By mathematical induction, we've proven that the given recursive algorithm computes factorials correctly, i.e., fact(n) = n! for all n ∈ N.
How to solveTo prove the algorithm using mathematical induction, we'll proceed in two steps: the base case and the inductive step.
Base case: n = 0
By definition, 0! = 1. According to the given algorithm, when n = 0, fact(0) = 1. Therefore, the base case holds, as fact(0) = 0! = 1.
Inductive step:
Assume that the algorithm is correct for some arbitrary non-negative integer k, i.e., fact(k) = k!. We want to show that the algorithm is also correct for k + 1, i.e., fact(k + 1) = (k + 1)!.
According to the algorithm:
fact(k + 1) = (k + 1) * fact(k)
Using the induction hypothesis that fact(k) = k!, we can substitute this into the equation:
fact(k + 1) = (k + 1) * k!
By definition of the factorial function, (k + 1)! = (k + 1) * k!, so:
fact(k + 1) = (k + 1)!
We've shown that if fact(k) = k!, then fact(k + 1) = (k + 1)!. Thus, the inductive step is complete.
By mathematical induction, we've proven that the given recursive algorithm computes factorials correctly, i.e., fact(n) = n! for all n ∈ N.
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A bag contains 3 red marbles, 6 blue marbles and 4 green marbles. If three marbles are drawn out of the bag, what is the probability, to the nearest 1000th, that all three marbles drawn will be blue?
please help
Answer:
0.077
Step-by-step explanation:
If a man was born in 1898 how old was he in 1945
Answer: He could be 47 or 46
Step-by-step explanation: I need the exact date of the time he was born (just the month and the day) to tell you how old he really is, but with the information given, he could be 47 or 46.
He would be 47 years [tex]1945\\-\\1898\\=\\47\\[/tex]
Pls answer question with explanation I will give u branliest if u help (find area of the triangle)
Answer:
1. 90
4. 468
Step-by-step explanation:
Answer:
234
Step-by-step explanation:
If your asking about number 4, the formula for the area of a triangle is height times base divided by 2. If we plug in, we get 30 times 15.6 which is 468. Divide by 2 and you get 234.
Graph the function.
f(x)=√x-4
Plot four points on the graph of the function: the leftmost point and three additional points. Then click on the graph-a-function button.
The four points on the graph of f(x) = √x-4 including the left most point are: (4,0), (5,1), (8,2), and (20,4).
Finding four points on the graph of the functionTo find points on the graph of f(x) = √x-4, we can choose values of x and find the corresponding values of y.
The leftmost point on the graph is when x is the smallest possible value, which is x = 4 (since we can't take the square root of a negative number). At x = 4, f(x) = √0 = 0, so the leftmost point is (4,0).
To find three additional points, we can choose values of x that are greater than 4 and find the corresponding values of y:
When x = 5: f(5) = √1 = 1, so the point is (5,1)
When x = 8: f(8) = √4 = 2, so the point is (8,2)
When x = 20: f(20) = √16 = 4, so the point is approximately (20.4)
So, four points on the graph of f(x) = √x-4 are: (4,0), (5,1), (8,2), and (20,4).
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The surface area of a prism can be determined by the formula S = 2B + Ph, where B is
the area of the base, P is the perimeter of the base and h is the height of the prism. How
can this equation be rearranged to solve for the area of the base, B?
the equation rearranged to solve for B is: B = 1/2 * (-S + Ph)
what is equation ?
An equation is a mathematical statement that shows that two expressions are equal. It typically contains one or more variables and can be solved to find the value(s) of the variable(s) that make the equation true.
In the given question,
To solve for the area of the base (B), we need to isolate it on one side of the equation.
First, we can simplify the equation by distributing the 2 to the terms inside the parentheses:
S = 2B + Ph
S = 2B + P * h
Next, we can move the term containing B to the other side of the equation by subtracting 2B from both sides:
S - 2B = P * h
Then, we can isolate B by dividing both sides of the equation by 2:
(S - 2B) / 2 = P * h / 2
Simplifying the right side of the equation:
(S - 2B) / 2 = Ph/2
Finally, we can isolate B by subtracting S/2 and dividing by -1:
(S/2 - Ph/2) = B
Therefore, the equation rearranged to solve for B is:
B = 1/2 * (-S + Ph)
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Please help
A space station is being built from 24 cubic modules. The space station can be any shape but the modules must be placed together so that entire faces match up with each other. Choose a tool to create two different plans for the space station. Explain why you chose the tool you selected.
