Is the following an example of theoretical probability or empirical probability? A fisherman notes that eight out of ten times that he uses a certain lure he catches a fish within an hour. He concludes that the probability that the lure will catch a fish on his fishing next trip is about 80%

There are 56 possible permutations of 8 items taken 2 at a time. To express the number of permutations of 8 items taken 2 at a time, we use the notation P(8,2) or sometimes written as 8P2.

Permutations are arrangements of items in a specific order where the order matters.
To calculate the number of permutations, we use the formula:
P(n, r) = n! / (n - r)!
In our case, n is the total number of items (8) and r is the number of items taken at a time (2). Using the formula:
P(8, 2) = 8! / (8 - 2)!
= 8! / 6!
Now, let's calculate the factorials:
8! = 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1
6! = 6 × 5 × 4 × 3 × 2 × 1

Divide 8! by 6!:
P(8, 2) = (8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (6 × 5 × 4 × 3 × 2 × 1)
Notice that the 6!, which is (6 × 5 × 4 × 3 × 2 × 1), cancels out from both the numerator and the denominator, leaving us with:
P(8, 2) = 8 × 7
P(8, 2) = 56
So, there are 56 possible permutations of 8 items.

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Answer: For the extra large box, the scale factor is 4. The volume of the single serving box is:

V1 = l x w x h = 1.2 cubic inches

The volume of the extra large box, which is a dilation of the single serving box with a scale factor of 4, is:

V2 = (4l) x (4w) x (4h) = 64(l x w x h) = 64V1

So the extra large box will contain:

1.2 x 64 = 76.8 ounces of cereal.

For the family size box, the volume is 18.75 ounces, which is 15.6 times larger than the single serving box. The scale factor is the cube root of this ratio:

scale factor = (18.75/1.2)^(1/3) = 2.5

So the family size box will be a dilation of the single serving box with a scale factor of 2.5.

For the soup boxes, the largest box is a dilation of the smallest box with a scale factor of 2. The volume of the smallest box is 15 cubic inches, so the volume of the largest box is:

V2 = (2l) x (2w) x (2h) = 8(l x w x h) = 8 x 15 = 120 cubic inches.

We can choose any dimensions that satisfy this volume requirement. For example, we could use:

The volume of this box would be:

V = 10 x 3 x 4 = 120 cubic inches

Therefore, a set of dimensions for the largest soup box is 10 inches by 3 inches by 4 inches, or any other dimensions that satisfy the volume requirement of 120 cubic inches.

The  probability that in a random sample of 45 people who took the remedy, 40 of them had a reduction of symptoms within a day is approximately 0.674, rounded to three decimal places.

This problem can be solved using the binomial distribution.

Let X be the number of people out of 45 who have a reduction of symptoms within a day. Then X follows a binomial distribution with n = 45 and p = 0.88.

The probability of exactly 40 people having a reduction of symptoms is given by:

P(X = 40) = (45 choose 40) * (0.88)^40 * (1-0.88)^(45-40)

We can use a calculator or software to evaluate this expression. Alternatively, we can use the normal approximation to the binomial distribution, since n is large and p is not too close to 0 or 1.

Using the normal approximation, we can approximate X as a normal distribution with mean mu = n*p = 39.6 and standard deviation sigma = sqrt(n*p*(1-p)) = 1.964.

Then the probability of X being less than or equal to 40 can be approximated by the cumulative distribution function of the standard normal distribution:

P(X <= 40) = P((X - mu)/sigma <= (40.5 - 39.6)/1.964) = P(Z <= 0.458)

Using a standard normal table or software, we can find that P(Z <= 0.458) is approximately 0.6736.

Therefore, the probability that in a random sample of 45 people who took the remedy, 40 of them had a reduction of symptoms within a day is approximately 0.674, rounded to three decimal places.

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Comparing statistics from different stores can provide valuable insights into price variations and help make informed purchasing decisions.

When comparing statistics from different stores, it's important to look at a variety of factors, such as the average price, the range of prices, and any discounts or promotions being offered. It can also be helpful to look at historical trends and compare prices over time to identify any patterns or changes in pricing. Additionally, it's important to consider other factors that may influence purchasing decisions, such as the quality of the product, the reputation of the store, and the availability of customer support or warranties. By carefully comparing these factors and weighing the pros and cons of each store, consumers can make informed decisions and maximize the value of their purchases.

