Find the length of QR

Answer:

a

Step-by-step explanation:

divide the number of tests by the number of weeks

The probability that a randomly chosen subject comples more than the expected number of puzzles in the five minute is equals to the 0.40. So, option (b) is right one.

We have a Random variables X denotes the number of puzzles complete by random choosen subject. The above table contains probability distribution of random variable X. The expected number of puzzles in the 5-minutes period while listening to smoothing music is calculated by following formula, [tex]E(x) = \sum x_i p(x_i)[/tex]

= 1(0.2) + 2(0.4) + 3(0.3) + 4(0.1)

= 0.2+ 0.8+ 0.9+ 0.4

= 2.3

Now, the probability that a randomly chosen subject completes more than the expected number of puzzles in the 5-minute period while listening to soothing music, that is possible value values of X are 1,2,3,4 but for X > 2.3 only 3 and 4. So, P(X>2.3)= P(X=3) + P(X=4)

=0.30+ 0.10=0.40

Hence, required value is 0.40.

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Complete question:

The above table complete the question

what is the probability that a randomly chosen subject comples more than the expected number of puzzles in the five minute period while losing music

a. 0.1

b. 0.4

c. 0.8

d. 1

e. Cannot be determined

In triangle ABC, ∠B is 17.7°, ∠C is 84.3° and a is 138.3 units.

To solve triangle ABC, we can use the law of sines and the fact that the sum of angles in a triangle is 180 degrees.

From the law of sines, we have:

a/sin(78) = b/sin(B) = c/sin(C)

Substituting the given values, we get:

a/sin(78) = 66/sin(B) = 32/sin(C)

Solving for sin(B), we get:

sin(B) = (asin(78))/66

Solving for sin(C), we get:

sin(C) = (asin(78))/32

Using the fact that sin(B) + sin(C) = sin(180 - B - C), we get:

(asin(78))/66 + (asin(78))/32 = sin(B+C)

Simplifying and solving for a, we get:

a = (6632sin(78))/(66sin(78) + 32sin(B+C))

To find angle B, we can use the fact that the sum of angles in a triangle is 180 degrees:

B = 180 - 78 - C

Substituting this into the law of sines equation, we get:

a/sin(78) = 66/sin(B)

Solving for sin(B), we get:

sin(B) = (66sin(78))/a

Substituting the value of a we found above, we get:

sin(B) = (66sin(78))/(66sin(78) + 32*sin(C))

Using a calculator to evaluate sin(C) and then sin(B), we get:

sin(C) = 0.478

sin(B) = 0.902

Substituting these values into the law of sines equation, we get:

a/sin(78) = 66/sin(B)

Solving for a, we get:

a = (66sin(78))/sin(B)

Using a calculator to evaluate a, we get:

a = 138.3

Therefore, the length of side a is 138.3 units, and angle B is approximately 17.7 degrees and angle C is approximately 84.3 degrees.

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Answer:

Team Members In A Workplace: An Analysis of Their Importance and Contribution to Organizational Success

In today's fast-paced business environment, organizations rely heavily on the collaborative efforts of teams to achieve their goals and objectives. The success or failure of an organization often depends on the effectiveness of its team members. Therefore, team members play a crucial role in the overall success of any workplace. This essay aims to explore the importance of team members in a workplace and their contribution to organizational success.

To start with, team members are highly valuable to an organization due to their diverse skill sets and knowledge. Individuals working in a team come from different backgrounds, experiences, education levels, and expertise, which can be effectively utilized to tackle complex issues and challenges. This diversity in team members allows for the exchange of ideas, brainstorming, and the creation of innovative solutions to problems. The unique perspectives and ideas of team members can also contribute to the development of new business strategies or products, which can give the organization a competitive edge.

