{an}={((-1)^(n+1))/n} please how do you prove if this is convergent or divergent?

The area of a rectangle is given by the formula:

Area = length x width

In this case, the length of the park is x + 5 and the width is x - 2. So, we can substitute these expressions into the formula to get:

Area = (x + 5) x (x - 2)

Expanding the product, we get:

Area = x^2 + 5x - 2x - 10

Simplifying, we have:

Area = x^2 + 3x - 10

Therefore, the area of the rectangular park is given by the expression x^2 + 3x - 10.

The functiοn is increasing and has a dοmain οf (1, ∞) is οptiοn D f(x) = lοg (x - 1) + 1.

Every equatiοn with οne οr mοre ratiοnal expressiοns is said tο be ratiοnal. A fractiοn whοse numeratοr and denοminatοr are pοlynοmials is knοwn as a ratiοnal expressiοn. A ratiο οf twο pοlynοmials is, in οther wοrds, a ratiοnal expressiοn.

The LCD (Least Cοmmοn Denοminatοr) οf the fractiοns, eliminatiοn οf the denοminatοrs, and simplificatiοn οf the resultant equatiοn are methοds fοr sοlving ratiοnal equatiοns. Sοlving the resultant equatiοn—which can require factοring, simplificatiοn, οr the use οf οther algebraic strategies—will prοvide the answers tο a ratiοnal equatiοn.

Fοr the given functiοn:

Optiοn A:

f(x) = -lοg(x - 1) + 2

Here, lοg is negative, hence fοr increasing value οf x f(x) will decrease.

Optiοn B:

f(x) = -lοg(x - 2) + 1

Here, lοg is negative, hence fοr increasing value οf x f(x) will decrease.

Optiοn C:

f(x) = lοg(x - 2) + 1

Here, (x - 2) > =

x > 2

The dοmain dοes nοt satisfy.

Optiοn D:

f(x) = lοg (x - 1) + 1

Here, (x - 1) > 0

x > 1 tο infinity.

Hence, the functiοn is increasing and has a dοmain οf (1, ∞) is οptiοn D f(x) = lοg (x - 1) + 1.

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The complete function is:

The value of (f o g)(3) using composite function is 35

Composite function refers to the combining of functions in a manner where the output from one function becomes the input for the next function.

To calculate the composite function (f o g)(3), we use the techniques below.

From the question,

Given:

Note that,

Step 1: Find g(3)

Therefore,

Step 2: To find f(g)(3), we subetitute the value of x in f(x) by 12

Hence, (f o g)(3) is 35

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The probability that the contestant wins at least one car is 13/35.

A television game show has 15 doors, of which the contestant must pick 3. Behind 3 of the doors are expensive cars, and behind the other 12 doors are consolation prizes

probability of an event =total favorable outcome for event/total possible outcomes

P(contestant wins at least one car) will be:

= 1 - 12/15 × 11/14

=13/35

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Jane won 90% of the video games. The solution has been obtained by using the arithmetic operations.

What are arithmetic operations?

The four fundamental operations, sometimes known as "arithmetic operations," are believed to adequately characterise all real numbers. Division, multiplication, addition, and subtraction are followed by the mathematical operations quotient, product, sum, and difference.

We are given that Jane won 9 out of the 10 video games she played.

Let the percentage be x.

Using the multiplication operation, we get,

⇒ 10 * x% = 9

⇒ 10 * [tex]\frac{x}{100}[/tex] = 9

⇒  [tex]\frac{x}{10}[/tex] = 9

⇒ x = 90

Hence, Jane won 90% of the video games.

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Since, there are multiple questions so, the question answered above is:

Question: Jane won 9 out of the 10 video games she played. What percent of the video games did Jane win?