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What statistical measures can Braden use to compare the prices of DVD players from two different stores?

Answer: An upward shift of 4 units.

Step-by-step explanation:

The transformation of the function F(x) = 3x to F(x) = 3x + 4  indicates a vertical translation or shift of the graph of f(x) upward by 4 units.

More specifically, the original function f(x) = 3x has a y-intercept at (0,0) and a slope of 3, meaning that for every 1 unit of increase in x, there is a corresponding increase of 3 units in y. Adding 4 to the function shifts all the y-values up by 4 units, so the new function f(x) = 3x + 4 has a y-intercept at (0,4) and a slope of 3, which means that for every 1 unit of increase in x, there is a corresponding increase of 3 units in y, starting from the new y-intercept at (0,4).

If you meet all the criteria for an independent t-test but do not satisfy the assumption of normality, you should technically do a non-parametric test instead of an independent t-test.

So, the correct answer is D.

Non-parametric tests do not require the data to be normally distributed, unlike parametric tests such as t-tests. Non-parametric tests include the Mann-Whitney U test, Wilcoxon rank-sum test, and Kruskal-Wallis test.

These tests are based on ranks rather than the actual values of the data and are generally less powerful than parametric tests, meaning that they may require larger sample sizes to achieve the same level of statistical significance.

It is important to choose the appropriate statistical test for your data to ensure accurate results and conclusions. Simply redoing the experiment or ignoring the violation of the normality assumption is not a valid solution.

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A hypothesis is a tentative explanation for a phenomenon that is based on limited evidence and requires further investigation to be confirmed or rejected. When a researcher conducts a study, they formulate a hypothesis that they seek to test through their research.

In scientific research, a researcher aims to disprove or reject their hypothesis rather than prove it. This is because it is impossible to prove a hypothesis with complete certainty, but it is possible to gather evidence that contradicts it. By trying to disprove or reject their hypothesis, a researcher can either find evidence to support their hypothesis or identify flaws in their reasoning that they can address in future research.

For example, a researcher might hypothesize that a new drug will improve memory in Alzheimer's patients. To test this hypothesis, the researcher would administer the drug to a group of Alzheimer's patients and compare their memory performance to a control group that received a placebo. If the results show that the drug did not improve memory performance, the researcher would reject their hypothesis and revise their understanding of the drug's effects.

In conclusion, a hypothesis is a tentative explanation that a researcher tries to disprove or reject through their research. By doing so, they can either find evidence to support their hypothesis or identify flaws in their reasoning that they can address in future research.

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The function f is continuous on the closed interval (1,7). If 7 to 1 f(x) dx= 42 and 3 to 7 f(x) dx= -32, then 3 to 1 2f(x) dx=-148 is not correct. The correct answer is option a.

We can use the linearity property of definite integrals to evaluate 3 to 1 2f(x) dx as follows:

3 to 1 2f(x) dx = 2 times the definite integral of f(x) from 1 to 3

+ 2 times the definite integral of f(x) from 3 to 7

- 2 times the definite integral of f(x) from 7 to 1

(Note that we have reversed the limits of integration for the last term.)

Using the given values, we have:

3 to 1 2f(x) dx = 2 times the definite integral of f(x) from 1 to 3

+ 2 times the definite integral of f(x) from 3 to 7

- 2 times (7 to 1 f(x) dx)

= 2 times (-32) + 2 times 42 - 2 times 42

= -64

Therefore, the answer is A. -148 is not correct.

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Answer:

270 minutes or 4.5 hours

Step-by-step explanation:

You need to find the unit rate in order to solve this problem with the given information. In order to find the unit rate (in minutes) let's find each minute. Do 3600/180=20. You use 180 because that's the amount of minutes in 3 hours. Then, you take 5400 and divide it by 20 to get 270. Convert back to hours and rounded you get 4.5 hours.

4/5 + 2/9 is rational and can be expressed as the ratio of two integers: 46/45.

To show that 4/5 + 2/9 is rational, we need to find a common denominator and add the fractions. The least common multiple of 5 and 9 is 45, so we can rewrite the fractions with 45 as the denominator:

4/5 = 36/45

2/9 = 10/45

Now we can add the fractions:

4/5 + 2/9 = 36/45 + 10/45 = 46/45

Therefore, 4/5 + 2/9 is rational and can be expressed as the ratio of two integers: 46/45.

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The remaining players have scored an average of 14 runs each.