Select the correct answer from each drop-down menu.
Chris purchased a new mattress for $1,399. He will need to pay $78 each month on his interest-free loan to pay off the total price of the mattress in 18
months. Separately, Chris sets aside $50 each month in a savings account, which currently shows a balance of $250.
Create an augmented matrix from a system of equations to help Chris determine when he can use monthly payments and savings to pay off the total
purchase price of the mattress,
Row 1
Row 2
Column 1
Column 2
>
Reset
Column 3
Next
>
>
Rewriting the equations in matrix form, we get:
[1 78/1399 | 1]
[50 1 | 1149/1399]
What is square matrix?
A square matrix is a matrix that has the same number of rows and columns. That is, a matrix A is square if A has dimensions n x n, where n is a positive integer.
To create an augmented matrix, we need to represent the system of equations in matrix form. Let x be the number of months it will take Chris to pay off the total purchase price of the mattress using monthly payments and savings. Then the system of equations is:
1399/18 + 78x = 1399
50x + 250 = 1399
The first equation represents the fact that the total price of the mattress must be paid off in 18 months using monthly payments, and the second equation represents the fact that Chris sets aside $50 each month in savings to put towards the purchase.
Rewriting the equations in matrix form, we get:
[1 78/1399 | 1]
[50 1 | 1149/1399]
This is the augmented matrix for the system of equations, where the leftmost columns represent the coefficients of the variables (1 for x in the first equation, 50 for x in the second equation, and 78/1399 for the monthly payment in the first equation), and the rightmost column represents the constants on the right-hand side of each equation.
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Column 1 Column 2 Column 3
Row 1-- 50 -1 -250
Row 2-- 78 1 1,399
I got it right on my test.
Your Welcome:)
How could expanded land acreage and pricing be viewed as some to be a bad thing
for the recovering Great Plains?
Increased land prices and acreage on the Great Plains could be seen as negative owing to potential ecological impact, as a disruption for Traditional communities and unsustainable resource use.
Why expanded land acreage and pricing in Great plains is considered negative?Ecological impact: Increased land acreage and price may cause habitat loss, biodiversity loss, soil erosion, and water pollution, all of which might be detrimental to the area's natural ecosystem.Disruption of traditional farming and ranching communities: Increasing land prices and greater parcel sizes may make it harder for small-scale farmers and ranchers to compete, which might result in the consolidation of land ownership and the eradication of traditional farming and ranching communities.Unsustainable resource use: Increasing land area might result in more water being used for irrigation, thereby worsening the Great Plains' water shortage problems.Loss of grasslands: The Great Plains are renowned for their distinctive grassland ecosystems, which are essential for biodiversity and carbon sequestration. More land being used for construction or agriculture might lead to the extinction of grasslands, which would have detrimental ecological effects.Degradation of the soil: Intensive agricultural techniques, such as monoculture and extensive chemical usage, may result in the loss of fertility and accelerated erosion of the soil. Long-term, this can have a negative effect on the sustainability and production of agricultural areas.Increased use of chemical inputs: Increasing land area may result in higher usage of chemical pesticides and fertilisers, which might impact regional ecosystems and contribute to environmental degradation.Danger of overproduction: Increased land acreage and pricing may lead to overproduction of agricultural products, which might lead to unstable prices, unstable farmer livelihoods, and potential market saturation.Potential for land speculation: Increasing land prices may encourage speculative investment, which might result in land being bought and retained for financial gain rather than for productive use. This could push up land prices even more and restrict access to land for local farmers and ranchers.Risks to wildlife and biodiversity: Include habitat fragmentation, the loss of wildlife corridors, and the displacement of native species, all of which might have an effect on the biodiversity and animal populations of the area.Learn more about land acreage and pricing here:
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A private high school advertised that last year their freshman students, on average, had a score of 80 on the high school entrance exam. Assuming that average refers to the mean, which of the following claims must be true based on this information?
Note: More than one statement could be true. If none of the statements is true, mark the appropriate box.
A. All of the freshman students scored 80 or higher on the exam.
B. Most of the freshman students scored 80 or higher on the exam.
C. None of the freshman students scored 80 or higher on the exam.
A. All of the freshman students scored 80 or higher on the exam.
B. Most of the freshman students scored 80 or higher on the exam.
Please help with the answer!
Answer:
Step-by-step explanation:
2.1 so a
A bag contains 9 green marbles, 6 purple marbles, 4 red marbles, 10 blue marbles, and 7 orange marbles. One marble is randomly selected from the bag.