Triangle 1:
A = 45 degrees, C = 40 degrees, c = 4.2
First, we need to find angle B.
B =[tex] 180 - (A + C) = 180 - (45 + 40) = 95[/tex] degrees

Now, we can use the Law of Sines to find side b.
[tex]b/sin(B) = c/sin(C)[/tex]
b =[tex] (c * sin(B)) / sin(C)[/tex]
b =[tex] (4.2 * sin(95)) / sin(40)[/tex]
b ≈ [tex]5.6[/tex]

Triangle 2:
N = 45 degrees, M = 180 - N = 135 degrees, m = 2
To find side n, we can use the Law of Sines again.
[tex]n/sin(N) = m/sin(M)[/tex]
n =[tex] (m * sin(N)) / sin(M)[/tex]
n =[tex] (2 * sin(45)) / sin(135)[/tex]
n ≈ [tex]1.4[/tex]

So, the solutions for the two triangles are:
Triangle 1: A = 45 degrees, B = 95 degrees, C = 40 degrees, b ≈ 5.6, c = 4.2
Triangle 2: N = 45 degrees, M = 135 degrees, n ≈ 1.4, m = 2

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Answer:

x = 4,5

Step-by-step explanation:

These two triangles are similar (according to 3 sides):

The three sides of one triangle are proportional to the three sides of the other triangle:

[tex] \frac{7.5}{5} = \frac{6}{4} = \frac{x}{3} [/tex]

[tex] \frac{6}{4} = \frac{x}{3} [/tex]

Use the property of proportion to find x (cross-multiply):

[tex]4x = 6 \times 3[/tex]

[tex]4x = 18[/tex]

Divide both sides by 4 to make x the subject:

[tex]x = 4.5[/tex]

The value of z that lies in a given area through which the given complex number expression is satisfied is z = 1.6

What about complex number?

A complex number is a number that can be expressed in the form a + bi, where "a" and "b" are real numbers and "i" is the imaginary unit, which is defined as the square root of -1. The real part of a complex number is "a" and the imaginary part is "b".

Complex numbers are important in many areas of mathematics, science, and engineering. They are used to represent and manipulate quantities that involve both real and imaginary components, such as alternating currents in electrical engineering and wave functions in quantum mechanics. Complex numbers also have a geometric interpretation, where they can be represented as points in the complex plane, which is a two-dimensional coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

Define area:

Area is a measure of the size of a two-dimensional region or surface. It is usually measured in square units, such as square meters, square centimeters, or square feet.

The area of a shape can be calculated using different formulas depending on the shape's characteristics.

According to the given information:

⇒ Here,  the z scores be z1 and z2

⇒ In which Z1 = 0.5 - 0.8904/2 = 0.0548

⇒ And,       Z2 = 0.5 + 0.8904/2 = 0.9452

⇒ Using standard normal distribution table,

⇒ We, have that, Z1 = -1.6

⇒ Z2 = 1.6

⇒ So, the value of Z = 1.6

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Answer:

Question 1 C. 25 1/3

Question 2 C. 19

Step-by-step explanation:

brief :

1. 6 1/3 = 19/3

19/3 * 4 = 76/3

76/3 = 25 1/3

To solve this problem, we need to use the information provided and do some simple arithmetic.

Question 1:

The candy store sold 6 1/3 pounds of white chocolate. Since it sold 4 times as much milk chocolate as white chocolate, we can find the amount of milk chocolate sold by multiplying 6 1/3 by 4:

6 1/3 * 4 = 25 1/3

Therefore, the candy store sold 25 1/3 pounds of milk chocolate last week.

Answer: C. 25 1/3

Question 2:

To find the difference between the amount of milk chocolate and white chocolate sold, we can subtract the amount of white chocolate from the amount of milk chocolate:

25 1/3 - 6 1/3 = 19

Therefore, the candy store sold 19 more pounds of milk chocolate than white chocolate.

Answer: C. 19

chatgpt

1)The number of milk chocolate that sold last week is = [tex]25\frac{2}{3}[/tex] B is correct

2)white chocolate at the candy store last week is 19.

