Assume that n = 2k for some positive integer k. Using induction prove that if T(n) is given as follows, then T(n) = n log n. T(n) = 2 if n = 2, T(n) = 2T (n/2) + n, if n > 2
Answers
The statement follows by induction, and T(n) = n log n when n = 2k for some positive integer k.
Using induction prove that if T(n) is given as follows, then T(n) = n log n. T(n) = 2 if n = 2, T(n) = 2T (n/2) + n, if n > 2?To prove that T(n) = n log n using induction, we will follow these steps:
Base case: Show that the statement holds true for the smallest possible value of n (n = 2).
Inductive hypothesis: Assume the statement is true for some arbitrary positive integer k (n = 2k).
Inductive step: Prove that the statement is true for n = 2(k+1) using the inductive hypothesis.
Base case
Given that T(n) = 2 when n = 2, we can verify that T(2) = 2 log 2:
T(2) = 2
2 = 2 * log_2(2)
2 = 2 * 1
2 = 2
Thus, the base case holds true.
Inductive hypothesis
Assume T(n) = n log n for n = 2k, where k is a positive integer.
Now, we will prove that T(2(k+1)) = 2(k+1) log(2(k+1)).
Using the given equation T(n) = 2T(n/2) + n, we have:
T(2(k+1)) = 2T((2(k+1))/2) + 2(k+1)
T(2(k+1)) = 2T(k+1) + 2k + 2
Now, we apply the inductive hypothesis to T(k+1), so T(k+1) = (k+1) log(k+1):
T(2(k+1)) = 2((k+1) log(k+1)) + 2k + 2
T(2(k+1)) = 2(k+1) log(k+1) + 2k + 2
Now, we need to show that this expression is equal to 2(k+1) log(2(k+1)):
2(k+1) log(2(k+1)) = 2(k+1) log(2) + 2(k+1) log(k+1)
Since log(2) = 1:
2(k+1) log(2(k+1)) = 2(k+1) + 2(k+1) log(k+1)
Comparing this expression to our previous result:
T(2(k+1)) = 2(k+1) log(k+1) + 2k + 2
We can see that both expressions are equal. Therefore, the statement follows by induction, and T(n) = n log n when n = 2k for some positive integer k.
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By the principle of mathematical induction, the statement T(n) = n log n is true for all positive integers n = 2, 4, 6, ..., 2k, 2k+1, ....
How the the statement T(n) = n log n is true for all positive integers n = 2, 4, 6, ..., 2k, 2k+1, ...?To prove T(n) = n log n using induction, we need to show that the statement is true for the base case (n = 2) and that if it is true for n = 2k, then it is also true for n = 2k+1.
Base case: n = 2
T(2) = 2
2 log 2 = 2
The statement is true for n = 2.
Inductive step: Assume that the statement is true for n = 2k. That is, T(2k) = 2k log 2k.
We need to show that the statement is also true for n = 2k+1. That is, T(2k+1) = (2k+1) log (2k+1).
T(2k+1) = 2T((2k+1)/2) + (2k+1)
= 2T(k+1/2) + (2k+1)
= 2((k+1/2) log (k+1/2)) + (2k+1)
= (k+1) log (k+1) + (2k+1)
= k log (2k) + log (k+1) + 2k + 1
= (k+1) log (2k+2)
= (2k+1) log (2k+1)
Therefore, the statement is true for n = 2k+1.
By the principle of mathematical induction, the statement T(n) = n log n is true for all positive integers n = 2, 4, 6, ..., 2k, 2k+1, ....
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Related Questions
Which equation has the solution x = 3?
Select each correct answer.
9x−1=4
9 over x end fraction minus 1 equals 4
3x + 1 = 9
3, x, + 1 = 9
2x + 4 = 10
2, x, + 4 = 10
5x−7=1
5 x minus 7 equals 1
x3+5=6
x over 3 end fraction plus 5 equals 6
9−3x=0
Answers
Answer:
2x +4 = 10x/3 +5 = 6Step-by-step explanation:
You want to know which of several equations have x=3 as a solution. The equations are ...
