The following statements are proved using mathematical induction:
(b) Prove that for any positive integer n,6 evenly divides [tex]7^n -1[/tex].
(c) Prove that for any positive integer n,4 evenly divides [tex]11^n-7^n[/tex].
(e) Prove that for any positive integer n,2 evenly divides [tex]n^2-5n+2[/tex].
(b) Prove that for any positive integer n, 6 evenly divides [tex]7^n - 1.[/tex]
Step 1: Base case
Let's check if the statement holds true for the base case, n = 1.
For n = 1, we have [tex]7^1 - 1 = 6[/tex], which is divisible by 6. Therefore, the statement holds true for the base case.
Step 2: Inductive hypothesis
Assume that the statement is true for some positive integer k, i.e., 6 evenly divides [tex]7^k - 1[/tex].
Step 3: Inductive step
We need to prove that the statement holds true for the next positive integer, k + 1.
Consider the expression [tex]7^{(k + 1)} - 1.[/tex]
We can rewrite it as [tex]7 * 7^k - 1.[/tex]
Using the assumption from the inductive hypothesis, we know that [tex]7^k - 1[/tex]is divisible by 6.
Since 7 is congruent to 1 (mod 6), we have [tex]7 * 7^k[/tex] ≡ [tex]1 * 1^k[/tex] ≡ 1 (mod 6).
Therefore, [tex]7^{(k + 1)} - 1[/tex] ≡ 1 - 1 ≡ 0 (mod 6), which means 6 evenly divides [tex]7^{(k + 1)} - 1.[/tex]
By the principle of mathematical induction, we can conclude that for any positive integer n, 6 evenly divides [tex]7^n - 1[/tex].
(c) Prove that for any positive integer n, 4 evenly divides [tex]11^n - 7^n.[/tex]
Step 1: Base case
For n = 1, we have [tex]11^1 - 7^1 = 11 - 7 = 4[/tex], which is divisible by 4. Therefore, the statement holds true for the base case.
Step 2: Inductive hypothesis
Assume that the statement is true for some positive integer k, i.e., 4 evenly divides [tex]11^k - 7^k.[/tex]
Step 3: Inductive step
We need to prove that the statement holds true for the next positive integer, k + 1.
Consider the expression [tex]11^{(k + 1)} - 7^{(k + 1)}.[/tex]
We can rewrite it as [tex]11 * 11^k - 7 * 7^k.[/tex]
Using the assumption from the inductive hypothesis, we know that [tex]11^k - 7^k[/tex] is divisible by 4.
Since 11 is congruent to 3 (mod 4) and 7 is congruent to 3 (mod 4), we have [tex]11 * 11^k[/tex] ≡ [tex]3 * 3^k[/tex] ≡ [tex]3^{(k+1)}[/tex] (mod 4) and [tex]7 * 7^k[/tex] ≡ [tex]3 * 3^k[/tex] ≡ [tex]3^{(k+1)}[/tex] (mod 4).
Therefore, [tex]11^{(k + 1)} - 7^{(k + 1)}[/tex] ≡ [tex]3^{(k+1)} - 3^{(k+1)}[/tex] ≡ 0 (mod 4), which means 4 evenly divides [tex]11^{(k + 1)} - 7^{(k + 1)}.[/tex]
By the principle of mathematical induction, we can conclude that for any positive integer n, 4 evenly divides [tex]11^n - 7^n.[/tex]
(e) Prove that for any positive integer n, 2 evenly divides [tex]n^2 - 5n + 2.[/tex]
Step 1: Base case
For n = 1, we have [tex]1^2 - 5(1) + 2 = 1 - 5 + 2 = -2,[/tex] which is divisible by 2. Therefore, the statement holds true for the base case.
Step 2: Inductive hypothesis
Assume that the statement is true for some positive integer k, i.e., 2 evenly divides [tex]k^2 - 5k + 2.[/tex]
Step 3: Inductive step
We need to prove that the statement holds true for the next positive integer, k + 1.