Last week, a candy store sold [tex]5\frac{1}{3}=\frac{16}{3}[/tex] pounds of white chocolate. It sold 4 times as much milk chocolate as white chocolate.

The number of milk chocolate did the candy store sell last week.

Since the candy store sold 4 times as much milk chocolate as white chocolate, it sold:

[tex]4 *\frac{ 19}{3 } =\frac{ 76}{3}\\ pounds\ of\ milk\ chocolate.= 25 \frac{2}{3}[/tex]

option B is correct.

2)

To find out how many more pounds of milk chocolate were sold than white chocolate at the candy store last week, we can subtract the amount of white chocolate sold from the amount of milk chocolate sold:

[tex]\frac{76}{3} - \frac{19}{3} \\= \frac{57}{3}\\ = 19\ pounds.[/tex]

option C is correct

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The value of j is equal to 9√3 meters.

In Mathematics and Geometry, a 30-60-90 triangle is also referred to as a special right-angled triangle and it can be defined as a type of right-angled triangle whose angles are in the ratio 1:2:3 and the sides are in the ratio 1:√3:2.

This ultimately implies that, the length of the hypotenuse of a 30-60-90 triangle is double (twice) the length of the shorter leg (adjacent side), and the length of the longer leg (opposite side) of a 30-60-90 triangle is √3 times the length of the shorter leg (adjacent side).

Note: j is the side opposite to the 60° angle.

The side opposite to the 60° angle, j = 9 × √3

The side opposite to the 60° angle, c = 9√3 meters.

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[tex]~~~~~~ \textit{Simple Interest Earned Amount} \\\\ A=P(1+rt)\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$9000\\ r=rate\to 4\%\to \frac{4}{100}\dotfill &0.04\\ t=years\dotfill &1 \end{cases} \\\\\\ A = 9000[1+(0.04)(1)] \implies A = 9360[/tex]

Therefore, only the points (4, 1) and (3, 8) can be used to find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem.

In mathematics, a coordinate is a set of numbers that uniquely determines the position of a point or an object in space. The most common coordinate system is the Cartesian coordinate system, which uses two or three numerical values to locate points on a plane or in space.

Here,

To find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem, we need to use the coordinates of these two points to form a right-angled triangle, with the line segment connecting the two points as its hypotenuse. Therefore, the points that can be used to find the distance between the points (4, 1) and (3, 8) using the Pythagorean theorem are:

(4, 1) and (3, 8)

The point (0, 0) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.

The point (4, 8) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.

The point (4, 3) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.

The point (1, 8) cannot be used to find the distance between the two given points using the Pythagorean theorem because it is not on the line segment connecting the two points.

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Step-by-step explanation:

I = E/R    multiply both sides of the equation by R

I R = E / R  * R

I R = E     Now divide both sides by I

R  = E/ I

Hence, by multiplying the number of airports in California by a factor of three-quarters, we may get the number of airports in Georgia.

To find how many airports Georgia has, we need to use division. We can divide the total number of airports in California by 3 to find the number of airports in Georgia.

The key math word in the problem is "three times."

Let's say the number of airports in California is "C". Then, we can represent the number of airports in Georgia as "G," and we know that:

C = 3G

To solve for G, we can divide both sides of the equation by 3:

C/3 = G

Therefore, the number of airports in Georgia is equal to one-third of the number of airports in California.

The key math word in the problem is "three times."

We need to divide in order to determine the number of airports in Georgia. The number of airports in Georgia can be calculated by dividing the total number of airports in California by 3.

Let's say the number of airports in California is "C". Then, we can represent the number of airports in Georgia as "G," and we know that:

C = 3G

To solve for G, we can divide both sides of the equation by 3:

C/3 = G

Therefore, the number of airports in Georgia is equal to one-third of the number of airports in California.

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The dilation of the black figure to blue figure is a reduction with a scale factor of 0.5

A dilation is a transformation that produces an image that is the same shape as the original, but is a different size.

