PLEALSLE ANSWER FOR 70 POINTS
prove: the square of a number that is two more than a multiple of 3 is one more than a multiple of 3

The arithmetic sequence is solved and the nth term is aₙ = 5n - 4

Given data ,

The formula for the nth term of an arithmetic sequence is:

aₙ = aₙ1 + (n - 1)d

where a₁ is the first term, d is the common difference, and n is the term number.

In this case, a₁ = 1 and d = 5, so the formula becomes:

aₙ = 1 + (n - 1)5

Simplifying, we get:

aₙ = 5n - 4

Hence ,  the formula for the nth term of the arithmetic sequence with first term 1 and common difference 5 is aₙ = 5n - 4

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We can see here that converting 11/15 to decimal and a percent, we have:

Converting 15/11 to decimal and percent, we have:

A decimal is a number that uses the decimal system, which is a system of representing numbers using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, and a decimal point.

Thus, converting 11/15 to decimal and a percent:

Converting 15/11 to decimal and percent, we have:

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PART A: Linda has $4.41 more than Enrique after one year.

PART B: Enrique has $12.10 more than Linda after the second year.

PART A: After one year, we can calculate the amount of money each person has in their respective accounts.

For Linda's account:

Principal amount = $610

Interest rate = 1.2%

[tex]Interest $ earned = Principal $ amount \times (Interest rate/100) = $610 \times (1.2/100) = $7.32[/tex]

[tex]Total $ amount after one year = Principal amount + Interest earned = $610 + $7.32 = $617.32[/tex]

For Enrique's account:

Principal amount = $590

Interest rate = 3.9%

[tex]Interest $ earned = Principal amount \times (Interest rate/100) = $590 \times (3.9/100) = $22.91[/tex]

Total amount after one year = Principal amount + Interest earned = $590 + $22.91 = $612.91

Comparing the two amounts, after one year Linda has more money.

The difference in the amount is:

$617.32 - $612.91 = $4.41

Therefore, Linda has $4.41 more than Enrique after one year.

PART B: After a second year, we need to calculate the amounts again.

For Linda's account:

Principal amount = $617.32

Interest rate = 1.2%

[tex]Interest earned = Principal amount \times (Interest rate/100) = $617.32 \times (1.2/100) = $7.41[/tex]

Total amount after the second year = Principal amount + Interest earned = $617.32 + $7.41 = $624.73

For Enrique's account:

Principal amount = $612.91

Interest rate = 3.9%

[tex]Interest earned = Principal amount \times (Interest rate/100) = $612.91 \times (3.9/100) = $23.92[/tex]

Total amount after the second year = Principal amount + Interest earned = $612.91 + $23.92 = $636.83

Comparing the two amounts, after the second year Enrique has more money.

The difference in the amount is:

$636.83 - $624.73 = $12.10

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The solution is:

The area of the rectangular field is 2831.25 sq. yards.

The perimeter of the rectangular field is 153.1 yards

Here,

the length of the field =  62. 5 yards

The width of the field = 45.3 yards

PERIMETER OF A RECTANGLE = 2 (LENGTH + WIDTH)

⇒ The perimeter of the yard = 2 ( 62.5 + 45.3)

                                               = 2 x (107.8) = 153.1 yards

or the perimeter of the rectangular field is 153.1 yards

Area of a Rectangle = (LENGTH x WIDTH)

⇒ The area of the yard =  ( 62.5 x 45.3)

                                      = 2831.25 sq. yards

or the area of the rectangular field is 2831.25 sq. yards.

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complete question:

John has a rectangular-shaped field whose length is 62.5 yards and width is 45.3 yards.

The area of the field is square yards. The perimeter of the field is yards.

If a subset A of natural numbers (N) satisfies the conditions 0 ∈ A and if n ∈ A, then n+1 ∈ A, then A must be equal to the set of all natural numbers (N).

We will use a proof by contradiction to demonstrate this.

Assume that there exists a subset A of natural numbers that satisfies the given conditions, but A is not equal to N.

This means there exists a natural number k that is not in A.

Let B = N - A be the set of natural numbers that are not in A.

B is non-empty because it contains at least k.

Since 0 ∈ A, but k ∉ A, there must be a smallest natural number in B. Let's denote it as b.

Now, according to the given conditions, if n ∈ A, then n+1 ∈ A.

We can apply this repeatedly to obtain n+1, (n+1)+1, (n+1)+1+1, and so on.

Considering b-1, it is a natural number that is not in B because b is the smallest natural number in B. So, b-1 ∈ A.

Using the condition that if n ∈ A, then n+1 ∈ A, we can apply it repeatedly to b-1 to get (b-1)+1, (b-1)+1+1, and so on.

This implies that b, b+1, b+2, and so on, all belong to A.