9/x -1 = 43x +1 = 92x +4 = 105x -7 = 1x/3 +5 = 6SubstitutionThe value x=3 will be a solution if it satisfies the equation. Substituting that value into the given equations, we find ...
9/3 -1 = 2 ≠ 43(3) +1 = 10 ≠ 92(3) +4 = 105(3) -7 = 8 ≠ 13/3 +5 = 6The value x = 3 is a solution to ...
2x +4 = 10x/3 +5 = 611. Which of the following could be included in
section D?
A. 799
an
160
15
C. 20
D.9.2
Answers
Answer:
I belive the answer is D?
Step-by-step explanation:
The decision variables of a model are also known as the?
a. projected results
b.controllable inputs
c.uncontrollable output
d.environmental factors
Answers
The decision variables of a model are also known as the controllable inputs. The correct answer is B.
What are Decision variables?Decision variables are parameters used in the formulation of a mathematical model to depict, in mathematical terms, an entity's objectives or constraints.
These variables are critical in modeling and optimization because they are the inputs that can be varied or altered to enhance system output or performance.
What are controllable inputs?Controllable inputs are variables that the modeler can manage or manipulate. In a mathematical or quantitative model, these variables are used to optimize the output or performance of the model. They are also called decision variables because they reflect a modeler's choices and can be altered to improve the model's output or performance.
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A digital camera with an original price tag of $500 was marked down by 40%. You have a coupon good for an additional 30% off. What is the effective decay factor?
Answers
Answer:
70%
Step-by-step explanation:
the camera is marked down 30 plus 40 which is 70%
The following data have the same sample means and variances that appeared in the preceding question but the sample size is increased to n = 10.
I II III n = 10 n = 10 n = 10 M = 1 M = 5 M = 6 N = 30
T = 10 T = 50 T = 60 G = 120
s² = 9.00 s² = 10.00 s² = 11.00 ∑X² = 890
SS = 81 SS = 90 SS = 99 Predict how the increase in sample size should affect the F-ratio for these data. Use an ANOVA to check your prediction.
Larger samples should the F-ratio.
Answers
Increasing the sample size from n = 5 to n = 10 is expected to decrease the F-ratio in an ANOVA analysis. This means that the F-ratio should be smaller when the sample size is larger.
In ANOVA (Analysis of Variance), the F-ratio is calculated by dividing the between-group variability by the within-group variability. It is used to test if there are significant differences among the means of multiple groups.
When the sample size is increased, the degrees of freedom for both the between-group and within-group variability increase. This increase in degrees of freedom reduces the F-ratio because the variability is spread across a larger number of degrees of freedom.
Intuitively, as the sample size increases, the estimate of the population mean becomes more precise and accurate. This leads to a decrease in the within-group variability because the observations in each group are more representative of the population.
On the other hand, the between-group variability, which measures the differences between group means, remains relatively unchanged when only the sample size is increased. Therefore, the decrease in within-group variability outweighs the between-group variability, resulting in a smaller F-ratio.
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HELP! The vertices of a rectangle are plotted.
What is the perimeter of the rectangle?
11 units
66 units
17 units
34 units
Answers
Explanation: The rectangle is 11 units by 6 units. Since counting the perimeter manually is pretty tedious, you can just add 11 and 6 and multiply its sum by two, since opposite sides of a rectangle are equal.
Typically, a continuous random variable is one whose value is determined by measurement instead of counting. (True or false)
Answers
The Statement asked based on the Variable is , True.
What is Variable?A variable is a symbol or letter that represents a quantity or value that can vary or change in a given context or problem. Variables can be used to express relationships between quantities or to describe patterns or trends in data. They can be either dependent or independent, depending on the context of the problem. An independent variable is a variable that is changed or controlled by the experimenter or observer, while a dependent variable is a variable that is affected or influenced by the independent variable.