It can either be an enlargement or a reduction.

An enlargement means a scale factor greater than 1 while a reduction is a scale factor less than 1.

Thus, this dilation is a reduction.

Coordinates of initial figure are:

(-8, 10), (10, 10), (8, -8), (2, -8)

Coordinates of dilated figure are:

(1, -4), (4, -4), (-4, 5), (5, 5)

Thus, it is clear that the scale factor is 0.5

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To find the total units of water collected for day 3, we can simply add up the units of water collected on each day, starting with the 1 unit of water we already have:

Day 1: 1 unit of water

Day 2: 1 unit of water

Day 3: 1 unit of water

1. Total units of water collected for day 3 = 1 + 1 + 1 = 3 units of water

2. Since we only need to consume 1 unit of water per day to survive, we would have consumed a total of 3 units of water in 3 days.

3. To find the amount of water leftover, we can subtract the units of water consumed from the total units of water collected:

Total units of water collected = 3 units

Units of water consumed = 3 units

Water leftover = Total units of water collected - Units of water consumed

= 3 - 3

= 0 units of water leftover

4. We could collect water in a number of ways, depending on what resources were available on the island. We could collect rainwater using containers, or we could find a freshwater stream or well and collect water from there. We could also try to collect dew or moisture using a cloth or other absorbent material.

5. The tools we would use to collect water would depend on the method we chose. For example, if we were collecting rainwater, we would need containers to catch the water, such as buckets or barrels. If we were collecting water from a stream or well, we would need a way to transport the water back to our shelter, such as a jug or canteen. If we were collecting moisture, we would need a cloth or other absorbent material to collect the water.

D. Reflection across the y-axis, translation 4 units to the right. sequence of transformation is most likely applied to figure QRSTU to form Q'R'S'T'U'.

What is transformation?

To determine the sequence of transformations applied to figure QRSTU to form Q'R'S'T'U', we can compare the corresponding vertices of the two pentagons:

Q and Q': Q' is to the right of Q and has a positive x-coordinate.

R and R': R' is to the left of R and has a negative x-coordinate.

S and S': S' is above S and has a positive y-coordinate.

T and T': T' is above T and has a positive y-coordinate.

U and U': U' is to the right of U and has a positive x-coordinate.

Based on these comparisons, it appears that the figure has been reflected across the y-axis and then translated 4 units to the right. Therefore, the most likely sequence of transformations applied to QRSTU to form Q'R'S'T'U' is:

D. Reflection across the y-axis, translation 4 units to the right.

What are vertices?

A vertex (plural: vertices) is a point where two or more line segments or curves meet. Vertices are the corner points of a polygon or the endpoints of a line segment or ray. In three-dimensional geometry, vertices are the corner points of a polyhedron or the endpoints of an edge.

Vertices are important in defining the properties of a shape, such as its angles, sides, and symmetry. They are also used in graph theory to represent points in a graph or network.

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Complete question is: D. Reflection across the y-axis, translation 4 units to the right. sequence of transformation is most likely applied to figure QRSTU to form Q'R'S'T'U'.

The test statistic is less than the critical value, so, we reject the null hypothesis and conclude that there is a statistically significant difference between the population means.

A two-sample t-test is used to compare the means of two samples to determine if there is a statistically significant difference between them.

To answer this question, we must conduct a two-sample t-test.

In this case, the null hypothesis is that there is no difference between the population means, and the alternative hypothesis is that there is a difference between the population means.

The test statistic is calculated using the following formula:

t = (x₁ - x₂) / (√((s₁²/n₁) + (s₂²/n₂)))

Where x₁ and x₂ are the sample means of the two samples, s₁ and s₂ are the sample standard deviations, and n₁ and n₂ are the sample sizes.

Plugging in the sample data, we get:

t = (22 - 26) / (√((4.3²/11) + (4.4²/8)))

= -1.98

The critical value of t for a two-tailed test at the 0.01 significance level is 2.58.