Moreover, team members in a workplace offer mutual support to each other, which is essential for the success of the organization. Working as a team creates a sense of belonging and camaraderie, which translates to a positive work environment. Team members can also provide feedback, encouragement, and motivation to their colleagues, which helps to establish a productive and efficient work culture. A collaborative work culture can lead to higher job satisfaction, employee retention, and overall performance of the workplace.

Another important contribution of team members in a workplace is their ability to hold each other accountable. When working in a team, members can hold each other responsible for their actions, which ensures that everyone is doing their part to achieve the team's goals. This kind of accountability not only improves individual performance but also ensures that the team is on track to achieve its objectives.

Additionally, team members in a workplace can contribute to the development of leadership skills. Working in a team can give individuals the opportunity to lead a project or work on a specific initiative, which can help develop their leadership skills. The skills developed in such scenarios include communication, decision-making, problem-solving, delegation, and conflict resolution, which are all essential skills required for effective leadership.

As a facilitator, there are several other things I would do to ensure the team runs smoothly and efficiently. Firstly, I would establish clear communication channels to ensure that team members can easily communicate with each other. This includes setting up regular team meetings and encouraging open and honest communication. Secondly, I would ensure that each team member has a clear understanding of their roles and responsibilities within the team. This means setting clear expectations and goals for each team member and providing them with the necessary resources and support to achieve these goals. Finally, I would encourage collaboration within the team by promoting a culture of trust, respect, and teamwork. This involves fostering an environment where team members feel comfortable sharing ideas, giving feedback, and working together towards a common goal.

In conclusion, team members are a critical component of any workplace, and their contributions cannot be overstated. The diverse skill sets, mutual support, accountability, and leadership skills of team members significantly impact the overall success of the organization. Therefore, it is important for organizations to create a work culture that encourages teamwork and collaboration, as it ultimately leads to higher productivity, efficiency, and satisfaction among employees.

Step-by-step explanation:

Answer:

Step-by-step explanation:

The general formula is y=mx+b

Calculate the slope, m.

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

m=(3-5)/(1-2)

m=(-2)/(-1)=2

The slope is 2.

So plug that into the equation y=mx+b

y=2x+b.

Now let's use one of the points we were given to figure out the y intercept (b).

y=2x+b, plug in (1,3)

3=2(1)+b

3=2+b

1=b

So the intercept is 1.

Now we know the equation is y=2x+1.

You can plot the graph using a graphing calculator and check that both points are on the line. You can also substitute the points into the equation to check that it's correct.

Using the definition of nth roots, we can prove that a = the nth root of a^n over a^2, or \sqrt[n]{a^n}/a^2.

Let x be the nth root of a^n, so x^n = a^n. Using the definition of nth roots, we can write x as:

x = a^(1/n)

Substituting this into x^n, we get:

(a^(1/n))^n = a^n

Simplifying, we get:

a = x^n

Substituting x with a^(1/n), we get:

a = (a^(1/n))^n

Now, we can simplify the expression \sqrt[n]{a^n}/a^2 using the value we just found for x:

\sqrt[n]{a^n}/a^2 = x/a^2

= (a^(1/n))/a^2

= a^(1/n - 2)

Since x = a^(1/n), we can rewrite the expression as:

a^(1/n - 2) = (a^(1/n))/(a^2)

Therefore, we have shown that a = the nth root of a^n over a^2, or \sqrt[n]{a^n}/a^2, using the definition of nth roots.

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For a non-constant member function of class test, the this pointer has type D. const Test * const. The this pointer is a special pointer in C++ that points to the object whose member function is being executed.

It is a hidden parameter that is passed to all non-static member functions. The type of the this pointer depends on the const-ness of the member function and the const-ness of the object on which the member function is being called.
In this case, the member function is non-constant, which means it can modify the object on which it is being called. Therefore, the this pointer is a pointer to a constant object of type Test. This is because the object on which the member function is being called is being treated as constant inside the member function, even though it may not actually be constant.
The const keyword before Test indicates that the object pointed to by the this pointer is constant, and the const keyword after Test indicates that the this pointer itself is constant and cannot be modified. Therefore, the correct answer is D. const Test * const.
In summary, the type of the this pointer for a non-constant member function of class test is a pointer to a constant object of type Test, which is itself a constant pointer that cannot be modified.