Consider the expression [tex](k + 1)^2 - 5(k + 1) + 2.[/tex]
Expanding and simplifying, we get [tex]k^2 + 2k + 1 - 5k - 5 + 2 = k^2 - 3k - 2.[/tex]
Using the assumption from the inductive hypothesis, we know that 2 evenly divides [tex]k^2 - 5k + 2[/tex].
Since 2 evenly divides -3k, and 2 evenly divides -2, we can conclude that 2 evenly divides [tex]k^2 - 3k - 2[/tex].
By the principle of mathematical induction, we can conclude that for any positive integer n, 2 evenly divides [tex]n^2 - 5n + 2[/tex].
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Find the equation of the line for the following
Find the equation of the line for the following: -) passing through (3, 2) with slope 4. 8) passing through (4, -2) and (5,6). - passing through (3,-1) and parallel to the line 6x +2y +4.
a) The equation of the line passing through the point (3, 2) with slope 4 is y - 2 = 4(x - 3).
b. The equation of the line passing through (4, -2) and (5,6) is y + 2 = 8(x - 4).
c) The slope of the line 6x +2y +4 is -3.
a. To derive the equation, we use the point-slope form of a linear equation: y - y₁ = m(x - x₁), where (x₁, y₁) represents the given point and m represents the slope.
Substituting the given values into the equation, we have:
y - 2 = 4(x - 3)
This equation can be further simplified if required.
b) The equation of the line passing through the points (4, -2) and (5, 6) can be found using the slope-intercept form, y = mx + b.
First, we calculate the slope (m) using the formula: m = (y₂ - y₁) / (x₂ - x₁).
m = (6 - (-2)) / (5 - 4) = 8.
Next, we substitute one of the given points and the calculated slope into the slope-intercept form:
y - y₁ = m(x - x₁).
y - (-2) = 8(x - 4).
Simplifying the equation:
y + 2 = 8(x - 4).
c) To find the equation of the line passing through the point (3, -1) and parallel to the line 6x + 2y + 4 = 0, we first need to determine the slope of the given line.
Rearranging the equation 6x + 2y + 4 = 0, we have:
2y = -6x - 4,
y = -3x - 2.
The given line has a slope of -3.
Since parallel lines have the same slope, the line we are looking for will also have a slope of -3. Using the point-slope form with the given point (3, -1), the equation becomes:
y - (-1) = -3(x - 3).
Simplifying:
y + 1 = -3(x - 3).
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due soon! Please help :)
Answer:
Dining Hall = 24 square meters
Kitchen = 48 square meters
Together = 72 square meters
1. What number is 20% of 40?
Answer:
8
Step-by-step explanation:
0.20 x 40 = 8
Therefore, 20% of 40 is 8
The first term of a geometric sequence is 8 and the fourth term is 216. What is the sum of the first 12 terms of the corresponding series? A. 2,125,760 B. 6,377,288 C. 236,192 D. 708,584
Answer:
2,125,760
Step-by-step explanation:
The first term (a) is 8
The fourth term is 216
Hence the sum of the first 12 term can be calculated as follows
= 8-8(3)^12/1-3
= 8-24^12/-2
= 2,125,760
The sum of first 12 terms is 2,125,760
Sum of the first 12 term = 2,125,760
For geometric sequence,
aₙ = arⁿ⁻¹
where
a = first term
r = common ratio
n = number of terms
Therefore,
a = 8
a₄ = 216
let's find the common ratio
216 = 8 × r⁴⁻¹
216 = 8 × r³
r³ = 216 / 8
r³ = 27
r = [tex]\sqrt[3]{27}[/tex]
r = 3
Let's find sum of the first 12 terms.
Sₙ = a (rⁿ - 1) / r - 1
S₁₂ = 8(3¹² - 1) / 3 - 1
S₁₂ = 8(531440) / 2
S₁₂ = 4251520 / 2
S₁₂ = 2,125,760
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If the point (x,y) is in Quadrant IV, which of the following must be true?
x<0 and y>0
x>0 and y<0
The statement x > 0 and y < 0 must be true for any point (x, y) located in Quadrant IV.
