones are -

[tex]$1.\;\;\int\limits^3_3 {f^{2} (x)} \, dx = 0[/tex]

[tex]$2.\;\int\limits^3_8 {g(x)-f(x)} \, dx =[/tex][tex]$\int\limits^3_8 {g(x)} \, dx - \int\limits^3_8 {f(x)} \, dx = -2 - 6 = -8[/tex]

[tex]$3.\;\;\int\limits^3_8 {4f(x)} \, dx - \int\limits^3_8 {2g(x)} \, dx = 24 - 4 = 20[/tex]

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Answer:

  C.  ∫(f·g)dx = -12 is incorrect

Step-by-step explanation:

You want to know which of the integral expressions is incorrect given ...

[tex]\displaystyle \int_3^8{f(x)}\,dx=6\ \ \text{ and }\int_3^8{g(x)}\,dx=-2[/tex]

Expression A has the same value for upper and lower limits of the integral. The area being integrated has zero width, so its value will be zero, as shown in expression A.

Expression E has the limits reversed from those in expression B, so the value of that integral will be the opposite of the value shown for B. Both of these integral expressions (B, E) are correct.

The integral of a sum is the sum of the integrals. This is shown in expressions B, D, and E, all of which are correct.

In general, the integral of a product will have no relation to the product of integrals. Hence expression C is NOT necessarily correct.

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Answer: (-∞, -5)

Step-by-step explanation:

When plotted on a line graph there will be an open circle on negative five and your line would be going infinitely off to the left.

Answer:

164

Step-by-step explanation:

4 x 10 = 40

40 x 2 = 80 (there are two sides with the same measurements)

---

3 x 4 = 12

12 x 2 = 24 (there are two sides with the same measurements)

----

10 x 3 = 30

30 x 2 = 60 (there are two sides with the same measurements)

60 + 80 + 24 = 164

Answer:

A, D

Step-by-step explanation:

8/5 + 2/5y

Factor out the 2/5, which gives you:

2/5(4 + 1y) = 2/5(5 + y)

or,

Factor out 1/5, which gives you:

1/5(8 + 2y)

So both A and D are equivalent

Answer:

2: 0.6

3: 0.1

Step-by-step explanation:

The difference in the medians is 1/3 times the IQR.

Mathematically, interquartile range (IQR) of a data set and it is typically calculated as the difference between the first quartile (Q₁) and third quartile (Q₃):

Interquartile range (IQR) for Speed Slide = Q₃ - Q₁

Interquartile range (IQR) for Speed Slide = 12 - 6

Interquartile range (IQR) for Speed Slide = 6.

For the interquartile range (IQR) for Wave Machine, we have:

Interquartile range (IQR) for Wave Machine = 14 - 8

Interquartile range (IQR) for Wave Machine = 6.

For the required expression, we have the following:

2 = 6x

x = 2/6

x = 1/3.

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Answer:

The difference in the medians is 1/3 times the IQR.

Step-by-step explanation:

hope this helps!

Based on the definition of similar triangles, the length of PR is 16 units.

In similar triangles, corresponding sides are in proportion. This means that if two triangles are similar, their corresponding sides will have the same ratio.

Therefore, we will have the proportion:

8/6 = (x + 7 + x - 1) / (x + 7)

4/3 = (2x + 6) / (x + 7)

Cross multiply:

3(2x + 6) = 4(x + 7)

6x + 18 = 4x + 28

6x - 4x = -18 + 28

2x = 10

x = 5

PR = x + 7 + x - 1

PR = 2x + 6 = 2(5) + 6

PR = 16 units

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Verifying trig identities can be difficult because they involve manipulating complex equations with multiple trigonometric functions that have various associated angles and coefficients.

For example, the Pythagorean Identity states that sin2θ + cos2θ = 1, which must be verified by using the identity sin2θ = 2sinθcosθ and substituting it into the equation. This results in a complex equation: 2sinθcosθ + cos2θ = 1, which must then be simplified and rearranged in order to show that it is true for all values of θ. Furthermore, trig identities can involve multiple steps that involve multiple functions and angles, which can make verifying them even more difficult.