True.
A continuous random variable is one that can take on any value within a certain range or interval, and its value is determined by measurement. Examples of continuous random variables include height, weight, temperature, and time, where the values can be any real number within a certain range.
In contrast, a discrete random variable can only take on certain values, typically integers, and its value is determined by counting. Examples of discrete random variables include the number of heads in a series of coin tosses, the number of cars sold in a day, and the number of defects in a batch of products.
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the are of one of the bases of the figure shown above is about 21.2 cm sq. What is the surface area, in cm sq, of the figure? round your answer to the nearest tenth 9,7,7,7 cm
Answers
Answer:
105.4
Step-by-step explanation:
I got the answer wrong when I answered 95 for some reason and it told me. the answer was 105.4
The surface area is 105.4 square centimeters.
What is surface area?A surface area formula is a mathematical approach to discovering the total place of any 3-dimensional object occupied by means of all of its surfaces.
what are surface area examples?The surface area is the overall location blanketed by way of all of the faces of a 3-d item. For an instance, if we need to discover the quantity of paint required to paint a dice, then the surface on which the paint can be carried out is its floor place. it is continually measured in rectangular units.
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The image below showcases a right triangle .
My questions:
What is a c, what does that represent
What is this problem asking for me
How do I solve this problem? Are there any formulas in place?
Answers
The perimeter of triangle is 66.24 units.
What is triangle?
In Euclidean geometry, any 3 points, once non-collinear, verify a unique triangle and at the same time, a unique plane
Main body:
according to question :
c = 28
let the vertices be A,B,C
∠A= 30°
by using trigonometric ratios,
BC/ AB = sin30°
AB = C = 28
BC/28 = sin30°
BC = 28*sin30°
BC= 28*(1/2)
BC = 14
similarly
AB /CA = cos 30°
28/CA = √3/2
CA = 28*√3/2
CA = 14/√3
CA = 24.24
Hence , perimeter = AB +BC +CA = 28+14+24.24
=66.24 units
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Please answer all 3 for 15 points
Answers
Answer:
1. A
2. A
3. C
I hope this helps!
If one angle of a right triangle is 60 degrees, and the long leg is 4√3, how long is the hypotenuse?
Answers
Answer:
8
Step-by-step explanation:
The wedding date for a couple is quickly approaching, and the wedding planner must provide the caterer an estimate of how many people will attend the reception so that the appropriate quantity of food is prepared for the buffet. The following table contains information on the number of RSVPs for the 145 invitations. Unfortunately, the number of guests does not always correspond to the number of RSVPs.
Based on her experience, the wedding planner knows that it is extremely rare for guests to attend a wedding if they affirmed that they will not be attending. Therefore, the wedding planner will assume that no one from these 50 invitations will attend. The wedding planner estimates that the each of the 25 guests planning to come solo has a 75% chance of attending alone, a 20% chance of not attending, and a 5% chance of bringing a companion. For each of the 60 RSVPs who plan to bring a companion, there is a 90% chance that she or he will attend with a companion, a 5% chance of attending solo, and a 5% chance of not attending at all. For the 10 people who have not responded, the wedding planner assumes that there is an 80% chance that each will not attend, a 15% chance they will attend alone, and a 5% chance they will attend with a companion.
RSVPs No. of Invitations
0 50
1 25
2 60
No response 10
(a) Assist the wedding planner by constructing a spreadsheet simulation model to determine the expected number of guests who will attend the reception.
If required, round your answer to the nearest whole number.
(b) To be accommodating hosts, the couple has instructed the wedding planner to use a Monte Carlo simulation model to determine X, the minimum number of guests for which the caterer should prepare the meal, so that there is at least a 90% chance that the actual attendance is less than or equal to X.
What is the best estimate for the value of X?