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The solution set of the given system of equations is {(8/3, -1/6)}.

We are given the following system of linear equations:

2x + 2y = 5 ...(1)

x - 2y = 3 ...(2)

We can use the method of elimination by substitution to solve this system. First, we solve equation (2) for x in terms of y:

x = 2y + 3

Substituting this value of x in equation (1), we get:

2(2y + 3) + 2y = 5

Simplifying the above expression, we get:

6y + 6 = 5

6y = -1

y = -1/6

Substituting this value of y in equation (2), we get:

x - 2(-1/6) = 3

x + 1/3 = 3

x = 8/3

Therefore, the solution set of the given system of equations is:

{(8/3, -1/6)}

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The equation representing visual models is z+1/5=3/5.

Since it is given in the question that the hanger image represents a balanced equation therefore equating LHS with RHS it can be written as follows.

LHS = 1/5 + z

RHS = 3/5+ z

Equating both the equations it can be written as

LHS = RHS

1/5+z=3/5

or z=3/5-1/5

Therefore, z=2/5

Hence, for  z=2/5 the hanger will represent a balanced equation.

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Answer:

38.9°

Step-by-step explanation:

a) a² = b² + c² - 2 bc cos A

2 bc cos A = b² + c² - a²

cos A = (b² + c² - a²) ÷ 2 bc

(shown)

b) cos theta =

[tex] \frac{23 {}^{2} + 11 {}^{2} - 16 {}^{2} }{2 \times 23 \times 11} = \frac{394}{506} = \frac{197}{253} [/tex]

Theta = cos¯¹ 197/253 = 38.8623 = 38.9° (1 dp)

Answer:

the answer is B=0,5

Step-by-step explanation:

True: When a form is created based on two or more tables, a relationship must be defined between queries.

True. When a form is created based on two or more tables, a relationship must be defined between queries in order to ensure that the form displays accurate data. Queries are used to pull data from multiple tables and present it in a single view, so it is important to define the relationships between these tables to avoid inconsistencies or errors in the displayed data. This ensures that the data from the related tables can be properly displayed and managed within the form.

When you drag and search a field from an "other" (unrelated) table, a new one-to-many relationship is created from the table in the list and the table from which you dragged the field. This relationship is established by Access, which does not enforce integrity by default.

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The FICO is a type of federal payroll tax

Given data ,

One type of federal payroll tax is the Federal Insurance Contributions Act (FICA) tax. This tax is a combination of Social Security and Medicare taxes, which are used to fund these programs. The current Social Security tax rate is 12.4%, with 6.2% paid by the employee and 6.2% paid by the employer.

The Medicare tax rate is 2.9%, with 1.45% paid by the employee and 1.45% paid by the employer. However, there is an additional 0.9% Medicare tax that is paid by employees who earn over a certain amount.

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To prove by induction on n ≥ 1 that if a tree T has n vertices, then it has exactly n-1 edges, we will use the theorem about leaves in trees.

To prove by induction on n ≥ 1 that if a tree T has n vertices, follow the given steps :

1. Base Case: For n = 1, there is only one vertex in the tree T and no edges. Since 1-1 = 0, the statement holds true for n = 1.

2. Inductive Hypothesis: Assume that the statement is true for some n = k, i.e., if a tree T has k vertices, then it has k-1 edges.

3. Inductive Step: We need to prove that the statement is true for n = k+1, i.e., if a tree T has k+1 vertices, then it has k edges.

Consider a tree T with k+1 vertices. By the theorem about leaves in trees, we know that T has at least one leaf (a vertex with degree 1). Let v be a leaf in T, and let u be its only adjacent vertex. Remove the vertex v and the edge connecting u and v from the tree. The resulting tree T' has k vertices.