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The perimeter of the parallelogram is 24 units

The formula for calculating the perimeter of a parallelogram is given as;

P = 2( a+ b)

Given that the parameters are;

From the diagram shown, we have the values as the distance between points, we have ;

a = 4 - 1

a = 3

b = 4

Substitute the values

Perimeter , P = 2(3×4)

expand the bracket

Perimeter, P = 24 units

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65 kg is equivalent to approximately 143.3 pounds and both are unit of measurement of weight.

To convert kilograms to pounds, we multiply the weight in kilograms by the conversion factor of 2.20462. Therefore, to convert 65 kg to pounds, we can multiply 65 by 2.20462, which gives us approximately 143.3 pounds.

It's important to note that kilograms and pounds are both units of weight, but they are used in different parts of the world. Kilograms are the standard unit of weight in most countries, while pounds are commonly used in the United States and some other countries.

Converting between units of measurement is a valuable skill, especially when communicating or traveling internationally. Knowing how to convert from kilograms to pounds, or vice versa, can help you understand weight measurements in different contexts.

Online conversion tools or calculators can be useful resources to quickly and accurately convert between units of measurement as needed.

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When you horizontally stretch the parent function f(x)=x² by a factor of 4, the new function becomes f(x)= (1/4) x².

The function can be stretched or shrunk in different directions to produce new functions that are similar in shape but differ in size and orientation.

When you horizontally stretch the function f(x)=x² by a factor of 4, the function becomes

=> f(x)= (1/4) x².

To understand this, let's look at the equation of the parent function in detail.

The equation of the parent function f(x)=x² can be written as

=> y=x².

The x values represent the horizontal distances from the origin, and the y values represent the vertical distances from the origin.

When you stretch the function horizontally by a factor of 4, it means that the x values will be multiplied by 4. In other words, the distance from the origin in the x direction will be four times larger. The new equation of the stretched function will be

=> y= (1/4) (4x)².

We can simplify this equation to y= (1/4) x².

In mathematical terms, the horizontal stretching of a function is equivalent to multiplying the independent variable (x) by a constant factor. In this case, the constant factor is 1/4. The constant factor of 1/4 in the equation

=> y= (1/4) x²

represents the horizontal stretch by a factor of 4. The effect of this horizontal stretch is that the parabolic shape of the function will be wider and flatter compared to the parent function. The shape of the parabolic graph of the stretched function will be similar to the parent function, but the x values will be multiplied by 4.

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The answers to all parts shown below.

Radical - The √ symbol that is used to denote square root or nth roots. Radical Expression - A radical expression is an expression containing a square root. Radicand - A number or expression inside the radical symbol.

Given:

We have to find the Square/Cube Factors

1. 5/6√3

Now, rationalizing

5/6√3 x √3/ √3

= 5√3 / 18

2. √3/ (7- 3√5)

Now, rationalizing

= √3/ (7- 3√5) x (7+ 3√5) / (7+ 3√5)

= √3(7+ 3√5) / 4

3. 8/ (4 + 5√2)

Now, rationalizing

= 8/ (4 + 5√2) x  (4 - 5√2)/  (4 - 5√2)

= -8 (4 - 5√2)/ 34

4. 7 ∛(48 [tex]m^{18} n^{13}[/tex])

= 14 [tex]m^6[/tex][tex]n^4[/tex]∛6n

5.[tex]2\sqrt{\left(-300x^{18}\right)}[/tex]

=[tex]20\sqrt{3}\sqrt{-x^{18}}[/tex]

6. (4+ 7√10)²

= 4² + (7√10)²+ 2 (4)( 7√10)

= 16 + 490 + 56√10

= 506 + 56√10

7. (6- 7√3) x (6+ 7√3)

= 6² - (7√3)²

= 36 - 147

= -111

8. 11√50[tex]x^{12}y^5[/tex]

= 55[tex]x^6[/tex]y²√2y

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The product of all integers from -99 to -90, including -99 and -90, will be positive.

The product of an even number of negative integers will be positive, while the product of an odd number of negative integers will be negative.

In this case, we have 10 negative integers, which is an even number. Therefore, the product of all these integers will be positive.

However, the product of a positive and a negative integer is negative, so we need to keep track of the sign changes that occur when we multiply the integers.