What is the probability that the marble selected is blue or purple?
Answer as a Fraction, not simplified.
So, the answer as a fraction is 16/36.
What is fraction?A fraction is a mathematical expression that represents a part of a whole or a ratio between two quantities. It is written as one number (the numerator) placed above another number (the denominator) and separated by a horizontal line. For example, 3/4 is a fraction where 3 is the numerator and 4 is the denominator. The numerator represents the number of equal parts being considered, while the denominator represents the total number of equal parts that make up the whole. Fractions can also be written in decimal or percentage form. Fractions are commonly used in everyday life, such as in cooking recipes, measurements, and financial calculations.
There are 6 purple marbles and 10 blue marbles, so there are 6 + 10 = 16 marbles that are either blue or purple.
The total number of marbles in the bag is 9 + 6 + 4 + 10 + 7 = 36.
Therefore, the probability of selecting a blue or purple marble is 16/36.
So, the answer as a fraction is 16/36
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Evaluate the expression.
20.4 0.2
Write your answer as an integer or a decimal. Do not round.
After solving the given expression 2 - 0.4 ÷ 0.2 the resultant answer is an integer which is 0.
What is an integer?Zero, a positive natural number, or a negative integer denoted by a minus sign are all examples of integers.
The inverse additives of the equivalent positive numbers are the negative numbers.
The boldface Z is a common mathematical symbol for the set of integers.
Numbers that are negative: -1, -2, -3, -4, -5, and so on indefinitely.
Non-negative integers include 0 and all whole numbers that are positive, such as 6, 7, 8, 9, 10, and so forth.
Positive integers include 1, 2, 3, 4, 5, and on indefinitely.
So, we have the expression:
= 2 - 0.4 ÷ 0.2
Now, evaluate using the BODMAS rule as follows:
= 2 - 0.4 ÷ 0.2
= 2 - 2
= 0
Therefore, after solving the given expression 2 - 0.4 ÷ 0.2 the resultant answer is an integer which is 0.
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Order each amount from least to greatest
1 pint, 2 gallons, 3 quarts, 3 cups
What is the length of side AB?
The length of side AB is given as follows:
[tex]AB = \frac{6}{\sqrt{3}}[/tex]
What are the trigonometric ratios?The three trigonometric ratios are the sine, the cosine and the tangent, and they are defined as follows:
Sine of angle = length of opposite side to the angle divided by the length of the hypotenuse.Cosine of angle = length of adjacent side to the angle divided by the length of the hypotenuse.Tangent of angle = length of opposite side to the angle divided by the length of the adjacent side to the angle.For the angle of 30º, we have that:
AB is the opposite side.AC is the adjacent side.Hence the length of side AB is obtained as follows:
tan(30º) = AB/6
AB = 6 x tan(30º)
[tex]AB = 6 \times \frac{1}{\sqrt{3}}[/tex]
[tex]AB = \frac{6}{\sqrt{3}}[/tex]
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The value of the side AB is calculated as: AB = 6/√3
How to use trigonometric ratios?There are different trigonometric ratios used in right angle triangles which are:
sin x = opposite/hypotenuse
cos x = adjacent/hypotenuse
tan x = opposite/adjacent
Looking at the given triangle, the value of AB can be gotten via the trigonometric ratio tan x. Thus:
AB = 6 * tan 30
AB = 6 * 1/√3
AB = 6/√3
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The half-life of a radioactive isotope is the time it takes for a quantity of the isotope to be reduced to half its initial mass. Starting with 155 grams of a radioactive isotope, how much will be left after 5 half-lives?
Use the calculator provided and round your answer to the nearest gram.
Answer:
i think its 11.2
Step-by-step explanation:
Triangle D has been dilated to create triangle D'. Use the image to answer the question.
112
02
24
3
D
Determine the scale factor used.
2.1
3.8
4.8
D
4.2
Since triangle D has been dilated to create triangle D', the scale factor used include the following: A. 1/2.
What is scale factor?In Mathematics and Geometry, the scale factor of a geometric figure can be calculated by dividing the dimension of the image (new figure) by the dimension of the pre-image (original figure):
Scale factor = Dimension of image (new figure)/Dimension of pre-image (original figure)
By substituting the given parameters into the formula for scale factor, we have the following;
Scale factor = Dimension of image/Dimension of pre-image
Scale factor = 2.4/4.8
Scale factor = 1/2.
