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The largest integer that always divides (n+1)(n+3)(n+5)(n+7)(n+9) for all positive even integers n is 15.

Positive integers are numbers such as 1, 2, 3, etc. They are also called countable or natural numbers. A positive number is any number greater than zero.

Let P(n) = (n+1)(n+3)(n+5)(n+7)(n+9) be a polynomial for all positive even integers, n . We have to calculate the largest integer that always divides the P(n) .

Let "l" be the largest integer that divides P(n) for each n∈N . Then, l must divide both P(1) and P(2)

i.e, P(1) = (1+1)(1+3)(1+5)(1+7)(1+9) = 2×4×6×8×10

= 2^8×5×3

P(2) = 3×5×7×9×11 , hence , l divides 15.

Since, n + 5 ≡ (n+2) mod(3), one (and exactly one) of the numbers n+1 , n+3 , n+5 must be multiple of 3 . Thus, 3/P(n) for each n∈N.

Since, n+7 ≡ (n+2)mod(5) and n+9 ≡ (n+4)mod(5),

one (and exactly one) of the numbers n+1 , n+3 , n+5 , n+7 and n+9 , must be multiple of 5.

Thus, 5/P(n) for each n∈N.

Therefore, 15/p(n) for each n∈N , so that l = 15 .

So, required value of n is 15.

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The Example of addition of Rational Expressions is "Add 2/3 and 6/7" .

The Rational Expression is defined as the expression , that is represented in the form of p/q , where q ≠ 0 .

Example of Addition of Rational Expression is

Add 2/3 + 6/7 .

In order to add the above two rational expression , we first make the denominator of both the terms same by taking LCM ,

the LCM of denominators 3 and 7 is 21 .

So , the expression becomes  ⇒ 14/21 + 18/21 .

⇒ (14 + 18)/21

Simplifying further ,

we get ;

⇒ 32/21

Therefore , the example of addition of rational expressions is explained above .

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Answer: ( x + 5, y + 11)

Step-by-step explanation:

Let's pick point S at ( -9, -9); it changes to point S' at (-4, 2)

We see the x increase by 5 and the y increase by 11, so the answer is

( x + 5, y + 11)

The top 25% of cars have a typical gas mileage of at least 33.5 miles per gallon.

To calculate this, you need to use the interquartile range (IQR) formula.

The IQR is calculated by subtracting the 25th percentile (Q1) from the 75th percentile (Q3).

IQR = Q3 - Q1

In this case, the 25th percentile (Q1) is 28.4, and the 75th percentile (Q3) is 37.9. So:

IQR = 37.9 - 28.4 = 9.5

The top 25% of cars have a gas mileage of 33.5 miles per gallon, which is Q3 + 1.5*IQR.

33.5 = Q3 + 1.5*IQR

33.5 = 37.9 + 1.5*9.5

33.5 = 37.9 + 14.25

33.5 = 52.15

Therefore, the top 25 percent of cars have a typical gas mileage of at least 33.5 miles per gallon.

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I can borrow $17717.48from E-Loan.

The interest I will pay for the loan is $1,686.52

Now, According to the question:

We know,

The formula of present value is

PV = PMT [tex](\frac{ 1-(1+i)^-^n}{i} )[/tex]

PMT = The monthly payment = $539

i= Rate of interest per period  = 6%/12 = 0.005

n = The number of month = 36 months

Therefore,

PV = 539[tex](\frac{ 1-(1+0.005)^-^3^6}{0.005} )[/tex]

    ≈$17717.48

Therefore I can borrow $17717.48 from E-Loan.

The interest = Paid amount - Loan amount

                      =$[(539×36)-17717.48]

                      = $1,686.52

The interest I will pay for the loan is $1,686.52

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Harvard in-text citations allow for the inclusion of up to three authors. When there are four or more authors, et al. is used to shorten the citation.

In-text citations in APA style are used to quickly describe the sources of your information within the text of your research report.