We can start by multiplying the two extremes, -99 and -90. The product of these two integers is positive, so at this point, the answer is positive. However, as we multiply more and more of the integers between -99 and -90, we'll eventually reach an even number of positive integers

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Answer: The relationship between the distance a car has traveled in miles, d, and the amount of gas left in the tank in gallons, x, can be represented by the equation:

d = r * m

where r is the fuel efficiency of the car in miles per gallon and m is the amount of gas consumed in gallons.

Since x is the amount of gas left in the tank, we can write:

x = m_0 - m

where m_0 is the initial amount of gas in the tank.

Combining the two equations, we get:

d = r * (m_0 - x)

This equation represents the relationship between the distance the car has traveled in miles, d, and the amount of gas left in the tank in gallons, x.

Step-by-step explanation:

Answer:

Let -5x+y=-3 be the first equation, and 3x-8y=24 be the second. Let's manipulate the first equation by adding 5x to both side; resulting in y=-3+5x, or 5x-3. Now let's plug in 5x-3 for wherever y has been in the second formula. 3x-8(5x-3)=24. This is now pretty straightforward to solve for x. We distribute the 8 into (5x-3) and get 3x-40x-24=24. -48=27x. x=-48/27, x=-16/9.

Now let's plug in the x for either one of the equations to get y. 3(-16/9)+8y=24. -16/3+8y=24. 8y=24+16/3. 24y=72+16. 24y=88. y=88/24, y=11/3.

Step-by-step explanation:

Jodie spent a total of $2,166 last month on groceries, utilities, and other expenses.

To find out how much Jodie spent in all, we need to add up all of her expenses. We can do this by using the following formula:
Total expenses = Groceries + Utilities + Other expenses

Plugging in the given values, we get:
Total expenses = $178 on groceries + $238 on utilities + $1,750

Adding these values together, we get:
Total expenses = $2,166

Therefore, Jodie spent a total of $2,166 last month.

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Answer:

To avoid cutting out a fraction of an inch, we need to choose a value of x that is a factor of both 36 inches and 48 inches, since those are the lengths of the sides of the rectangular sheet of cardboard we are using.

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48.

The largest factor that both 36 and 48 have in common is 12. However, we can't choose x to be 12 inches because that would mean the length and width of the rectangular box would also be 12 inches, which is too small for our purposes.

Therefore, we should choose x to be the next largest factor that both 36 and 48 have in common, which is 6 inches. This means we will cut six 6-inch by 8-inch rectangles from the cardboard sheet.

The dimensions of the rectangular box will be:

Length = 2x + 2h = 2(6 in) + 2(8 in) = 12 in + 16 in = 28 in

Width = 2x + 2w = 2(6 in) + 2(12 in) = 12 in + 24 in = 36 in

Height = h = 8 in

Therefore, the dimensions of the rectangular box are 28 inches by 36 inches by 8 inches

The 8 fluid ounces(oz) of water is equivalent to 236.588 milliliters.

To convert 8 ounces (oz) to milliliters (ml), we need to know the density of the substance. The conversion factor will vary depending on what we are measuring. Here are some common conversion factors:

1 fluid ounce of water weighs 29.5735 grams, which is equivalent to 29.5735 ml.

1 fluid ounce of milk weighs 30 grams, which is equivalent to 28.4131 ml.

1 fluid ounce of honey weighs 21 grams, which is equivalent to 20.0784 ml.

For example, let's say we want to convert 8 fluid ounces of water to milliliters. We know that 1 fluid ounce of water weighs 29.5735 grams, and that 1 milliliter of water also weighs 1 gram (since water has a density of 1 g/mL). Therefore, we can use the following formula to convert 8 fluid ounces of water to milliliters:

8 oz x 29.5735 g/oz x 1 mL/1 g = 236.588 ml

So 8 fluid ounces of water is equivalent to 236.588 milliliters.

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Yes, Paul's Online Math Notes is a good enough source for self-teaching calculus 3 and ODE.

The site is well-organized, with clear explanations and plenty of examples and practice problems. It also includes a section on differential equations, which is helpful for those studying ODE.

However, it is always a good idea to supplement your learning with additional resources, such as textbooks, online videos, and practice problems from other sources.

This will help you gain a deeper understanding of the material and ensure that you are well-prepared for exams.

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