By the inductive hypothesis, T' has k-1 edges. Since we removed a leaf and its connecting edge, we can conclude that the original tree T with k+1 vertices has (k-1)+1 = k edges.

Thus, the statement holds true for n = k+1.

By using mathematical induction on n ≥ 1, we have proved that if a tree T has n vertices, then it has exactly n-1 edges.

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The length of ZW is 45 inches.

We have,

A parallelogram is a member of quadrilateral which has a pair of opposite sides to be equal, and a pair of slant opposite sides.

In the given information, we have;

WX = YZ (a pair of opposite side of a parallelogram are equal)

2x + 15 = 4x - 21

collect like terms,

21 + 15 = 4x - 2x

36 = 2x

x = 36/2

x = 18

So that;

WX = 2x + 15

     = 2(18) + 15

WX = 36 + 15

     = 51

XY = x + 27

    = 18 + 27

    = 45

Therefore,

ZW = XY  (a pair of opposite sides are equal)

ZW = 45

The length of ZW is 45 inches.

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Answer:

A1. The rectangular coordinates for the point (5, 5TT ) are

(5, 5т).

A2. The question is incomplete and does not provide any information about the point (SI).

A3. To find the rectangular coordinates for the point (3,

-120°), we need to use the formula x = r cos(e) and y = r sin(e), where r is the distance from the origin to the point and O is the angle the line connecting the point with the origin makes with the positive x-axis.

A4. To find the rectangular coordinates for the point (2, π/4), we need to use the formula x = r cos(θ) and y = r sin(θ), where r is the distance from the origin to the point and θ is the angle the line connecting the point with the origin makes with the positive x-axis.

We have r = 2 and θ = π/4.

So, x = r cos(θ) = 2 cos(π/4) = √2 and y = r sin(θ) = 2 sin(π/4) = √2.

Therefore, the rectangular coordinates for the point (2, π/4) are (√2, √2).

A5. To find the rectangular coordinates for the point (1/4, π/2), we need to use the formula x = r cos(θ) and y = r sin(θ), where r is the distance from the origin to the point and θ is the angle the line connecting the point with the origin makes with the positive x-axis.

We have r = 1/4 and θ = π/2.

So, x = r cos(θ) = (1/4) cos(π/2) = 0 and y = r sin(θ) = (1/4) sin(π/2) = 1/4.

Therefore, the rectangular coordinates for the point (1/4, π/2) are (0, 1/4).

A6. To find the rectangular coordinates of (5,240), we need to use the formula x = r cos(θ) and y = r sin(θ), where r is the distance from the origin to the point and θ is the angle the line connecting the point with the origin makes with the positive x-axis.

We have r = 5 and θ = 240°. Converting 240° to radians, we get θ = 4π/3.

So, x = r cos(θ) = 5 cos(4π/3) = -2.5 and y = r sin(θ) = 5 sin(4π/3) = -4.330.

Therefore, the rectangular coordinates for the point (5,240) are (-2.5, -4.330).

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Step-by-step explanation:

The correct answer to the question is option a) influential observation. An influential observation is an observation that has a significant impact on the regression results, meaning that if it is removed, the regression equation and the coefficients can change significantly.

In other words, it can have a strong effect on the fit of the regression model.
For instance, an influential observation could be an outlier, a point that deviates significantly from the general pattern of the data. This point can affect the slope and intercept of the regression line, and therefore the predictions and inference based on the model. Another example of an influential observation could be a point that has a high leverage, meaning that it has a high leverage on the estimated coefficients due to its position in the predictor space.
Therefore, it is essential to detect and address influential observations when building regression models to ensure that the results are reliable and valid. Techniques such as Cook's distance and leverage plots can be used to identify influential observations and various methods can be employed to deal with them, such as removing them, transforming the data, or using robust regression techniques.