The reader is directed to more detailed information in the Reference list by succinct in-text citations.

The author's last name and the year of publication are used in APA in-text citations,: (Field, 2005). Include the page number when using direct quotations, as in the following example: (Field, 2005, p. 14).

Reduce the source title or organization name, excluding any articles, to the first word or phrase in the in-text citation if it is longer than four words (a, an, and the).

The alphabetized word for the source in the Works Cited should be the first word in the abbreviated title or organization name.

Therefore, Harvard in-text citations allow for the inclusion of up to three authors. When there are four or more authors, et al. is used to shorten the citation.

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The 3 different options for horizontal asymptotes are:

This refers to a horizontal line y = b where the graph approaches the line as the inputs approach infinity (∞) or negative infinity (–∞).

The 3 different options for horizontal asymptotes

Note: If n=m+1, there is an oblique or slant asymptote.

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answer :

d÷d = 0

then 10 remain

The discriminant of the quadratic equation 2x² + 3x - 5 = 0 is 49 and the nature of the roots are -3/4 + 7/4 and -3/4 - 7/4.

The discriminant of the quadratic equation 2x² + 3x - 5 = 0 is b² - 4ac = 9 - 4(2)(-5) = 49.

The nature of the roots of the equation can be determined by the discriminant. If the discriminant is positive, then the roots are real and distinct; if it is zero, then the roots are real and equal; and if it is negative, then the roots are complex and conjugate. In this case, the discriminant is positive, so the roots are real and distinct.

To calculate the roots, we use the quadratic formula:

x = [-b ± √b² - 4ac]/2a.

For this equation, a = 2, b = 3, and c = -5,

so x = [-3 ± √49]/2(2) = [-3 ± 7]/4 = -3/4 ± 7/4.

This means the two roots of the equation are -3/4 + 7/4 and -3/4 - 7/4.

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To solve a 7 step equation, identify the equation, isolate the variable to one side of the equation, simplify the equation if necessary, divide both sides of the equation by the coefficient of the variable.

1. Identify the equation.

2. Limit the variable to only one side of the problem.

3. Simplify the equation, if necessary.

4. Divide both sides of the equation by the coefficient of the variable, if necessary.

5. Continue carrying out steps 2-4 until the equation is resolved.

6. Check your solution by substituting the solution back into the equation.

7. Write the solution in the form of an equation.

To solve a 7 step equation, identify the equation, isolate the variable to one side of the equation, simplify the equation if necessary, divide both sides of the equation by the coefficient of the variable, repeat steps 2-4 until the equation is solved, check your solution by substituting it back into the equation, and then write the solution in the form of an equation.

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The equation for the quadratic function in standard form is f(x) = x² - 11x + 24

From the question, we are to write an equation for the quadratic function in standard form.

From the given quadratic graph,

The roots of the function are 3 and 8.

That is,

x = 3 and x = 8

Then,

The factors of the quadratic function are (x -3) and (x - 8)

Thus,

We can write that

(x - 3)(x - 8) = 0

x² - 8x - 3x + 24 = 0

Simplify

x² - 11x + 24 = 0

Hence, the equation is f(x) = x² - 11x + 24

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Answer:

D, [tex]y=\frac{-5}{2x}+3[/tex]

An observational study is used to answer a research question based purely on what the researcher observes.

An observational study is a statistical study that does not have any treatment or intervention done by the researchers.

Observational research allows the researcher to see what their subjects really do when confronted with various choices or situations. The term refers to the study of non-experimental situations in which behaviour is observed and recorded.

Statistics is the process of collecting data about a group of objects to draw conclusions about populations of those objects. For example, a statistician may take a sample group of workers from company XYZ and have them rate their job satisfaction with the company. Based on the results of that sample, the statistician can make educated assumptions about all the employees of company XYZ and their job satisfaction.

However, observational studies can only establish that significant associations exist between predictor and outcome variables. Observational studies cannot establish that the associations identified represent cause-and-effect relationships.