The total runs scored by the 11 players are:

11 players * 23 runs/player = 253 runs

If the first player scored 113 runs, then the remaining 10 players must have scored:

253 total runs - 113 runs scored by the first player = 140 runs

To find the average runs scored by the remaining players, we divide the total runs scored by the remaining 10 players by the number of players:

Average runs scored by remaining 10 players = 140 runs / 10 players = 14 runs/player

Therefore, the average runs scored by the remaining players is 14 runs per player.

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a) The distance of the cabin from the peak is A = 5.8 units

b) The distance of the fire tower from the lake is B = 8.6 units

c) The distance of the lake from the peak is C = 7.1 units

d) The distance from the lake is D = 11.7 units

Given data ,

Let the coordinates of the point be represented as C , P , L , F

where C ( -3 , 0 ) , P ( 0 , -5 ) , L ( 5 , 0 ) , F ( 0 , 7 )

And , from the distance formula ,

distance between A and B is D , and the distance D is

Distance D = √ ( x₂ - x₁ )² + ( y₂ - y₁ )²

Now , distance of the cabin from the peak is A

A = √ ( -3 )² + ( 5 )²

A = √34

A = 5.8 units

And , distance of the fire tower from the lake is B

B = √ ( 5 )² + ( 7 )²

B = √74

B = 8.6 units

And , distance of the lake from the peak is C

C = √ ( 5 )² + ( 5 )²

C = √50

C = 7.1 units

And , distance from the lake is D from ( -5 , -6 )

D = √ ( 5 - ( -5 ) )² + ( 6 )²

D = √( 100 + 36 )

D = √136

D = 11.7 units

Hence , the distances are solved

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Answer:

To calculate Juan's tax liability, we need to first determine his taxable income. We can do this by subtracting his exemptions and deductions from his total income:

Total income: $98,066 + $2,454 = $100,520

Exemption amount: $4,000

Standard deduction for Head of Household: $9,350 (for 2017)

Itemized deductions: $5,864 + $6,487 + $6,846 + $8,147 = $27,344

Total deductions: $9,350 + $27,344 = $36,694

Taxable income: $100,520 - $4,000 - $36,694 = $59,826

Using the tax tables for 2017, we can calculate Juan's tax liability:

Taxable income: $59,826

Tax liability: $10,517.50

Juan's W-2 shows that $11,799 was withheld for federal taxes, which is more than his calculated tax liability of $10,517.50. Therefore, Juan will receive a refund.

Refund amount: $11,799 - $10,517.50 = $1,281.50

Therefore, Juan will receive a refund of $1,281.50.

Answer:

Using the information provided, we can calculate Juan's taxable income and then use the tax table to determine his federal income tax liability. Here are the calculations:

Juan's total income:

W-2 income: $98,066

Interest and dividends: $2,454

Total: $100,520

Juan's adjustments:

Exemption amount: $4,000

Adjusted Gross Income (AGI):

Total income - Exemption amount

$100,520 - $4,000 = $96,520

Juan's deductions:

State and local taxes: $5,864

Property tax: $6,487

Mortgage interest: $6,846

Charitable contributions: $8,147

Total deductions: $27,344

Juan's taxable income:

AGI - Deductions

$96,520 - $27,344 = $69,176

Using the tax table for Head of Household filers, Juan's federal income tax liability is approximately $12,998.

Juan's federal tax withheld from his W-2 is $11,799. Since this amount is less than his tax liability of $12,998, he will owe additional taxes of approximately $1,199.

Therefore, Juan will owe an additional $1,199 in federal income taxes when he files his tax return.

Step-by-step explanation:

An illustration showing a pentagon lying on ray AB, which when extended forms an exterior angle and is marked as 2x plus 14 degrees, the correct statements are (2x + 14)° + (7x + 13)° = 180°, (7x + 13)° = 132°, x = 7°.

In the given illustration, we have a pentagon with base AB lying on a line or ray AB. Let's call the first interior angle to the right of the exterior angle marked as 2x + 14 degrees as angle C.

We can see that the sum of the measures of all interior angles of a pentagon is 180 degrees. Hence, the sum of all interior angles of this pentagon is:

C + angle D + angle E + angle F + angle G = (5 - 2) x 180 degrees (since a pentagon has five sides and angles)

(2x + 14)° + (7x + 13)° = 180° (sum of interior angles of a pentagon)

(7x + 13)° = 132° (simplifying the above equation)

x = 7° (solve for x by subtracting 14 from both sides of the first equation)

Therefore, options 2, 3, and 4 are correct.