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Solve for X
Please show step by step
(X - 4)^2 = 25
Answer: x = -1 and x = 9
Step-by-step explanation:
Lets solve this equation step by step
1. Simplify both sides of the equation
x^2 - 8x + 16 = 25
2. Subtract 25 from both sides
x^2 - 8x + 16 - 25 = 25 - 25
3. Factor the left side of the equation
(x + 1) (x - 9) = 0
4. Set the factors equal to zero
x + 1 = 0 or x - 9 = 0
Thus your answers are x = -1 and x = 9
You can check by inputting the values into the original equation
(-1 - 4)^2 = 25 and (9 - 4)^2 = 25
Francisco invested $5,000 into an account that earns 3% compounded
quarterly. How many times will Francisco earn interest after keeping the money in
the account for 5 years?
What will be the value of his account in 5 years if he does not make any
additional deposits or withdrawals?
Answer: To calculate how many times Francisco will earn interest after keeping the money in the account for 5 years, we need to determine the number of quarters in 5 years.
Since there are 4 quarters in a year, the total number of quarters in 5 years is:
4 quarters/year x 5 years = 20 quarters
Therefore, Francisco will earn interest 20 times over the 5-year period.
To calculate the value of his account in 5 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (initial investment)
r = the annual interest rate (as a decimal)
n = the number of times the interest is compounded per year
t = the time the money is invested (in years)
In this case, P = $5,000, r = 0.03, n = 4 (since the interest is compounded quarterly), and t = 5. Substituting these values into the formula, we get:
A = $5,000(1 + 0.03/4)^(4x5)
A = $5,000(1.0075)^20
A = $5,866.02
Therefore, the value of Francisco's account in 5 years, if he does not make any additional deposits or withdrawals, will be $5,866.02.
Step-by-step explanation:
A bowl contained 33.4 grams of salt. Then, Charlie poured in another 58.09 grams. How much salt does the bowl contain now?
Solve x/ 2 < 4. Graph the solution
Answer:
The answer is x<8
Step-by-step explanation:
x/2<4
x<8
2. Kurt designs a square flower garden that has the same area as a rectangular one. The length of the rectangular
garden is 10 feet longer than the side of the square garden. The width of the rectangular garden is 5 feet shorter
than the side of the square. Find the side length of the rectangular garden.
low
The side length of the the rectangular garden are 20 feet by 5 feet.
Calculating the size of a Rectangular GardenLet
s = side length of the square garden
l = length of the rectangular garden
w = width of the rectangular garden
From the problem statement,
area of the square garden = area of the rectangular garden
Since the area of a square is just the side length squared, we can write:
s² = l * w
We also know that the length of the rectangular garden is 10 feet longer than the side of the square garden, and that the width of the rectangular garden is 5 feet shorter than the side of the square garden. We can express them as:
l = s + 10
w = s - 5
Now we can substitute these expressions for l and w into our first equation:
s² = (s + 10) * (s - 5)
Expanding the right-hand side, we get:
s² = s² + 5s - 50
0 = 5s - 50
5s = 50
Dividing both sides by 5, we get:
s = 10
Since the side length of the square garden is 10 feet.
We can use this to find the dimensions of the rectangular garden:
l = s + 10 = 10 + 10 = 20
w = s - 5 = 10 - 5 = 5
So the dimensions of the rectangular garden are 20 feet by 5 feet.
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Liam is making
8 sculptures. Each
sculpture needs 2/3 yard of wire for the
base and another piece of wire for the
top. He uses 10 2/3 yards of wire in all.
How much wire is needed for the top of
each sculpture?
Mathematically, the quantity of wire Liam needs for the top of the 8 sculptures he is making is 5¹/₃ yards.
How is the quantity of wire for the top determined?The quantity of wire Liam needs for the top of the sculptures that he makes is determined using mathematical operations.
The four basic mathematical operations include addition, subtraction, division, and multiplication.
The total quantity of wire used = 10²/₃ yards.
The quantity of wire needed for each base = ²/₃ yards
The quantity of wire needed for each top = ²/₃ yards
The number of sculptures Liam is making = 8
²/₃ yards + ²/₃ yards = 1¹/₃ yards
8 x 1¹/₃ yards = 10²/₃ yards
8 x ²/₃ yards = 5¹/₃ yards
5¹/₃ yards x 2 = 10²/₃ yards
Therefore, the quantity of wire needed for the top will be 5¹/₃ yards.
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Find the area? For this shape pleae
Justin says that x^-2and -x² are equivalent. Is he
correct? Justify your answer.