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Jing's initial pay rate is $12.50 per hour. She will receive a pay raise of $0.50 per hour every year.

Max's initial pay rate is 10 dollars per hour. He will receive a pay raise of 1 dollar per hour every year.

To determine how many years it will take for Max to earn more per hour than Jing, we need to find the point at which the two linear equations intersect. This can be done by setting the two equations equal to each other and solving for x.

Jing's equation is: $12.50 + 0.50x

Max's equation is: x + 10

Setting the equations equal to each other and solving for x, we get:

$12.50 + 0.50x = x + 10

This simplifies to:

$0.50x = -2.50

Dividing both sides by 0.50, we get:

x = -5

Since the value of x cannot be negative, this means that Max will never earn more per hour than Jing.

A transition of AB6 units to the right followed by a transition 6 unit down this transformations could be used to prove that the line segments are congruent.

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The three main types of forecasting are qualitative, quantitative, and time series.

Forecasting is a predictive tool used to make decisions and plan for the future. It involves analyzing past trends and data to predict future events and outcomes. Forecasting helps businesses and organizations prepare for potential market changes, plan ahead for budgeting, and make informed decisions. It is also an invaluable tool for understanding consumer behavior and market trends.
Qualitative forecasting is a subjective approach that relies on expert opinion, intuition, and past experiences. Quantitative forecasting is a more structured approach that uses statistical techniques and data-driven models to make predictions. Time series forecasting is a form of quantitative forecasting that uses historical data to predict future values. It is especially useful for predicting cyclical trends and seasonal variations. All three approaches can be used together to create a comprehensive forecasting strategy for a business.

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Answer:

Yes, you should

Step-by-step explanation:

Because the probability of winning the dollar increases from \frac{1}{3} 3 1​ to \frac{2}{3} 3 2​ by switching.

The value of the absolute expression |a - b| + |b - a| will be 1.

When the relevant components and basic processes of a numerical method are given values, the expression's result is the result of the computation it depicts.

The equation is given below.

a = b - 1/2

The expression is given as,

⇒ |a - b| + |b - a|

The value of (a - b) is given as,

a = b - 1/2

a - b = - 1/2

And the value of (b - a) is given as,

a = b - 1/2

b - a = 1/2

Then the value of the expression is calculated as,

⇒ |a - b| + |b - a|

⇒ |-1/2| + |1/2|

⇒ 1/2 + 1/2

⇒ 1

The worth of the outright articulation |a - b| + |b - a| will be 1.

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y = MX + b is the required linear graph formula.

Linear equations with two variables have straight-line graphs.

There are numerous ways to express linear equations: Y = mx+ b is the slope-intercept form.

The slope is m and the y-intercept is the point in an equation of the type y = MX + b, such as y = 2x 3. (0, b).

Write the equation in the form y = MX + b to determine the slope m and the y-intercept.

This will allow you to graph the equation using the slope and y-intercept (0,

b).

Plot the y-intercept next.

3) Depending on whether the slope is positive or negative, go up, down, or left or right from the y-intercept.

Therefore, y = MX + b is the required linear graph formula.

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Step-by-step explanation:

Velocity is a measure of the rate at which an object moves through space. It is a vector quantity, meaning that it has both magnitude (size) and direction. The standard unit for velocity is meters per second (m/s).

Of the quantities listed above, the following represent velocities:

A. 25 m/s: This represents a velocity of 25 m/s, because it gives both the speed (magnitude) and direction of the object's movement.

C. 15 mi/h eastward: This represents a velocity of 15 miles per hour (mi/h) in an eastward direction, because it gives both the speed and direction of the object's movement.

D. 3 mi/h: This represents a velocity of 3 mi/h, because it gives both the speed and direction (implied to be eastward or westward) of the object's movement.

B. 10 km/min: This does not represent a velocity, because it gives the distance traveled (10 km) but not the time over which the distance was traveled. Without the time, it is not possible to determine the speed or rate of movement.