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The point on the plane 4x - y + 4z = 40 nearest the origin is (-3.048, -0.762, 6.467)

Given data ,

To find the point on the plane 4x - y + 4z = 40 nearest the origin, we need to minimize the distance between the origin and the point on the plane.

The normal vector to the plane 4x - y + 4z = 40 is given by (4,-1,4). To find the perpendicular distance from the origin to the plane, we need to project the vector from the origin to any point on the plane onto the normal vector. Let's choose the point (0,0,10) on the plane:

Vector from origin to (0,0,10) on the plane = <0-0, 0-0, 10-0> = <0,0,10>

Perpendicular distance from the origin to the plane = Projection of <0,0,10> onto (4,-1,4)

= (dot product of <0,0,10> and (4,-1,4)) / (magnitude of (4,-1,4))

= (0 + 0 + 40) / √(4^2 + (-1)^2 + 4^2)

= 40 / √(33)

To find the point on the plane nearest the origin, we need to scale the normal vector by this distance and subtract the result from any point on the plane. Let's use the point (0,0,10) again:

Point on the plane nearest the origin = (0,0,10) - [(40 / √(33)) / √(4^2 + (-1)^2 + 4^2)] * (4,-1,4)

= (0,0,10) - (40 / √(33)) * (4/9,-1/9,4/9)

= (0,0,10) - (160/9√(33), -40/9√(33), 160/9√(33))

= (-160/9√(33), -40/9√(33), 340/9√(33))

Hence , the point on the plane 4x - y + 4z = 40 nearest the origin is approximately (-3.048, -0.762, 6.467)

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The turntable turns through a 45.0 degree angle, the radial line also rotates by 45.0 degrees. Therefore, the direction of the ladybug's displacement vector is 45.0 degrees counterclockwise from the positive x-axis.

Part A:
The radius of the turntable is half of the diameter, which is 6.0 in. The ladybug crawls in a straight radial line to the edge, which means its displacement vector is equal to the radius of the turntable. Therefore, the magnitude of the ladybug's displacement vector is 6.0 in.

Part B:
The direction of the ladybug's displacement vector is the same as the direction of the radial line from the center of the turntable to the edge. This direction can be described by the angle between the positive x-axis and the radial line.

Since the turntable turns through a 45.0 degree angle, the radial line also rotates by 45.0 degrees. Therefore, the direction of the ladybug's displacement vector is 45.0 degrees counterclockwise from the positive x-axis.

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The two theoretical quantities of ANOVA (Analysis of Variance) are:
1. Between-group variance.

2. Within-group variance:

1. Between-group variance.

This is the variance that can be attributed to differences between the group means.

It is calculated by comparing the mean of each group to the overall mean of all the data points.

The larger the between-group variance, the more likely there are significant differences between the groups.
2. Within-group variance:

This is the variance that can be attributed to differences within each group, i.e., the individual differences among the data points in each group.

It is calculated by comparing the individual data points in each group to their respective group mean.

The smaller the within-group variance, the more likely the groups are homogeneous.
In ANOVA, these two quantities are compared using an F-ratio.

If the between-group variance is significantly larger than the within-group variance, it indicates that there are significant differences between the group means, and the null hypothesis can be rejected.

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It provides character details that indicate Mrs. This function Faxon's interaction with this character serve in the given excerpt. Therefore, the correct option is option A.

An excerpt is indeed a quotation from that other resource that the author introduces with fresh language in a sentence. This phrase is taken from Thomas Jefferson's 1801 inaugural speech: As he warned his audience, "I may often go wrong by defect of judgement," Jefferson admitted the fallibility of human nature. "I shall frequently err through lack of judgement" is the excerpt. It provides character details that indicate Mrs. This function Faxon's interaction with this character serve.

Therefore, the correct option is option A.

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The new dimensions of the pictures are 2.5 inches by 3.5 inches.

Chris did image resizing of Career Week, which originally measured 5 inches by 7 inches, by a factor of 1/2. This means that he reduced each dimension of the pictures by half of its original size.

To calculate the new dimensions, he multiplied each original dimension by the scaling factor of 1/2.  

If the pictures were resized by a factor of 1/2, it means that each dimension was reduced to half of its original size.

The original dimensions of the pictures were 5 inches by 7 inches.