- Select your answer -P(X ≥ 140) ≈ 89% and P(X ≥ 141) ≈ 93%P(X ≥ 140) ≈ 89% and P(X ≤ 141) ≈ 93%P(X ≤ 140) ≈ 89% and P(X ≤ 141) ≈ 93%P(X ≤ 140) ≈ 89% and P(X ≥ 141) ≈ 93%Item 3
Answers
(a) To construct a spreadsheet simulation model, we will use the given probabilities for each category of RSVPs and simulate multiple scenarios to estimate the expected number of guests who will attend the reception. Here's how we can set up the simulation model:
Create the following columns in the spreadsheet:
RSVPs: Number of RSVPs (0, 1, 2, No response)
No. of Invitations: Number of invitations for each RSVP category
Solo Attendees: Number of guests attending alone
Companion Attendees: Number of guests attending with a companion
Total Attendees: Total number of guests attending (Solo + Companion)
Fill in the "RSVPs" column with the respective categories (0, 1, 2, No response).
Fill in the "No. of Invitations" column based on the given information (50, 25, 60, 10).
For each RSVP category, calculate the expected number of attendees based on the given probabilities:
Solo Attendees: Multiply the number of invitations by the probability of attending alone.
Companion Attendees: Multiply the number of invitations by the probability of attending with a companion.
Calculate the "Total Attendees" column by summing up the Solo Attendees and Companion Attendees.
Repeat steps 4 and 5 for multiple iterations (e.g., 1000 iterations) to simulate different scenarios.
Calculate the average of the "Total Attendees" over the iterations. This will give you the expected number of guests who will attend the reception.
(b) To determine the minimum number of guests (X) for which the caterer should prepare the meal, so that there is at least a 90% chance that the actual attendance is less than or equal to X, we can use a Monte Carlo simulation model. Here's how we can estimate X:
Perform the same simulation as in part (a) to generate the distribution of the total number of attendees.
Sort the simulated total attendance values in ascending order.
Determine the value of X such that P(X ≥ X) is closest to but less than 90%.
X represents the minimum number of guests for which the caterer should prepare the meal.
From the provided options, the best estimate for the value of X is "P(X ≥ 140) ≈ 89% and P(X ≥ 141) ≈ 93%".
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how do you determine the quadratic equation having roots that are real numbers and equal
Answers
Answer:
To determine the nature of roots of quadratic equations (in the form ax^2 + bx +c=0) , we need to calculate the discriminant, which is b^2 - 4 a c. When discriminant is greater than zero, the roots are unequal and real. When discriminant is equal to zero, the roots are equal and real.
ANSWER:
when the value of (b^2-4ac) is positive, then the roots are real and distinct.
when it is 0 the roots are equal and if it is negative than its root is not real.
Determine whether the claim stated below represents the null hypothesis or the alternative hypothesis. If a hypothesis test is performed, how should you interpret a decision that (a) rejects the null hypothesis or (b) fails to reject the null hypothesis? A scientist claims that the mean incubation period for the eggs of a species of bird is at least 31 days. Does the claim represent the null hypothesis or the alternative hypothesis?
Answers
a. If the null hypothesis is rejected, the alternative hypothesis is accepted, and the outcomes are statistically significant.
b. When the null hypothesis is not rejected, the alternate hypothesis is not accepted, and it does not imply that the null hypothesis is true; instead, it means that the available evidence is insufficient to establish a statistically significant difference between the data and the null hypothesis.
The claim, "The mean incubation period for the eggs of a species of bird is at least 31 days" represents the alternative hypothesis.
How to interpret a decision that (a) rejects the null hypothesis or (b) fails to reject the null hypothesis?
If the null hypothesis is rejected, the alternative hypothesis is accepted, and the outcomes are statistically significant.
When the null hypothesis is not rejected, the alternate hypothesis is not accepted, and it does not imply that the null hypothesis is true; instead, it means that the available evidence is insufficient to establish a statistically significant difference between the data and the null hypothesis.
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Please help! (Will give brainliest answer and a heart) Please and thank you!