In geometry, parallel lines are the lines which maintain the same distance apart from each other and never intersect at any point.

What is parallel class 6?

In geometry, parallel lines are the lines which maintain the same distance apart from each other and never intersect at any point.

All rungs are perpendicular to the left post is a line perpendicular to one of a set of parallel lines, it is perpendicular to all parallel lines.

The rungs are parallel

lines that are perpendicular to the same line are parallel.

The rungs are perpendicular to both posts and the posts are parallel  to each other.

if a line is perpendicular to one of a set of parallel lines, it is perpendicular to all the parallel lines.

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This is based on Conditional Probability.

P(a football, then a baseball) is 0.08333 and P(two baseballs)

is 0.15

The chance of an event happening given that another event has already occurred (via assumption, supposition, statement, or evidence) is known as conditional probability in the field of probability theory.  This particular approach depends on event B happening in some way connected to another event A. In this case, a conditional probability analysis with respect to event B is possible. The probability of A under the condition B is typically expressed as P(A|B) or PB if the event of interest is A and the event B is known to have occurred or is assumed to have done so (A). This can alternatively be thought of as the percentage of probability that B crosses A:

P(A|B)=P(A∩B)/P(B)

1) P(a football, then a baseball)= ([tex]\frac{5}{25}[/tex])×([tex]\frac{10}{24}[/tex]) =0.08333

2) P(2 Baseballs) = ([tex]\frac{10}{25}[/tex])×([tex]\frac{9}{24}[/tex]) = 0.15

This is based on Conditional Probability.

P(a football, then a baseball) is 0.08333 and P(two baseballs)

is 0.15

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The equation of the parabola y = 5 · x² + 15 · x - 25 in vertex form is y + 145 / 4 = 5 · (x + 3 / 2)².

Herein we find the equation of a parabola in standard form, that is, an equation of the form:

y = a · x² + b · x + c

Where:

From this equation we must derive the vertex form, whose definition is shown below:

y - k = C · (x - h)²

Where:

The vertex form is found by algebra properties. First, write the equation of the parabola:

y = 5 · x² + 15 · x - 25

Second, use algebra properties until the vertex form is found:

y + 25 = 5 · x² + 15 · x

y + 25 = 5 · (x² + 3 · x)

y + 25 + 45 / 4 = 5 · (x² + 3 · x + 9 / 4)

y + 145 / 4 = 5 · (x + 3 / 2)²

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Answer:

Step-by-step explanation:

To solve the equation 5(3+6a) = -15+6(5+5a), we need to use the rules of algebra to isolate the variable "a" on one side of the equation and simplify the other side.

First, we can distribute the 5 on the left side of the equation to obtain 53 + 56a = -15+6(5+5a). This gives us 15 + 30a = -15+6(5+5a).

Next, we can distribute the 6 on the right side of the equation to obtain 15 + 30a = -15+65 + 65a. This gives us 15 + 30a = -15+30 + 30a.

Now we can combine like terms on both sides of the equation to obtain 30a = -15 + 30, which simplifies to 30a = 15. Finally, we can divide both sides of the equation by 30 to obtain a = 0.5.

Therefore, the solution to the equation 5(3+6a) = -15+6(5+5a) is a = 0.5

The process of forecasting future expenses and savings is called financial planning.

Forecasting is the process of predicting future events or outcomes based on past data, trends, and other information. It is an important tool for businesses, allowing them to plan ahead and make informed decisions. Forecasting can help organizations anticipate changes in the market, anticipate customer demand, plan investments, and more. If done properly, it can help organizations achieve their goals and maximize profits.

Financial planning is the process of creating a budget, setting goals, and developing strategies to help manage money and reach financial goals. It includes estimating future income and expenses, assessing current financial resources, and creating a plan for spending, saving, and investing. Financial planning can help you to stay on track towards meeting your financial goals and create a secure financial future.