Resizing each dimension by a factor of 1/2 gives:

5 inches x 1/2 = 2.5 inches

7 inches x 1/2 = 3.5 inches

Therefore, the new dimensions of the pictures are 2.5 inches by 3.5 inches.

This means that the new pictures are significantly smaller than the original ones, which may affect their resolution and quality.

However, depending on the intended use of the pictures, a smaller size may be suitable or even desirable.

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For all integers a and d, d ⫮ n if, and only if, d is a common divisor of n and a, for any integer a.

The notation d ⫮ n means that d is a common divisor of n, or in other words, both n and a are multiples of d. The statement "d ⫮ n if, and only if, d is a common divisor of n and a, for any integer a" is equivalent to saying that d is a divisor of n, and that any other common divisor of n and a must also be divisible by d.

This property is important in number theory and is often used in proofs involving greatest common divisors (GCD) and least common multiples (LCM). By understanding the relationship between d, n, and a and the idea of common divisors, we can gain insight into the properties of numbers and their factors.

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The angle between the net displacement vectors for the two birds is 23 degrees

Given that;

A bird flies 3.0 km due west and then 2.0 due north. Another bird flies 2.0 km due west and 3.0 KM due north.

Now, First we need to find the angle between west direction and vectors given.

The angle between west direction and vector 1 is,

arctan(2/3) = 33.69

And, The angle between west direction and vector 2 is,

⇒ arctan(3/2) = 56.31

Hence, Angle between vector 2 and vector 1 is,

56.31-33.69 = 22.62 degrees

                    = 23 degree

Thus, The angle between the net displacement vectors for the two birds is 23 degrees.

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The value of p in the given equation is -1.

Given is an equation p-3 = -4, we need to find the value of p,

So,

p-3 = -4

p = -4+3

p = -1

Hence, the value of p in the given equation is -1.

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Answer: To find Jordan's balance after he makes the minimum payment for May, we need to calculate his balance at the end of each month, taking into account his purchases, the interest charged, and the minimum payment made.

In March, Jordan's balance was $628.16 and he made a minimum payment of:

$628.16 x 3.96% = $24.92

His purchases in March totaled $50.81, so his balance after March was:

$628.16 + $50.81 = $678.97

The interest charged in March would be:

$678.97 x (10.59%/12) = $5.72

So his balance after interest was added in March would be:

$678.97 + $5.72 = $684.69

In April, Jordan's balance was $684.69 and he made a minimum payment of:

$684.69 x 3.96% = $27.13

His purchases in April totaled $77.36 + $32.40 + $49.20 = $159.96, so his balance after April was:

$684.69 + $159.96 = $844.65

The interest charged in April would be:

$844.65 x (10.59%/12) = $7.10

So his balance after interest was added in April would be:

$844.65 + $7.10 = $851.75

In May, Jordan's balance was $851.75 and he made a minimum payment of:

$851.75 x 3.96% = $33.77

His purchases in May totaled $25.79 + $79.39 + $79.08 = $184.26, so his balance after May would be:

$851.75 + $184.26 = $1,036.01

The interest charged in May would be:

$1,036.01 x (10.59%/12) = $8.71

So his balance after interest was added in May would be:

$1,036.01 + $8.71 = $1,044.72

Therefore, Jordan's balance after he makes the minimum payment for May would be $1,044.72, which is closest to answer (a) $1,094.10.

This pooled estimate provides a common probability of success for both populations under the null hypothesis H0: p1 = p2, and is used in the calculation of the test statistic for the hypothesis test.

To calculate the best pooled estimate p for the population probability of success using H0: p1 = p2, we first need to calculate the individual sample proportions for each population.

The sample proportion for the first population is r1/n1, and the sample proportion for the second population is r2/n2.

Then, we can calculate the pooled estimate p as:

p = (r1 + r2) / (n1 + n2)

This is the weighted average of the sample proportions, where the weight is proportional to the sample size.

Using this pooled estimate p, we can then calculate the test statistic for the hypothesis test of difference of proportions.

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Answers in bold:

S9 = 2

i = 20

R = -2

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

Explanation:

[tex]S_0 = 20[/tex] is the initial term because your teacher mentioned [tex]A_0 = I[/tex] as the initial term.

Then R = -2 is the common difference because we subtract 2 from each term to get the next term. In other words, we add -2 to each term to get the next term.