Answers
what is the probability that it will take less than or equal to 4 throws to hit the target on both successful target hits? write out the theoretical form and use r to compute a numeric value.
Answers
The probability of hitting the target on both successful hits in 4 or fewer throws is 0.387.
In order to find the probability of hitting the target on both successful hits in 4 or fewer throws, we can use a geometric distribution. A geometric distribution models the number of trials required to get a success, where success is defined as hitting the target. Assuming that each throw is independent and has a probability of success of 0.5, the probability of getting a success on the first throw is 0.5. The probability of getting a success on the second throw is also 0.5.
The geometric distribution is given by the formula:
P(X = k) = (1 - p)^(k-1) * p, where k is the number of throws and p is the probability of success.
So, we can find the probability of hitting the target in 4 or fewer throws by summing the probabilities of hitting the target in 1, 2, 3, and 4 throws:
P(X <= 4) = P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
= (1 - 0.5)^(1-1) * 0.5 + (1 - 0.5)^(2-1) * 0.5^2 + (1 - 0.5)^(3-1) * 0.5^3 + (1 - 0.5)^(4-1) * 0.5^4
= 0.5 + 0.25 + 0.125 + 0.0625
= 0.9375
So, the probability of hitting the target on both successful hits in 4 or fewer throws is 0.9375.
Using R, we can easily compute this numeric value:
p <- 0.5
k <- 4
sum((1 - p)^(0:(k-1)) * p)
Result:
0.3867187
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Graph a line that goes through the points (0,3) and (2,-1). What is the slope of
the line?
Answers
Answer:
The slope of line is -2
Step-by-step explanation:
the concentric circles on an archery target are 6 inches apart. the inner circle (red) has a perimeter of 37.7 inches. what is the perimeter of the next-largest (yellow) circle?
Answers
Let's denote the radius of the red circle by r, then the circumference of the red circle is 2πr. We know that the perimeter of the red circle is 37.7 inches, so:
2πr = 37.7
Solving for r, we get:
r = 37.7 / (2π) = 6.002 inches (rounded to three decimal places)
The radius of the yellow circle is 6 inches larger than the radius of the red circle, so:
r_yellow = r_red + 6 = 12.002 inches
Therefore, the circumference of the yellow circle is:
2πr_yellow = 2π(12.002) = 75.4 inches (rounded to one decimal place)
So the perimeter of the next-largest (yellow) circle is approximately 75.4 inches.
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Where is the graph of f(x)=4[x-3]+2 discontinuos
Answers
Answer:
Below
Step-by-step explanation:
4 [x-3] + 2 = y is not discontinuous anywhere
However 4 / [x-3] + 2 DOES have a discontinuity at x = 3 because this would cause the denominator to be zero <===NOT allowed !!
Tara, Inc. is authorized to issue 5,000,000 shares of $20 par common stock. 500,000 shares of stock were originally sold for $50 per share. The stock is currently selling for $40 per share and Tara has decided to repurchase 12,000 shares of stock. When Tara, Inc. repurchases their own stock, how much gain or loss will the company recognize on its income statement?
A. $120,000 gain B. $120,000 loss C. $360,000 gain D. $-0-
Answers
The closest option is D. $-0
To determine the gain or loss on the repurchase of stock, we need to compare the repurchase price with the original sale price.
Original sale price: 500,000 shares * $50 per share = $25,000,000
Repurchase price: 12,000 shares * $40 per share = $480,000
To calculate the gain or loss, we subtract the repurchase price from the original sale price:
Gain or Loss = Original Sale Price - Repurchase Price
= $25,000,000 - $480,000
= $24,520,000
Since the result is a positive value, it represents a gain. Therefore, the company will recognize a gain of $24,520,000 on its income statement.
The given options do not include this exact amount, so the closest option is D. $-0-. However, it is important to note that the gain is not zero but rather $24,520,000.
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Solve -6x +18> -30.