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The salaries have a mean of $70,000 and a standard deviation of $20,000, and the largest salary is originally $100,000. However, this value is mistakenly changed to $1,000,000. We are asked to determine the value of the standard deviation after this change.

To solve this problem, we can use the formula for standard deviation, which is the square root of the variance. We can find the variance of the original data set using the formula: variance = sum of (x - mean)^2 / n. Using the given values, we get a variance of $400,000,000. Next, we can find the new mean of the data set after the change by adding $900,000 to the original sum and dividing by 28. This gives us a new mean of $134,286. The new variance can be calculated using the same formula, but with the new mean and the new largest value of $1,000,000. This gives us a variance of $657,143,556, which, when taking the square root, gives us a standard deviation of approximately $25,640.

Therefore, the value of the standard deviation after the change is approximately $25,640.

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Based on the given regression equation and ∑ of squared errors, we performed a hypothesis test to determine if the slope is significantly different from zero at a 0.10 level of significance.

To perform the hypothesis test, we first need to calculate the standard error of the slope (SEb). This can be done using the following formula:

SEb = √(SSE / (n - 2)) / √(SSx)

where SSE is the ∑ of squared errors, n is the sample size, and SSx is the ∑ of squared deviations of x from its mean. In this case, we are given that SSE = 21.0333 and the sample size is not specified. We can calculate SSx using the formula:

SSx = ∑((x - x₁)²)

where x₁ is the mean of x. If we as∑e that the sample size is 10, then we can calculate SSx as:

SSx = ∑((x - x₁)²) = 10(11.5²) - (100²) / 10 = 115

Plugging in the values, we get:

SEb = √(21.0333 / 8) / √(115) = 0.268

Next, we calculate the t-statistic using the formula:

t = (b - 0) / SEb

where b is the estimated slope from the regression equation. In this case, b = 0.3. Plugging in the values, we get:

t = (0.3 - 0) / 0.268 = 1.119

Finally, we compare the t-statistic to the critical value from the t-distribution with n - 2 degrees of freedom (where n is the sample size).

For an alpha level of 0.10 and 8 degrees of freedom, the critical value is 1.860. Since our t-statistic of 1.119 is less than the critical value of 1.860, we fail to reject the null hypothesis.

This means that we do not have sufficient evidence to conclude that the slope is significantly different from zero at the 0.10 level of significance.

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James has written a correct if-then statement.

In an if-then statement the part following the "if" should be a hypothesis, and the part following the "then" should be a conclusion statement.

Considering the two of the given statements:

Cicely's statement: "If a triangle is a right triangle, then it has one right angle." is correct since the hypothesis and conclusion are both correct.

James's statement: "If a triangle has one right angle, then it is a right triangle." is also correct since the hypothesis and conclusion are both correct.

Now, comparing both the statements "If a triangle has one right angle" is a perfect hypothesis since it gives a condition which when followed gives the conclusion.

Hence, James has written a correct if-then statement.

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The proof by mathematical induction shows that the statement 1^3 + 2^3 + 3^3 + n^3 = (n^2(n+1)^2)/4 holds for all positive integers n. The base case n = 1 is true, and assuming the statement is true for n = k, we can prove it is true for n = k+1 by substituting k+1 for n and simplifying the expression.

To prove the statement 1^3 + 2^3 + 3^3 + ... + n^3 = (n^2(n+1)^2)/4 for all natural numbers n using mathematical induction, we proceed as follows:

Base case: Let n = 1. Then 1^3 = (1^2(1+1)^2)/4 = 1, which is true.

Inductive step: Assume the statement is true for some arbitrary value k, i.e., 1^3 + 2^3 + 3^3 + ... + k^3 = (k^2(k+1)^2)/4.

We need to show that the statement is also true for n = k+1, i.e., 1^3 + 2^3 + 3^3 + ... + (k+1)^3 = ((k+1)^2(k+2)^2)/4.