Here is the scratch work for computing terms S1 through S4.

[tex]\begin{array}{|l|l|}\cline{1-2}S_{n} = S_{n-1} - 2 & S_{n} = S_{n-1} - 2\\S_{1} = S_{1-1} - 2 & S_{2} = S_{2-1} - 2\\S_{1} = S_{0} - 2 & S_{2} = S_{1} - 2\\S_{1} = 20 - 2 & S_{2} = 18 - 2\\S_{1} = 18 & S_{2} = 16\\\cline{1-2}S_{n} = S_{n-1} - 2 & S_{n} = S_{n-1} - 2\\S_{3} = S_{3-1} - 2 & S_{4} = S_{4-1} - 2\\S_{3} = S_{2} - 2 & S_{4} = S_{3} - 2\\S_{3} = 16 - 2 & S_{4} = 14 - 2\\S_{3} = 14 & S_{4} = 12\\\cline{1-2}\end{array}[/tex]

Then here is S5 though S8

[tex]\begin{array}{|l|l|}\cline{1-2}S_{n} = S_{n-1} - 2 & S_{n} = S_{n-1} - 2\\S_{5} = S_{5-1} - 2 & S_{6} = S_{6-1} - 2\\S_{5} = S_{4} - 2 & S_{6} = S_{5} - 2\\S_{5} = 12 - 2 & S_{6} = 10 - 2\\S_{5} = 10 & S_{6} = 8\\\cline{1-2}S_{n} = S_{n-1} - 2 & S_{n} = S_{n-1} - 2\\S_{7} = S_{7-1} - 2 & S_{8} = S_{8-1} - 2\\S_{7} = S_{6} - 2 & S_{8} = S_{7} - 2\\S_{7} = 8 - 2 & S_{8} = 6 - 2\\S_{7} = 6 & S_{8} = 4\\\cline{1-2}\end{array}[/tex]

And finally we arrive at S9.

[tex]S_{n} = S_{n-1} - 2\\\\S_{9} = S_{9-1} - 2\\\\S_{9} = S_{8} - 2\\\\S_{9} = 4 - 2\\\\S_{9} = 2\\\\[/tex]

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

Because we have an arithmetic sequence, there is a shortcut.

[tex]a_n[/tex] represents the nth term

S9 refers to the 10th term because we started at index 0. So we plug n = 10 into the arithmetic sequence formula below.

[tex]a_n = a_1 + d(n-1)\\\\a_n = 20 + (-2)(n-1)\\\\a_n = 20 - 2(n-1)\\\\a_{10} = 20 - 2(10-1)\\\\a_{10} = 20 - 2(9)\\\\a_{10} = 20 - 18\\\\a_{10} = 2\\\\[/tex]

In other words, we start with 20 and subtract off 9 copies of 2 to arrive at 20-2*9 = 20-18 = 2, which helps see a faster way why [tex]S_9 = 2[/tex]

A quadrilateral with two pairs of opposite sides always congruent is a parallelogram. A parallelogram is a quadrilateral with two pairs of opposite sides that are parallel and always congruent.

In a parallelogram, the opposite angles are also congruent. The rectangle, square, and rhombus are all special types of parallelograms, but they have additional properties that make them unique. A rectangle has four right angles, a square has four right angles and all sides congruent, and a rhombus has all sides congruent but does not require right angles.

Kites, isosceles trapezoids, and trapezoids do not necessarily have two pairs of congruent opposite sides, so they do not fit the description.

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The new balance after one month, given the finance charge and the down payment would be C. $2,114. 56

First, find the monthly interest :

= 16 % / 12

= 0. 013333

Then, get the interest in the first month on the loan :

= 2, 200 x 0. 013333

= $ 29.33

The amount that goes to the principal as repayment is:

= Monthly payment - interest

= 114. 66 - 29. 33

= $ 85. 33

The remaining loan balance is then :

= 2, 200 - 85. 33

= $ 2, 114. 56

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The two statements are equivalent. Both "there exists an x such that x is an apple and x has spots" and "there exists an apple with spots" express the idea that there is at least one apple that has spots.


The statement "there exists an x such that x is an apple and x has spots" means that within a certain set or context, there is at least one example (x) that satisfies the conditions of being both an apple and having spots.
This statement can be summarized as "there exists an apple with spots," which essentially conveys the same meaning. In this summarized version, it is implied that there is at least one instance of an apple that has spots.

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