However, this contradicts the fact that b is the smallest natural number in B, and A is defined such that it includes all natural numbers greater than or equal to 0.

Therefore, our assumption that A is not equal to N is false, and we can conclude that A = N.

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The composition (f o g)(x) = 10/(x+10) and its domain is all real numbers except x = -10.

a) To find (f o g) (x), we need to substitute g(x) into f(x) wherever we see x. That is,

(f o g) (x) = f(g(x)) = f(10/x) = (10/x) / (10/x + 1)

b) To find the domain of (f o g) (x), we need to look for any values of x that make the denominator of the expression equal to zero, since division by zero is undefined. Thus, we solve the equation:

10/x + 1 = 0

10/x = -1

x = -10

Therefore, the domain of (f o g) (x) is all real numbers except x = -10, since that value makes the denominator equal to zero. In interval notation, the domain is:

(-∞, -10) U (-10, ∞)

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Answer:

x=3

Step-by-step explanation:

12/8=x/2

12×2=8x

24=8x

x=24/8

x=3

Using the model, at 150°F they can sell 304 cups.

We can see that the line that models this passes through (65, 35) and (80, 75)

Then the slope of the line is:

a = (75 - 35)/(80 - 65)

a = 2.66

y = 2.66*x + b

To find the value of b, we can use the fact that y = 35 when x = 65

35 = 2*65 + b

35 - 2*65 = b

-95

y = 2.66*x -95

IF the tempererature is 150°F, then:

y = 2.66*150 - 95

y = 304

They can expect to sell 304 cups.

And no, the 150° don't make sense, that is equivalent to 66°C.

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That value being n = 2

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

Explanation:

The only even prime number is n = 2. This leads to n^2+3 = 2^2+3 = 7 which is also prime. So far we found one such value of n that fits the criteria.

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If n is odd, then n^2 is also odd.

The quick proof goes like this: let n = 2k+1 for some integer k. This guarantees n to be odd. Then n^2 = (2k+1)^2 = 4k^2+4k+1 = 2(2k^2+2k)+1 = 2*(some integer)+1 = some odd integer.

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

If n is prime and n isn't equal to 2, then n must be odd.

If n was even and n > 2, then it would be a multiple of 2 by definition, and would rule out it being prime. Example: n = 16 is not prime since 2 is a factor.

So n must be odd if we want to find more numbers to fit the given criteria. That makes n^2 being odd as well from the previous section's proof.

But then we use the idea that:

odd + odd = even

I'll leave the proof of this to the reader.

The sum of any two odd numbers gives an even number. An example would be 3+5 = 8.

There are no odd values of n that are prime and n^2+3 is prime.

We therefore can conclude there's only one value of n that is prime and n^2+3 is also prime. That value being n = 2.

Answer:

[tex]\frac{13}{145}[/tex]

The length of the side AB of the given triangle is 210.5cm.

Let us assume that AB = BC, then we can use the Sine Rule to find the length of AB.

Sine Rule states that:

[tex]\frac{a}{SinA}[/tex] = [tex]\frac{b}{SinB}[/tex] = [tex]\frac{c}{SinC}[/tex]

where a, b, and c are the lengths of the sides of the triangle, and A, B, and C are the angles opposite those sides.

In this case, we have:

a = BC = c

b = AC = 44

c = AB = x

B = 12°

A + B + C = 180 (sum of angles in a triangle)

but b = c, therefore angle A = angle C

A + 12 + C = 180°

2C + 12 = 180°

C = (180 - 12)/2

C = 84°

Using the Sine rule,

b/SinB = c/SinC

44/Sin 12° = x/Sin 84°

x = (44 Sin 84°)/Sin12

x = (44)(0.9945)/0.2079

x = 43.754/0.2079

x = 210.476

Therefore the side AB = 210.5cm

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The equation that matched the graph of the greatest interger function is y = [x] - 4. The correct option is (A).

To know the graph, we recall the equation of a straight line which is:

y = mx + c

where m is the gradient and

c is the intercept on the y-axis

Checking the graph, we can find the the intercept made by the straight line curve on the y-axis is -4.

Then we can have an equation of line written as:

y = mx + (-4) = mx - 4

From the options provided, there is only 1 with intercept of -4

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Matrices are widely used in various fields, including solving word problems in mathematics because they provide a convenient way to organize and manipulate data, making complex problem-solving more manageable.

In word problem solutions, matrices serve as valuable tools for organizing and manipulating data, simplifying complex problem-solving endeavors. For instance, when faced with a system of linear equations, matrices provide a structured representation that enables efficient computation.

Consider a word problem involving multiple variables and equations, such as determining quantities and prices of diverse items. By formulating the problem as a matrix equation and employing matrix operations like Gaussian elimination or matrix inverses, we can decipher the unknown variables and unlock solutions.