Starting with the left-hand side of the equation, we have:

1^3 + 2^3 + 3^3 + ... + (k+1)^3

= (1^3 + 2^3 + 3^3 + ... + k^3) + (k+1)^3 // regrouping the last term

= (k^2(k+1)^2)/4 + (k+1)^3 // using the induction hypothesis

= (k+1)^2(k^2+4k+4)/4 // factoring out (k+1)^2

= ((k+1)^2(k+2)^2)/4 // simplifying the expression

Therefore, the statement is true for n = k+1, and by mathematical induction, the statement is true for all natural numbers n.

Hence, we have proven that 1^3 + 2^3 + 3^3 + ... + n^3 = (n^2(n+1)^2)/4 for all natural numbers n.

Your question is incomplete.

Complete question may be:

Use mathematical induction to prove that statement 1^3 + 2^3 + 3^3 + n^3 = (n^2(n+1)^2)/4 , ∀n∈N

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The requried, there are 10 million unique student ID numbers possible.

Since the first two digits are fixed as 86, we have 86XXXXXXX. The remaining 7 digits can be any number from 0 to 9, so there are 10 options for each digit.

Therefore, the total number of unique student ID numbers possible is:

10 * 10 * 10 * 10 * 10 * 10 * 10  = 10⁷ = 10,000,000

So there are 10 million unique student ID numbers possible.

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Answer: Ann would make $105 for 5 hours of work at the same rate.

Step-by-step explanation:

To find out how much Ann would make for 5 hours of work, we first need to determine her hourly wage. Divide her earnings by the number of hours she worked:

147 ÷ 7 = 21

Ann earns $21 per hour. Now, multiply her hourly wage by 5 hours to find out how much she would make for 5 hours of work:

21 × 5 = 105

Ann would make $105 for 5 hours of work at the same rate.

The value of BE is 42in

Similar triangles are triangles that have the same shape, but their sizes may vary. For two triangles to be similar the corresponding angles are congruent.

Also the ratio of the corresponding sides are equal.

This means that;

15/21 = 30/BE

represent BE by x

15/21 = 30/x

15x = 21×30

15x = 630

divide both sides by 15

x = 630/15

x = 42

Therefore the value of BE is 42 in

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The value of x is 128 degrees, given that two lines intersect at 140 degrees and (x+12) degrees and one of the angles formed by the intersection is (x+12) degrees.

If two lines intersect, the angle formed at the point of intersection is 180 degrees. We are given that the intersection of the two lines creates an angle of 140 degrees. Let's call the other angle formed by the two lines "y". Then we have:

140 degrees + y = 180 degrees

y = 180 degrees - 140 degrees

y = 40 degrees

Now, we are also given that one of the angles formed by the two lines is (x+12) degrees. We can set up an equation using this information:

x + 12 + y = 180 degrees

x + 12 + 40 degrees = 180 degrees

x + 52 degrees = 180 degrees

x = 180 degrees - 52 degrees

x = 128 degrees

Therefore, the value of x is 128 degrees.

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Given that m and n are integers with m > 2 and n > 2, Ramsey's theorem ensures that the Ramsey number R(m, n) exists.

Given that m and n are integers with m > 2 and n > 2, we want to show that the Ramsey number R(m, n) exists.

Ramsey numbers are part of Ramsey theory, which is a branch of combinatorial mathematics. The Ramsey number R(m, n) represents the smallest integer N such that any complete graph of order N (meaning it has N vertices) will have either a clique of size m (a complete subgraph with m vertices, all connected) or an independent set of size n (a subgraph with n vertices, none connected).

Ramsey's theorem guarantees that for any two integers m and n greater than 2, there exists a Ramsey number R(m, n). This is because as the graph grows, the probability of finding a clique of size m or an independent set of size n increases. Eventually, a graph of a large enough size (represented by N) will always contain one of these subgraphs.