Since the test statistic is less than the critical value, we reject the null hypothesis and conclude that there is a statistically significant difference between the population means.

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Question:

A random sample of 11 observations from one population revealed a sample mean of 22 and a sample standard deviation of 4.3. A random sample of 8 observations from another population revealed a sample mean of 26 and a sample standard deviation of 4.4. At the 0.01 significance level, is there a difference between the population means?

The width of the added strip is 10 meters.

What is Area?

Area is a measure of the amount of surface enclosed by a two-dimensional shape or figure. It is usually expressed in square units, such as square meters or square inches, and is calculated by multiplying the length of a side or a diameter by itself or by using an appropriate formula for the shape.

If the area of the skating rink is doubled by adding a strip of equal width along one side and another strip at one end, then the new dimensions of the skating rink would be

=>  (30 + x) × (20 + x),

where x is the width of the added strip.

According to the problem, the new area of the skating rink is double the original area, so we can set up the following equation:

=> (30 + x)(20 + x) = 2(30 ×20)

Expanding the left-hand side and simplifying, we get:

=> 600 + 50x + x² = 1200

Rearranging and solving for x, we get:

=> x² + 50x - 600 = 0

Factoring the left-hand side, we get:

=> (x + 60)(x - 10) = 0

Therefore, the possible values for x are x = -60 and x = 10. Since the width of the strip cannot be negative, we reject x = -60 and conclude that the width of the added strip is 10 meters.

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Answer: Let's assume that the width of the rectangular room is "x" meters. Then, according to the problem, the length of the room is 2 times the width, which means the length is "2x" meters.

The perimeter of the room is the sum of all four sides, so we can write:

Perimeter = 2 × (length + width)

Substituting the values, we get:

48 = 2 × (2x + x)

Simplifying the expression, we get:

48 = 2 × 3x

Dividing both sides by 2, we get:

24 = 3x

Solving for "x", we get:

x = 8

Therefore, the width of the room is 8 meters and the length is 2 times the width, which is 2 × 8 = 16 meters.

So the dimensions of the room are 8 meters by 16 meters.

Step-by-step explanation:

The maximum profit of manufacturing company is  = $3150

If an equation is in ax + b = c and there is only one variable in the equation then it is called linear equation in one variable. a , b , c are real number and a is not equal to 0.

To maximize  profit, we must determine the number of trick  and slalom skis to produce each day, subject to the constraints of available labor hours.

Let x be the number of  skis and y be the number of slalom skis produced per day.

 The objective function to be maximized is:

Profit = 60x50 a

With the following limitations:

Production hours: 9x 6 a ≤ 432

Final hours: x y ≤ 63

Non-negativity constraints: x ≥ 0, y ≥ 0

We can solve this linear programming problem using graphical or algebraic methods. Here we  use algebraic methods:

From the second limit, we get y ≤ 63 - x.

If this inequality is replaced by the first limit, we get:

9 x 6 (63 - x) ≤ 432

3x ≤ 54

x ≤ 18

This means that the maximum production rate is 18 pieces per day. Replacing x = 18 by the inequality y ≤ 63 - x, we get:

y ≤ 63 – 18 = 45

So the maximum production is 45 slalom skis  per day.

 To verify that this is the correct solution, we need to ensure that it satisfies the non-negativity constraints:

x ≥ 0 and y ≥ 0 are already given in the assignment statement.  

Therefore, the maximum profit is:

Profit = 60(18) 50(45) = $3150

The biggest profit is when 18 trick skis and 45 slalom skis are made per day.