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Answer:(-1,2)

Step-by-step explanation:

-(6x+y=-4)

6x+5y=4

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

-6x-y=4

6x+5y=4

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

4y=8

y=2

6x+2=-4

6x=-6

x=-1

The largest area you can enclose with 120 feet of fencing is 900 square feet.

To find the maximum area you can enclose with 120 feet of fencing, you can use the formula for the area of a rectangle, which is A = lw, where A is the area, l is the length, and w is the width. Since the length of the rectangle is bounded by a wall and cannot be changed, the width will be the variable we seek to maximize.

Since the perimeter of the rectangle is fixed at 120 feet, we can set up the following equation:

2l + 2w = 120

This equation represents the total length of the fence needed to enclose the rectangular field. We can solve for w in terms of l by rearranging the equation as follows:

2w = 120 - 2l

w = (120 - 2l)/2

To maximize the area of the rectangle, we want to maximize the value of w. We can do this by making l as small as possible, since this will make w as large as possible. Therefore, we can set l equal to the minimum possible value, which is the length of the wall.

Let's say the length of the wall is x feet. Then, we can substitute x for l in the equation for w to find the maximum possible width:

w = (120 - 2x)/2

To find the maximum area, we can then substitute this value for w back into the formula for the area of a rectangle:

A = lw

= x((120 - 2x)/2)

= (120x - 2x^2)/2

This is a quadratic function, and we can use the techniques for maximizing quadratic functions to find the maximum value of A. To do this, we take the derivative of the function and set it equal to zero:

A' = (120 - 4x)/2 = 0

120 - 4x = 0

x = 30

Substituting this value back into the equation for A, we find that the maximum possible area is:

A = (120 * 30 - 2 * 30^2)/2

= (3600 - 1800)/2

= 1800/2

= 900 square feet

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The equation of the line tangent  to the curve at the points , y = 8ex cos(x), (0, 8) as calculated is y' = 0.

We will differentiate y = 4ex cos(x) so as to find the required equation.

we get the differential as ,

y' = - 8eˣsin(x)

given the value of x as 0 and y as 8

y' = -8e⁰sin(0)

Now as we know that the value of sin( 0 ) is 0 , therefore the equation becomes,

y'= 0  

Differentiation is a branch of calculus which is used to calculate the rate of change in a given object in a specified time. The other  branch of calculus is integration.

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The expression expressed as a single fraction is -2y(yx-2)/3x(yx+5)

You should know that simplification involves making an expression look more simpler than it was

The given expression is  2/(x+5/y)÷3/(y-2/x)

First of all let us express the denominator as a single fraction

x+5/y

⇒(yx +5)/y

In the same way express the expression y-2/x = (yx-2)÷x

By substitution we have

(-2)/yx+5)/(y/3)÷(yx-2/x)

⇒(-2y)/(yx+5) ÷ (3x/ (yx -2)

Simplifying this further to get

(-2y)/(yx +5) * (yx -2)/(3x)

⇒-2y(yx-2)/3x(yx +5)

In conclusion, the expression expressed as a single fraction is -2y(yx-2)/3x(yx+5)

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Correct question:

Which expression is equivalent to the following complex fraction? -2/x+5/y/3/y-2/x

Answer:the answer is

Step-by-step explanation:

just did the test

Because the equation depicts a straight line in the Cartesian plane, it is called a linear equation.

A straight-line link between two variables is referred to statistically as a linear relationship (or linear association). Linear relationships can be represented graphically or mathematically as the equation y = mx + b.

In the variable separation if the unknown function and its derivatives have no exponent larger than one and there are no cross-expressions—that is, terms like f f′ or f′f′′ in which the function or its derivatives appear more than once—the partial differential equation is said to be linear.

Because the equation depicts a straight line in the Cartesian plane, it is called a linear equation. Through the use of logical and mathematical tools, it enables us to solve issues in a variety of scientific fields and academic disciplines as well as in real-world circumstances.

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