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The mean of the distribution in this problem is given as follows:

19.5625 pounds.

The mean of a data-set is given by the sum of all observations in the data-set divided by the cardinality of the data-set, which represents the number of observations in the data-set.

The dot plot shows the absolute frequency of each observation in the context of the problem.

Considering the dot plot, the sum of the observations in this problem is given as follows:

2 x 12 + 4 x 15 + 1 x 16 + 1 x 18 + 2 x 20 + 1 x 22 + 3 x 25 + 2 x 29 = 313.

The number of observations is given as follows:

2 + 4 + 1 + 1 + 2 + 1 + 3 + 2 = 16.

Hence the mean is given as follows:

313/16 = 19.5625 pounds.

The problem is given by the image presented at the end of the answer.

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Answer: add image

Step-by-step explanation:

If the workers' union at a particular university is quite strong.  the probability that exactly 4 of the workers interviewed are union members is 0.1699.

You must count all conceivable outcomes as well as favourable outcomes in order to compute probability. The number of favourable outcomes divided by the total number of possible outcomes yields the chance that an event will occur.

Using this formula to the probability

P(X = k) = (n choose k) * [tex]p^{k}[/tex] * [tex](1-p)^{n-k}[/tex]

Where:

X = random variable

n = 5

p = 0.96

k = 4

Let plug in the formula

P(X = 4) = (5 choose 4) *[tex]0.96^{4}[/tex] *[tex](1-0.96)^{5-4}[/tex]

= 5 *[tex]0.96^{4}[/tex] * 0.04

= 0.1699

Therefore the probability is 0.1699.

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Question 20:

the complex number in trigonometric form r(cos + i sin 8), with 8 in the interval [0". 360") is  3[cos 90+ i sin 90°).

Option B is correct.

Question 21.

The expression in the standard form a + bi.

[2(cos 15" + i sin 15°)] is [tex]3\sqrt{} 2 + 3\sqrt{2} i[/tex]

Option C is correct.

Question 22

The  complex roots of-21  is :

5√2 (cos 54° + i sin 54°), %√2 (cos 126° + i sin 126°),% √2 (cos 198° + i sin 198°), 5√2 (cos 270° + i sin 270°), 5√2 (cos 342° + i sin 342°).

Option B is correct.

Complex roots are described as  the imaginary roots of equations, which are represented as complex numbers.

The expression : 2(cos 15° + i sin 15°) = 2e^(i15°) can be written in standard from as

2e^(i15°) = 2(cos 15° + i sin 15°)

= 2cos 15° + 2i sin 15°

=2cos 15° + 2i sin 15°

=  [tex]3\sqrt{} 2 + 3\sqrt{2} i[/tex]

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Answer:

Initially, national security was defined as the government's ability to protect its citizens from military attacks. Today, this definition also includes other non-military areas such as defense against crime and terrorism, economic security, environmental security, food security, energy security and cyber security.

A quadrilateral with double arrows going in the same direction for one pair of the opposite sides is a Parallelogram, and it is the shape that has two pairs of parallel sides.

The shape that has two pairs of parallel sides is the third option: a quadrilateral with double arrows going in the same direction for one pair of the opposite sides.

This shape is called a parallelogram, which is a type of quadrilateral where opposite sides are parallel and congruent. In a parallelogram, each pair of opposite sides has the same length and the same arrow direction, which is what is represented by the double arrows in the description.

The other options do not fit the definition of a parallelogram. The first option describes a shape with four right angles, but only two pairs of parallel sides, which would make it a trapezoid. The second option describes a shape with only one pair of parallel sides, which is not enough to be a parallelogram. The fourth option describes a shape with sides of different lengths and in the shape of an arrow, which does not fit the definition of a parallelogram.

Parallelograms have many interesting properties, such as opposite angles being congruent and diagonals bisecting each other. They are also the basis for other geometric shapes, such as rectangles and rhombuses, which have additional properties due to their specific angles and side lengths.

A quadrilateral with double arrows going in the same direction for one pair of the opposite sides is a parallelogram, and it is the shape that has two pairs of parallel sides.

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The length of the hypotenuse of the triangle is b = x√2 units

Given data ,

Let the triangle be represented as ΔABC

And , the triangle is a 45 - 45 - 90 triangle

So , the two legs are congruent to one another and the non-right angles are both equal to 45 degrees

And , the sides are in the proportion x : x : x√2

Now , the length of the sides of the triangle are

The measure of base BC = a units = x

The measure of height AB = c units = x

So , the measure of the hypotenuse of the triangle is = x√2 units

Hence , the triangle is solved and hypotenuse AC = x√2 units

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Hannah needs to invest approximately $3173.79 now to have $6500 in 10 years at an interest rate of 7% compounded continuously.