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Answer:

Step-by-step explanation:

5 + x (where x represents the number)

x / (-2)

2x

x - 7

12x

x - 3

$15h (where h represents the number of hours worked)

C - $5 (where C represents the cost of the ticket)

x + 24

50 / n (where n represents the number of students)

(2/5)x

(1/2)x - 19

-4x - 11

8x + 5

x^2 - 14

(3/4)x + 17

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

Five added to a number

The result of dividing a number by -2

Two times a number

A number decreased by 7

The result of multiplying a number by 12

A number decreased by 3

Earning $15 per hour

The cost of the ticket minus $5

The sum of a number and 24

Dividing 50 markers among several students

Two-fifths times a number

Half of a number minus 19

The result of subtracting 11 from the product of a number and -4

The sum of eight times a number and five

The difference between the square of a number and 14

Three-fourths times a number plus 17

Answer:

x = 4 is a straight, vertical line at 4 on the x axis.

The value of y for the given quadrilateral is 5.

A closed quadrilateral has four sides, four vertices, and four angles. It is a form of polygon. In order to create it, four non-collinear points are joined. Quadrilaterals always have a total internal angle of 360 degrees.

The Latin words quadra, which means four, and latus, which means sides, are combined to form the English term quadrilateral. A quadrilateral does not always have to have equal lengths on each of its four sides. As a result, depending on the sides and angles, we might have several sorts of quadrilaterals.

We know that the opposite sides of the quadrilateral have a sum of 180 degrees.

Thus,

23y + 13y = 180

36y = 180

y = 5

Hence, the value of y for the given quadrilateral is 5.

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Answer:

[tex]\frac{12}{35}[/tex]

Step-by-step explanation:

[tex]\frac{2}{5}[/tex] x [tex]\frac{6}{7}[/tex] = [tex]\frac{12}{35}[/tex]

[tex]\frac{12}{35}[/tex] is already in simplest form, so the answer is [tex]\frac{12}{35}[/tex]

The equation of the line passing through points [tex]A(a,b)[/tex] and [tex]B(b,a)[/tex] is [tex]x + y = a + b[/tex]  , and the length of   [tex]AB[/tex] is  [tex]\sqrt(2(a - b)^2).[/tex]

To find the equation of the line passing through points A(a,b) and B(b,a), we can first find the slope of the line. using the two given points, and then use the point-slope form of a line equation.

The slope of a line passing through two points ([tex]x_{1} , y_{1}[/tex]) and ([tex]x_{2}, y_{2}[/tex]) is given by:

[tex]m = (y_{2} - y_{1} ) / (x_{2} - x_{1} )[/tex]

Applying this formula, we can find the slope of the line passing through points A(a,b) and B(b,a):

Slope [tex](m) = (a - b) / (b - a)[/tex]

Since a and b can take any real values, we can see that the slope of the line passing through points A and B is -1.

Now, let's find the equation of the line passing through points A and B using the point-slope form of a line equation:

Point-slope form: [tex]y - y_{1} = m(x - x_{1} )[/tex]

Substituting the slope (-1) and the coordinates of point A(a,b) into the point-slope form, we get:

[tex]y - b = -1(x - a)[/tex]

Expanding the equation, we get:

[tex]y - b = -x + a[/tex]

Rearranging the terms, we get the equation of the line passing through points A(a,b) and B(b,a):

[tex]x + y = a + b[/tex]

Now, let's find the length of AB using the distance formula:

The distance between two points [tex](x_{1} , y_{1} )[/tex] and  [tex](x_{2} , y_{2} )[/tex] is given by:

[tex]d = \sqrt((x_{2} - x_{1} )^2 + (y_{2} - y_{1} )^2)[/tex]

Applying this formula, we can find the length of AB using the coordinates of points A(a,b) and B(b,a):

Length of  [tex]AB (d) = \sqrt((b - a)^2 + (a - b)^2)[/tex]

Simplifying the expression, we get:

Length of [tex]AB (d) = \sqrt(2(a - b)^2)[/tex]

Therefore, The equation of the line passing through points [tex]A(a,b)[/tex] and [tex]B(b,a)[/tex] is [tex]x + y = a + b[/tex]  , and the length of   [tex]AB[/tex] is  [tex]\sqrt(2(a - b)^2).[/tex]

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