[tex]~\hspace{7em}\textit{negative exponents} \\\\ a^{-n} \implies \cfrac{1}{a^n} ~\hspace{4.5em} a^n\implies \cfrac{1}{a^{-n}} ~\hspace{4.5em} \cfrac{a^n}{a^m}\implies a^na^{-m}\implies a^{n-m} \\\\[-0.35em] ~\dotfill\\\\ x^{-2}\implies \cfrac{1}{x^2}\hspace{5em} {\Large \begin{array}{llll} \ne \end{array}}\hspace{5em}-x^2[/tex]
Wilson's Woodworking Shop produces wood chairs and sells them for $250. The company's weekly profit, (y), depends on the number of $1.50 price increases (x) for each chair. Use the graph below to answer the question.
What is the approximate number of price increases needed to cause zero profit?
A) 0
B) 40
C) 87
D) 97
The number of price increases that would be needed to cause zero profit as shown on the graph, is C) 87.
How to find the number of price increases ?The point on the graph that would show the number of price increases necessary for the company to record zero profit would be the point on the x - axis where the line crosses the x-axis. This is because this would show the 0 profit mark.
This point on the graph is shown to be 87. This means that if the company increases the price of the wood chairs they sell, 87 times, they will experience zero profit.
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Need help on this one.
The following recursive function computes the number of comparisons used in the worst case of merge sort.
M (1) = 0
M (2k) = 2M (2k−1) + 2k for all k > 0
Use mathematical induction to prove that M (2k) = k · 2k for all k ∈ N.
By the principle of mathematical induction, we have shown that
M(2k) = k 2k for all k ∈ N.
The Principle of Mathematical Induction:The principle of mathematical induction is a proof technique used to prove a statement that holds for all positive integers.
It consists of two steps:
Base case: Show that the statement is true for the smallest possible value of the positive integer, usually n = 1.
Inductive step: Assume that the statement is true for an arbitrary positive integer k, and then use this assumption to show that the statement must also be true for k + 1.
Here we have
The following recursive function computes the number of comparisons used in the worst case of merge sort.
M (1) = 0
M (2k) = 2M (2k−1) + 2k for all k > 0
To prove that M(2k) = k × 2k for all k ∈ N using mathematical induction, we will first show that the base case holds:
Base case: When k = 1,
we have M(2) = 2M(1) + 2¹ = 2(0) + 2 = 2, and k × 2k = 1 × 2¹ = 2.
Therefore, the base case holds.
Inductive hypothesis:
Assume that M(2k) = k × 2k holds for some positive integer k.
Inductive step:
We need to show that M(2k+1) = (k+1) × 2k+1.
Using the recursive definition of M, we have:
M(2k+1) = 2M(2k) + 2(2k+1)
= 2(k × 2k) + 2(2k+1) (using the inductive hypothesis)
= 2k(2k+2) + 2(2k+1)
= 2k(2(k+1)) + 2(2k+1)
= 2(k+1) × 2k + 2(2k+1)
= (k+1) × 2(2k+1)
Therefore, the inductive step holds.
Hence,
By the principle of mathematical induction, we have shown that
M(2k) = k 2k for all k ∈ N.
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What is the third quartile of this data set? 14, 18, 20, 21, 25, 32, 42, 46, 48
I need help with this
Answer:
13
Step-by-step explanation:
Show that by the uniqueness theorem the linear transformation,
Y = aX + b, is also a normal random variable.
We can show by the uniqueness theorem that the linear transformation, Y = aX + b, is also a normal random variable because the resultant probaility density fnction of Y equals: f(y) = (1/√(2πa^2σX^2)) * exp(-(y-aμX-b)^2/(2a^2σX^2)).
How to prove a normal random variableTo show that the linear transformation, Y = aX + b is a normal random variable, we need to demonstrate that it satisfies the properties of a normal distribution. This means that it should have a bell-shaped probability density function, mean, and variance.
We can prove that it meets the mean condition this way:
E(Y) = E(aX + b) = aE(X) + b = aμX + b
Next, we can prove that it meets the variance condition thus:
Var(Y) = Var(aX + b) = a^2Var(X) = a^2σX^2
Lastly, the probability density function is given as f(y) = (1/√(2πa^2σX^2)) * exp(-(y-aμX-b)^2/(2a^2σX^2)). This proves that the conditions for a normal random variable is met.
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HELP GIVING BRAINLY AND 20 POINTS CORRECTIONS DUE LATER
Answer:
Step-by-step explanation:
cos47=80/x
x=80/0.6820
x=117.3020 feet