If the point (x, y) is in Quadrant IV, it means that the x-coordinate is positive and the y-coordinate is negative. In Quadrant IV, the x-values are positive, as they are to the right of the y-axis, and the y-values are negative, as they are below the x-axis.
Therefore, the correct statement is:
x > 0 and y < 0.
In Quadrant IV, the x-values are greater than 0, indicating a positive x-coordinate, and the y-values are less than 0, indicating a negative y-coordinate. This is because in Quadrant IV, the x-axis is to the right of the y-axis, and the y-axis is below the x-axis.
Hence, the statement x > 0 and y < 0 must be true for any point (x, y) located in Quadrant IV.
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the path of a projectile launched from a 16-ft-tall tower is modeled by the equation y = −16t2 64t 16. which is the correct graph of the equation?
The graph is a downward-opening parabola opening downward with its vertex at (2,0). Therefore, the correct option is C.
The equation y = -16t^2 + 64t + 16 represents the path of a projectile launched from a 16-ft-tall tower. To determine the correct graph, we can analyze the equation. The coefficient of t^2 (-16) is negative, indicating a downward-opening parabola. The coefficient of t (64) determines the horizontal shift of the graph, and in this case, t = 4 represents the maximum height of the projectile. The constant term (16) represents the initial height of the tower.
Considering these factors, we find that the correct graph of the equation is option C. It depicts a downward-opening parabola with its vertex at (2,0). The parabola starts at an initial height of 16 ft (the tower's height) and descends symmetrically from its vertex.
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(- x * y) ^ 2 - 8x ^ - 7 * y ^ - 2
I need help fast!!!
What is the answer??
Answer:
answer is \/
Step-by-step explanation:
two positive numbers x and y with maximum value 4 add up to 5. what is the difference between the maximum and minimum value of the product x^2y^3
The difference between the maximum and minimum value of x^2y^3 is determined by finding the product at the extreme values.
Given that two positive numbers x and y with a maximum value of 4 add up to 5, we can solve for the minimum value of x and y. Since the maximum value for x and y is 4, we assume that one number is 4 and the other number is 1, as they add up to 5.
Calculating the product x^2y^3 for these values, we have (4^2)(1^3) = 16. To find the maximum value of x and y, we need to distribute the sum of 5 unevenly between them. By assigning x = 1 and y = 4, we get (1^2)(4^3) = 64.
Therefore, the difference between the maximum and minimum value of x^2y^3 is 64 - 16 = 48.
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A method of assigning probabilities based upon personal judgment is referred to as the:
a. subjective method b. classical method c, Crelative frequency method
d, . non-probabilistic method.
A method of assigning probabilities based upon personal judgment is referred to as the subjective method. The correct answer is a.
The subjective method of assigning probabilities is based on personal judgment or subjective beliefs about the likelihood of events occurring. It involves incorporating individual knowledge, experience, and intuition to estimate the probabilities of different outcomes.
Unlike the classical method, which assigns probabilities based on equally likely outcomes, or the relative frequency method, which uses observed frequencies to determine probabilities, the subjective method relies on individual opinions and subjective assessments of probabilities.
In the subjective method, probabilities are not determined through mathematical calculations or empirical data. Instead, they are based on an individual's expertise, opinions, and subjective reasoning.
This method is commonly used in situations where objective data or historical information is limited or unavailable, such as in decision-making under uncertainty or when dealing with complex and uncertain scenarios.
The correct answer is a.
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According to a recent survey, the probability that the driver in a fatal vehicle accident is female (event F) is 0.2868. The probability that the driver is 24 years old or less (event A) is 0.1828. The probability that the driver is female and is 24 years old or less is 0.0576.
(a) Find the probability of FUA
(b) Find the probability of F'UA.
a. Using intersection of events probability of FUA is 0.0165
b. Using complement rule, the probability of F'UA is 0.9835
What is the probability of FUA?To find the probabilities, we can use the given information and apply the appropriate probability rules.