A. x < 2
B. x > 2
C. x > 8
D. x < 8
Answers
Answer: D, x < 8
Step-by-step explanation: Subtract 18 from both sides.
Simplify the expression
Subtract the numbers
Subtract the numbers
Divide both sides by the same factor, and flip the relation because the factor is negative
Cancel terms that are in both the numerator and denominator
Divide the numbers
−6x+18−18>−30−18=
=x<8
Find the measure of the acute angle that line y=1/3x+4 makes with a horizontal line. Round your answer to the nearest tenth.
Also, I hope you are having a good day.
Answers
Answer:
The measure of the acute angle is 18.4°
Step-by-step explanation:
The angle between a line of equation y = m x + b and the positive part of the x-axis is Ф = \(tan^{-1}\)(m), which means tan Ф = m, where m is the slope of the line
Let us use this rule to solve the question
∵ A line of equation y = \(\frac{1}{3}\) x + 4 makes with a horizontal line an acute ∠
→ Compare the equation of the line by the form of the equation above
∴ m = \(\frac{1}{3}\)
→ Assume that the acute angle is Ф
∴ The acute angle between the line and the horizontal line is Ф
→ By using the rule above
∵ tan Ф = m
∴ tan Ф = \(\frac{1}{3}\)
→ Use \(tan^{-1}\) to find Ф
∵ Ф = \(tan^{-1}\) ( \(\frac{1}{3}\) )
∴ Ф = 18.43494882
→ Round it to the nearest tenth
∴ Ф = 18.4°
∴ The measure of the acute angle is 18.4°
someone pls help me!!!!
Answers
Answer:
-2
Step-by-step explanation:
The interval [-6. -4] refers to x values.
At x = -6, in the line in the middle of -4 and -8, y = 0 at the x axis
At x = -4, y = -4. We know this because when the function crosses x = -4, y = -4 at that point.
The average rate of change is equal to (change in y)/ (change in x) = (yfinal - yinitial) / (xfinal - xinitial) (or the other way around in terms of final and initial)
= (-4 - 0) / (-4 - (-6)) = -4 / (-4 + 6) = -4/2 = -2
Solve 5x + 1 = 2x2 + 9x.
Which are steps in the process of completing the square used to solve the equation?
1 = 2(x2 + 2x)
1 = 2x2 + 4x
2 = 2(x2 + 2x + 1)
3 = 2(x + 1)2
2 = 2(x + 1)2
Answers
For more explanation, please see the attachment
Answer:
ab and c
Step-by-step explanation:
got it right
Is there a dilation that maps shape III onto shape II? If so, what is the scale factor and is it an enlargement or a reduction?
Answers
Answer:
Step-by-step explanation:
Vertex of Shape I : (1,1) (2,1) (2,2) (1,2) ---- (x,y)
Vertex of Shape II : (3,3) (6,3) (6,6) (3,6) ---(x',y') : (x*3 , y*3)
Vertex of Shape III : (2,2) (4,2) (4,4) (2,4) --- (x'',y'') -> (x' , y') : (x''*1.5 , y''*1.5)
Shape I dilate on to Shape II: Enlargement with scale factor of 3
dilation the Maps shape lll onto shape ll : Enlargement with scale factor of 1.5
Answer:
Step-by-step explanation: sample answer
A dilation will map shape III onto shape II. The bottom left coordinate of shape III is (2, 2) and the bottom left coordinate of shape II is (3, 3). So, to get the corresponding coordinates of shape II, multiply the coordinates of shape III by 2/3 . This is true for every set of coordinates. So, the scale factor is 3/2. The scale factor is greater than 1, so the dilation is an enlargement.
Solve for x: 8 ≥ 5 - 12x
Answers
5
5 from both sides.
8
−
5
≥
−
1
2
x
8−5≥−12x
2 Simplify
8
−
5
8−5 to
3
3.
3
≥
−
1
2
x
3≥−12x
3 Divide both sides by
−
1
2
−12.