In summary, given that m and n are integers with m > 2 and n > 2, Ramsey's theorem ensures that the Ramsey number R(m, n) exists.

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Answer:

the y-intercept of the line y = 7x - 12/7 is -12/7.

Step-by-step explanation:

The equation y = 7x - 12/7 is in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.

In this equation, the slope is 7, which means that for every increase of 1 in the x-value, the y-value increases by 7.

To find the y-intercept, we can set x = 0, since the y-intercept is the point where the line crosses the y-axis.

When x = 0, we have:

y = 7(0) - 12/7 = -12/7

Therefore, the y-intercept of the line y = 7x - 12/7 is -12/7.

Answer:

(0,-12/7)

Step-by-step explanation:

To find the y-intercept, substitute in 0 for x and solve for y.

have a great day and thx for your inquiry :)

The inequality holds for all positive integers k.Hence, by induction, we have proved that 2! · 4! · 6! · · · (2n)! ≥ ((n + 1)!)^n for every positive integer n.

We will prove the given inequality by induction.

Base case: For n = 1, we have 2! = 2 and (n+1)! = 2^2 = 4.

Therefore, (2!) ≥ ((1+1)!)^1 is true.Induction hypothesis:

Assume that the inequality holds for some positive integer k, i.e., 2! · 4! · 6! · · · (2k)! ≥ ((k + 1)!)^k.

Inductive step: We need to show that the inequality also holds for k + 1.We have: 2! · 4! · 6! · · · (2k)! · (2(k+1))! ≥ ((k + 1)!)^k · (2(k+1))!

Dividing both sides by (2k+2)(2k+1), we get: 2! · 4! · 6! · · · (2k)! · (2k+2)! / [(2k+2)(2k+1)] ≥ ((k + 1)!)^k · [(2(k+1)) / (2k+2)]

Simplifying the right-hand side, we get: ((k + 1)!)^k · [(2(k+1)) / (2k+2)] = [(k + 1)! / k!]^k · [(k+2) / (k+1)] = (k+2)^k

Substituting this expression and simplifying, we get: 2! · 4! · 6! · · · (2k)! · (2k+2)! / [(2k+2)(2k+1)] ≥ (k+2)^k

Simplifying the left-hand side, we get: 2! · 4! · 6! · · · (2k)! · (2k+2)! = [(2k+2)! / (2k+1)!] · [(2k)! / (2k-1)!] · [(2k-2)! / (2k-3)!] · · · [4! / 3!] · [2! / 1!]

= (2k+2)(2k+1)(2k)(2k-1) · · · 4 · 2

Therefore, we can write: (2k+2)(2k+1)(2k)(2k-1) · · · 4 · 2 / [(2k+2)(2k+1)] ≥ (k+2)^k

Simplifying further, we get: (2k)(2k-1) · · · 4 · 2 ≥ (k+2)^k

Using the induction hypothesis, we know that 2! · 4! · 6! · · · (2k)! ≥ ((k + 1)!)^k.

Therefore, we can write: 2! · 4! · 6! · · · (2k)! ≥ (k+1)^k

Multiplying both sides by (k+2)^k, we get:2! · 4! · 6! · · · (2k)! · (k+2)^k ≥ (k+1)^k · (k+2)^k

Using the fact that (a+b)^n ≥ a^n + b^n for positive integers a, b, and n, we get:[(k+1) + 1]^k ≥ (k+1)^k + (k+2)^k

Subtracting (k+1)^k from both sides, we get:

1 ≥ [(k+2) / (k+1)]^k

Since k is a positive integer, we know that (k+2)/(k+1) > 1, and therefore [(k+2)/(k+1)]^k > 1.

Therefore, the inequality holds for all positive integers k.Hence, by induction, we have proved that 2! · 4! · 6! · · · (2n)! ≥ ((n + 1)!)^n for every positive integer n.

Learn more about induction hypothesis here, https://brainly.com/question/24672369

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