(a) FUA represents the event that the driver is female, 24 years old or less, and is involved in a fatal vehicle accident.
We can calculate this probability using the formula: P(FUA) = P(F ∩ A), where ∩ denotes the intersection of events.
P(FUA) = P(F) * P(A|F)
Given information:
P(F) = 0.2868 (probability that the driver is female)
P(A) = 0.1828 (probability that the driver is 24 years old or less)
P(A|F) = 0.0576 (probability that the driver is 24 years old or less given that the driver is female)
P(FUA) = 0.2868 * 0.0576 ≈ 0.0165
Therefore, the probability of FUA is approximately 0.0165.
(b) F'UA represents the event that the driver is not female, 24 years old or less, and is involved in a fatal vehicle accident.
We can calculate this probability using the complement rule: P(F'UA) = 1 - P(FUA).
P(F'UA) = 1 - P(FUA) ≈ 1 - 0.0165 ≈ 0.9835
Therefore, the probability of F'UA is approximately 0.9835.
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2 The table below shows the cost, s, to purchase p pounds of steak at a
local grocery store.
Cost of Steak
Number of
Pounds (p)
2.5
3.0
$23.98
$28.77
$33.57
$38.36
3.5
4.0
Which statement is true about the relationship shown in the table?
A The relationship is proportional, and the equation that represents this
relationship is s = 9.59 + p.
B. The relationship is proportional, and the equation that represents this
relationship is s = 9.59 + P.
The relationship is proportional, and the equation that represents this
relationship is s = 9.59p.
The relationship is not proportional.
Answer:
OMg whats there ikdd im tring
Step-by-step explanation: Im starting to solve it
does anyone here know how to write negative 1 on there
Answer:
you fo to the fourth box on there and put - then go to the box after that and put 1
Step-by-step explanation:
sorry if im wrong
If Lucy instead invested her $20,000 in an account with 5% simple interest,
how much would she end up with after 10 years?
Answer:
The 2009 Nobel Peace Prize was awarded to United States President Barack Obama for his "extraordinary efforts to strengthen international diplomacy and cooperation between people".
Step-by-step explanation:
The 2009 Nobel Peace Prize was awarded to United States President Barack Obama for his "extraordinary efforts to strengthen international diplomacy and cooperation between people".
An adult pass is 3 times as much as a child's pass. What is the cost of the adults pass? Write an expression for the situation
Answer:
however much the Child's pass is x 3 = Adult Pass cost
Step-by-step explanation:
plllllzzz helppppppp
Answer:
5. B
6. A
7. C
Step-by-step explanation:
Answer:
5.) b. 12 - 2x
6.) a. 8x
7.) c. 5d + 3c = 66
Step-by-step explanation:
uhm, I don't really have proof, i've just been doing this for a long time, you're just gonna have to trust me on this one
please help me with this one!!
Answer:
V = 960 cm^3
Step-by-step explanation:
Volume of a rectangular prism = L * W * H
V = 12 * 8 * 10
V = 960 cm^3
Answer:
d. 960 cm^3
Step-by-step explanation:
The formula for volume is length x width x height. To solve you have to multiply 12 by 8 by 10 to get 960
Graph y = -x^2 – 2. Identify the vertex of the graph. Tell whether it is a minimum or maximum.
Answer:
Maximum, and (0, -2)
Step-by-step explanation:
Adan took a test at school and completed it in an hour and a half. The test had two sections. If he took 45 minutes to complete the first Section, how much time did he use to finish the second section?
Answer:
45 minutes
Step-by-step explanation:
1 1/2 hours = 90 minutes
90 - 45 = 45
In a survey of 100 men and 120 women, it was found that 40 of the men had group A blood. If sex and blood type are assumed to be independent variables, what is the expected number of women in the survey who have group A blood?
The expected number of women in the survey who have group A blood is 48.
To calculate the expected number, we assume that sex and blood type are independent variables. This means that the proportion of women with group A blood would be the same as the proportion of men with group A blood.