Use the formula for continuous compounding to solve this problem:

[tex]A = Pe^{(rt)[/tex]

where A is the future value, P is the present value (the amount Hannah needs to invest), r is the annual interest rate (as a decimal), and t is the time in years.

We want to find P, so we can rearrange the formula as:

[tex]P = \dfrac{A}{e^{(rt)}}[/tex]

Substituting the given values, we get:

[tex]P = \dfrac{6500}{e^{(0.07\times10)}}[/tex]

P≈ $3173.79

Therefore, Hannah needs to invest approximately $3173.79 now to have $6500 in 10 years at an interest rate of 7% compounded continuously.

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The zeros of the polynomial p(x) = x³ + x² + 8x - 60 will be (- 2 + √4 i), (- 2 - √4 i) and 3.

Given that:

Polynomial, p(x) = x³ + x² + 8x - 60

A polynomial expression is an algebraic expression with variables and coefficients. Unknown variables are what they're termed. We can use addition, subtraction, and other mathematical operations. However, a variable is not divisible.

At x = 3, then we have

p(3) = 3³ + 3² + 8(3) - 60

p(3) = 60 - 60

p(3) = 0

The one zeros are 3. Then the other two zeros are given as,

(x³ + x² + 8x - 60) / (x - 3) = 0

[(x - 3)(x² + 4x + 20)] / (x - 3) = 0

x² + 4x + 20 = 0

x = [-4 ± √(4² - 4 × 1 × 20)] / (2 × 1)

x = - 2 ± √4 i

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a.) The number of students that took the Algebra 1 for the whole three years = 672

b.) The empirical probability that Sharon will receive an A would be =0.061

For question a.)

To calculate the total number of students that took the Algebra 1 for the 3 years, the students that got the various grades are being added up. That is;

Grade A= 41

Grade B =183

Grade C =265

Grade D =96

Grade F =85

Incomplete = 2

Total = 41+183+265+96+85+2 = 672

For b.)

The probability = possible outcome/sample space

possible outcome = 41/672=0.061

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Answer:

231002 = (2340)25 · 232 ≡ 125 · 529 = 529 ≡ 37 mod 41.

The area of the rectangle is 1632/3 square feet.

To calculate the area, we need to multiply the length and width of the given rectangle. The length is 51 feet, and the width is 10 2/3 feet, which can be expressed as an improper fraction as 32/3 feet.

Multiplying these values, we have:

Area = length × width

= 51 ft × (32/3) ft

= (51 × 32)/(3)

= 1632/3 square feet

So, the area of the rectangle is 1632/3 square feet.

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Decentralization is a management approach that involves the delegation and assignment of decision-making rights from top levels of management to lower levels of management. Option A

It is an effective way to empower employees and to improve decision-making processes within an organization. The benefits of decentralization include the following:

It empowers more employees at lower levels of management: Decentralization allows employees to take ownership of their work, to be more accountable, and to make decisions that are in the best interest of the organization. This, in turn, leads to improved job satisfaction, employee engagement, and better overall performance.

It allows for better and more timely decision-making: Decentralization ensures that decisions are made closer to where the work is being done. This leads to faster decision-making, improved responsiveness to changes in the market, and better coordination between different departments and teams.

It trains future managers: Decentralization provides employees with the opportunity to gain valuable management experience, which can help prepare them for future leadership roles within the organization.

However, one benefit of decentralization that is not accurate is that it forces top levels of management to focus on individual units. In fact, decentralization can free up top management to focus on the big picture and long-term strategy, rather than getting bogged down in day-to-day operations.

In conclusion, decentralization is a useful management approach that can help organizations improve decision-making, empower employees, and train future managers.

By delegating decision-making rights to lower levels of management, organizations can improve overall performance and become more adaptable to changes in the market.

So Option A is correct.

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complete question:

The benefits of decentralization i.e. delegation and assignment of decision rights include all of the following except:

a it forces top levels of management to focus on individual units

b it empowers more employees at lower levels of management

c it allows for better and more timely decision making

d it trains future managers

Answer:

24

Step-by-step explanation:

As x is now equal to 5, all you have to do is substitute it into were x is in the f(x) formula given.

x^2-1 turns into (5)^2-1=

25-1=

24

The exponential model for the population of the city t years after 2010 is given by:

The population in 2020 can be estimated to be 187,000 × 1.01^10 = 200,270, which is rounded to 200,000.

[tex]\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{initial amount}\dotfill &187000\\ r=rate\to 1\%\to \frac{1}{100}\dotfill &0.01\\ t=years \end{cases} \\\\\\ A = 187000(1 + 0.01)^{t} \implies A=y = 187000(1.01)^t \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{in 2020 is 10 years later}}{y=187000(1.01)^{10}}\implies y\approx 207000[/tex]