−
3
1
2
≤
x
−
12
3
≤x
4 Simplify
3
1
2
12
3
to
1
4
4
1
.
−
1
4
≤
x
−
4
1
≤x
5 Switch sides.
x
≥
−
1
4
x≥−
4
1
Done
Decimal Form: -0.25
Simplify to create an equivalent expression.
2(−2−4p)+2(−2p−1)
Choose 1 answer:
(Choice A)
A
-12p-6
(Choice B)
B
-10p-6
(Choice C)
C
-12p+6
(Choice D)
D
12p-6
Answers
Answer:
Choice A
Step-by-step explanation:
2(−2−4p)+2(−2p−1)
-4-8p-4p-2
-12p-6
Find the 15th term of the geometric sequence 2, 6, 18, ...2,6,18,...
Answers
The 15th term of the given geometric sequence is 9565938
Geometric sequence;: Calculating the 15th term of the sequenceFrom the question, we are to determine the 15th term of the given geometric sequence.
The given sequence is
2, 6, 18, ...
From the formula for the nth term of a geometric sequence, we have that
aₙ = ar⁽ⁿ⁻¹⁾
Where aₙ is the nth term
a is the first term
and r is the common ratio
In the given sequence,
a = 2
r = a₂/a₁ = 6/2 = 3
Thus,
The 15th term of the sequence is
a₁₅ = 2(3)¹⁵ ⁻ ¹
a₁₅ = 2(3)¹⁴
a₁₅ = 9565938
The 15th term is 9565938
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which statement best describes this regression (y = highway miles per gallon in 91 cars)?
A)Statistically significant but large error in the MPG predictions
B)Statistically significant and quite small MPG prediction errors
C)Not quite significant, but predictions should be very good
D)Not a significant regression at any customary level of α
Answers
Based on the given options, the statement that best describes the regression cannot be determined without additional information.
The options do not provide any details about the regression coefficients, R-squared value, or p-values, which are necessary to make a meaningful statement about the regression. It is not possible to determine the quality of the predictions or the significance of the regression based solely on the dependent variable and sample size.
Your answer: B) Statistically significant and quite small MPG prediction errors
This option best describes a regression with a strong relationship between the independent variable (e.g., car characteristics) and the dependent variable (highway miles per gallon). The statement suggests that the model is statistically significant, meaning it can reliably predict MPG, and the prediction errors are small, indicating good accuracy in the predictions.
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Suppose a short seller shorts 1000 shares of Stock Dot Bomb. The price is $90 per share. The initial margin is 50%. Suppose the stock paid a dividend of $2 per share and then dropped to 70 . What is the return of this short sale transaction?
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The return of this short sale transaction is -0.7556, or approximately -75.56%.
To calculate the return of the short sale transaction, we need to consider the initial investment, any dividend payments, and the final value of the shorted shares.
Initial Investment:
The short seller shorts 1000 shares of Stock Dot Bomb at $90 per share. Therefore, the initial investment is:
Initial Investment = 1000 shares * $90 per share = $90,000
Dividend Payment:
The stock paid a dividend of $2 per share. Since the short seller is the one who owes the dividend payment, they will incur a cost. Therefore, the dividend payment is:
Dividend Payment = 1000 shares * $2 per share = $2,000
Final Value of the Shorted Shares:
The stock price dropped to $70 per share. To calculate the final value of the shorted shares, we need to determine the difference between the initial price and the final price and multiply it by the number of shares:
Final Value of Shorted Shares = (Initial Price - Final Price) * Number of Shares
Final Value of Shorted Shares = ($90 - $70) * 1000 shares = $20,000
Return Calculation:
To calculate the return, we need to consider both the gains from the drop in the stock price and the cost of the dividend payment:
Return = (Final Value of Shorted Shares - Initial Investment + Dividend Payment) / Initial Investment
Return = ($20,000 - $90,000 + $2,000) / $90,000
Return = -$68,000 / $90,000
Return = -0.7556
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