Out of 100 men surveyed, 40 had group A blood. This corresponds to a proportion of 40/100 = 0.4.
Since sex and blood type are assumed to be independent, we can assume that the proportion of women with group A blood is also 0.4.
To find the expected number of women with group A blood, we multiply the proportion by the total number of women surveyed:
Expected number of women with group A blood = Proportion of women with group A blood * Total number of women surveyed
Expected number of women with group A blood = 0.4 * 120 = 48
Therefore, the expected number of women in the survey who have group A blood is 48.
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The graph shows the height(t) of a model rocket t seconds after it is launched from the ground at 48 feet per second. Where is the height of the rocket increasing? Where is it decreasing?
A. The height of the rocket is always decreasing
B. The height of the rocket is always increasing
C. The height of the rocket is increasing when 0< t < 3 and decreasing when 3 < t < 6.
D. The height of the rocket is increasing when 3 < t< 6 and decreasing when 0 < t < 3
The height of the rocket is increasing when 0< t < 3 and decreasing when 3 < t < 6.
Option C is correct.
Increasing function:From the graph,
It is observed that , In interval 0< t < 3 , slope of graph is positive. Therefore, the height of the rocket is increasing
It is observed that , In interval 3 < t< 6 , slope of graph is negative. Therefore, the height of the rocket is decreasing.
Therefore, option C is correct.
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if u know what to do pls help
Answer:
6.2 miles
Step-by-step explanation:
10 kilometers=.62 miles
.62*10= 6.2 miles
Suppose you have two similar trapezoids with a scale factor of 4. If the angle measures of trapezoid ABCD are 70,110,110,70, what is the answer?
Answer: 140, 220, 220, 140
Step-by-step explanation:
All you have to do is double the numbers.
Social Networking sites A recent survey of 10 social networking sites has a mean of 12.67 million visitors for a specific month. The standard deviation was 4 million. Find the 99% confidence interval of the true mean. Assume the variable is normally distributed. Round your answers to at least two decimal places.
______million <μ< _____million
The 99% confidence interval of the true mean is given as follows:
8.56 million < μ < 16.78 million.
What is a t-distribution confidence interval?
The t-distribution is used when the standard deviation for the population is not known, and the bounds of the confidence interval are given according to the equation presented as follows:
[tex]\overline{x} \pm t\frac{s}{\sqrt{n}}[/tex]
The variables of the equation are listed as follows:
[tex]\overline{x}[/tex] is the sample mean.t is the critical value.n is the sample size.s is the standard deviation for the sample.The critical value, using a t-distribution calculator, for a two-tailed 99% confidence interval, with 10 - 1 = 9 df, is t = 3.2498.
The parameter values for this problem are given as follows:
[tex]\overline{x} = 12.67, s = 4, n = 10[/tex]
The lower bound of the interval is then given as follows:
[tex]12.67 - 3.2498 \times \frac{4}{\sqrt{10}} = 8.56[/tex]
The upper bound of the interval is then given as follows:
[tex]12.67 + 3.2498 \times \frac{4}{\sqrt{10}} = 16.78[/tex]
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The mean diastolic blood pressure of 144 individuals is 61 mmHg with a standard deviation of 10 mmHg Construct a 95% confidence interval for p. Select one: O a. (61.25, 62.63) O b. (-59.37, 60.59) O c. (59.37, 62.63) O d. (143.85, 145.56) O e. (-143.85, 145.56) Determine the success rate of the student from the following course list Course Score of student Standard Mean, deviation, Physics Mathematics 60 58 55 49 11 16 Select one: a. PHYS>MATH O b. PHYS=MATH = O c. NONE C. O d. MATH>PHYS
The the success rate of the student is PHYS>MATH. Therefore, the correct answer is Option A.
We can calculate the success rate of the student by calculating the z-score of each of the courses. The z-score is calculated as (x-μ)/σ, where x is the individual score, μ is the mean of the data, and σ is the standard deviation of the data.
For Physics, we have x=60, μ=49, and σ=11. So the z-score is (60-49)/11=1.09.
For Mathematics, we have x=58, μ=55, and σ=16. So the z-score is (58-55)/16=0.19.
Since the z-score of Physics is greater than the z-score of Mathematics, the student's success rate in Physics is greater than the student's success rate in Mathematics.
Therefore, the correct answer is Option A.
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Peter makes 15 dollars an hour and he spends 25 dollars a day on transportation and food. Write an expression to describe his spendings and earnings in a day, where h is the number of hours that Peter works that day
15h - 25 dollars is an expression to describe his spendings and earnings in a day.
What is algebraic Expression?Any mathematical statement that includes numbers, variables, and an arithmetic operation between them is known as an expression or algebraic expression. In the phrase 4m + 5, for instance, the terms 4m and 5 are separated from the variable m by the arithmetic sign +.
Peter makes 15 dollars an hour, so if he works for h hours, he earns:
15h dollars
Peter spends 25 dollars a day on transportation and food, so his total spending can be expressed as:
25 dollars
Therefore, his total earnings minus his total spending can be expressed as:
15h - 25 dollars
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i'm really stuck on this part. once i figure out how i'll mark brainliest (however you spell it).
just the first one 11 ........
A person pushes a car with a force of 50 pounds. The car moves 5 feet into his garage. How much work was done?
Answer:
10
Step-by-step explanation:
Answer:
10
Step-by-step explanation:
Plzzz help me!!!!!!!!
Actual sales for January through April are shown below.
Month Actual Sales (Yt)
January 18
February 25
March 34
April 40
May -
Use exponential smoothing with α = .3 to calculate smoothed values and forecast sales for May from the above data. Assume the forecast for the initial period (January) is 18. Show all the forecasts from February through April along with the answer.
The forecasted sales for February through April are as follows:
February: 19.5, March: 25.65, April: 30.755. The forecasted sales for May is approximately 35.928.
Exponential smoothing is a time series forecasting method that assigns weights to past observations, with the weights decreasing exponentially as the observations get older. The smoothed value for a particular period is a weighted average of the previous smoothed value and the actual value for that period.
To calculate the smoothed values and forecast sales using exponential smoothing with α = 0.3, we start with the initial forecast for January, which is given as 18. Then, for February, we use the formula:
Smoothed value (February) = α * Actual sales (February) + (1 - α) * Smoothed value (January)
= 0.3 * 25 + 0.7 * 18 = 19.5
Similarly, for March:
Smoothed value (March) = α * Actual sales (March) + (1 - α) * Smoothed value (February)
= 0.3 * 34 + 0.7 * 19.5 = 25.65
And for April:
Smoothed value (April) = α * Actual sales (April) + (1 - α) * Smoothed value (March)
= 0.3 * 40 + 0.7 * 25.65 = 30.755
Finally, for the forecasted sales in May:
Forecasted sales (May) = Smoothed value (April) = 30.755
Therefore, the forecasted sales for May, using exponential smoothing with α = 0.3, is approximately 35.928.
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Which of the following polynomials is reducible over Q 4x3³ + x - 2 5x³ + 9x² - 3 This option 3x³ - 6x² + x - 2 This option 08 O This option None of choices This option Activ 74°F Sun
None of the options are reducible polynomial
How to determine the reducible polynomialFrom the question, we have the following parameters that can be used in our computation:
The list of options
The variable Q means rational numbers
So, we can use the rational root theorem to test the options
So, we have
(a) 4x³ + x - 2
Roots = ±(1, 2/1, 2, 4)
Roots = ±(1, 1/4, 2, 1, 1/2)
(b) 3x³ - 6x² + x - 2
Roots = ±(1, 2/1 ,3)
Roots = ±(1, 1/3, 2, 2/3)
(c) 5x³ + 9x² - 3
Roots = ±(1, 3/1 ,5)
Roots = ±(1, 1/5, 3, 3/5)
See that all the roots have rational numbers
And we cannot determine the actual roots of the polynomial.
Hence, none of the options are reducible polynomial
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