Identify the correct description for the formula u'(t) = -(2k)-Puſt – k)+(2k)-fu(t+k) from the following options: FFD1: forward finite difference with stepsize k for the first derivative of u att BFD1: backward finite difference with stepsize k for the first derivative of u att CFD1: central finite difference with stepsize k for the first derivative of u at t CFD2: central finite difference with stepsize k for the second derivative of u att None of the Above
Answers
The correct description for the formula u'(t) = -(2k)-Puſt – k)+(2k)-fu(t+k) is CFD1: central finite difference with stepsize k for the first derivative of u at t.
The given formula is a differential equation that describes the behavior of the function u(t) over time t. The derivative of u(t) with respect to t, denoted by u'(t), is given by the right-hand side of the equation:
u'(t) = -(2k)^(-1) * [Pu(t-k) - 2u(t) + Pu(t+k)] - k + (2k)^(-1) * [fu(t+k)]
Here, k is a positive constant, and P is a constant matrix. The term (2k)^(-1) represents the inverse of twice the step size k, which is used in a numerical method known as central finite difference with step size k, denoted as CFD1.
The central finite difference method is used to approximate the first derivative of a function u(t) at a point t, by computing the slope of a secant line passing through two neighboring points (t-k) and (t+k). The slope of this secant line is then divided by 2k to obtain the approximation of the first derivative at t.
In the given formula, the term [Pu(t-k) - 2u(t) + Pu(t+k)] represents the central finite difference approximation of the first derivative of u(t) at time t with step size k.
The term k subtracts a constant rate of decay from the function u(t) at every time step. The term (2k)^(-1) * [fu(t+k)] adds a contribution to the function u(t) at a future time t+k, where f is a given function.
Therefore, the formula u'(t) = -(2k)^(-1) * [Pu(t-k) - 2u(t) + Pu(t+k)] - k + (2k)^(-1) * [fu(t+k)] represents the behavior of a function u(t) using CFD1 method to approximate its first derivative at time t with step size k, along with a rate of decay and a contribution from a future time t+k.
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Related Questions
Set up the appropriate form of a particular solution y, for the following differential equation, but do not determine the values of the coefficients. y (4) +10y" +9y = 5 sin x + 5 cos 3x Which of the following is the appropriate form of a particular solution yp? O A. yp = (A+BX+Cx? + Dxº) e* OB. Yp = Ax cos x + Bx sin x + Cx cos 3x + Dx sin 3x Oc. Yo = (A + Bx) e*+Csin 3x + Dcos 3X OD. Yp = A cos x +B sin x +C cos 3x + D sin 3x Click to select your answer. BI Type here to search
Answers
The values of these coefficients to set up the appropriate form of the particular solution is A cos(x) + B sin(x) + C cos(3x) + D sin(3x)
The correct option is: Yp = A cos(x) + B sin(x) + C cos(3x) + D sin(3x)
To set up the appropriate form of a particular solution for the given differential equation, we need to first determine the type of the forcing function. The forcing function in this case is 5sinx + 5cos3x, which is a combination of sine and cosine functions with different frequencies. Therefore, the appropriate form of the particular solution would be a combination of sine and cosine functions with coefficients that need to be determined.
The general form of the particular solution can be written as:
yp = A cos(x) + B sin(x) + C cos(3x) + D sin(3x)
Here, A, B, C, and D are the coefficients that need to be determined using the method of undetermined coefficients or variation of parameters. We do not need to determine the values of these coefficients to set up the appropriate form of the particular solution.
Therefore, the correct option is:
D. Yp = A cos(x) + B sin(x) + C cos(3x) + D sin(3x)
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The appropriate form of a particular solution for a differential equation is the form of the solution that matches the form of the non-homogeneous term in the equation. In this case, the non-homogeneous term is 5sin(x) + 5cos(3x), which is a sum of trigonometric functions.
Therefore, the appropriate form of a particular solution would be a linear combination of trigonometric functions, as seen in options B and D. However, we cannot determine the values of the coefficients without further information. It is important to note that the choice of the appropriate form of a particular solution is crucial in finding the complete solution to a differential equation, as it allows us to separate the homogeneous and non-homogeneous parts and solve them separately.
To set up the appropriate form of a particular solution, yp, for the given differential equation y(4) + 10y'' + 9y = 5sinx + 5cos3x, you need to consider the terms on the right-hand side of the equation. Since the right-hand side contains sin(x) and cos(3x) terms, the particular solution should also include these trigonometric functions.
The appropriate form of a particular solution, yp, is: yp = A cos x + B sin x + C cos 3x + D sin 3x (option D). In this form, A, B, C, and D are coefficients to be determined later, and the solution contains the necessary trigonometric functions that match the right-hand side of the given differential equation.
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HELP! 30 POINTS!
Q4. An 8-sided fair die is rolled twice and the product of the two numbers obtained when the die is rolled two times is calculated.
(a) Draw the possibility diagram of the product of the two numbers appearing on the die in each throw (
b) Use the possibility diagram to calculate the probability that the product of the two numbers is
I) A prime number
ii) Not a perfect square
iii) A multiple of 5
iv) Less than or equal to 21
v) Divisible by 4 or 6
Answers
Answer: i) 12.5% ii) 87.5% iii) 23.4%
Step-by-step explanation:
There are two die, each with 8 sides so the total number of outcomes is 8²
i) Product is a prime number --> 2, 3, 5, 7
Since it is a prime we only have the option of one die being a 1 and the other being the prime number.
So possible successful outcomes is: (1, 2), (1, 3), (1, 5), (1, 7), (7, 1), (5, 1), (3, 1), and (2, 1) = 8 options.
\(\dfrac{successful\ outcomes}{total\ possible\ outcomes}=\dfrac{8}{8^2}=\dfrac{1}{8}=\large\boxed{12.5\%}\)
ii) Not a perfect square. Let's find the probability that it IS a perfect square and subtract that value from the total (64).
Product IS a perfect square --> 1, 4, 9, 16, 25, 36, 49, 64
Possible successful outcomes of a perfect square: (1, 1) (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 8) = 8 options.
Product is NOT a perfect square = 64 - 8 = 56 options
\(\dfrac{successful\ outcomes}{total\ possible\ outcomes}=\dfrac{56}{8^2}=\dfrac{7}{8}=\large\boxed{87.5\%}\)
iii) Product is a multiple of 5 --> 5, 10, 15, 20, ,25, 30, 35, 40,
Possible successful outcomes of a multiple of 5: (1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (7,5), (8, 5), (5, 8), (5, 7), (5, 6), (5, 4), (5, 3), (5, 2), (5, 1) = 15 options
\(\dfrac{successful\ outcomes}{total\ possible\ outcomes}=\dfrac{15}{8^2}=\dfrac{15}{64}=\large\boxed{23.4\%}\)
You are limited to 3 questions on Brainly so hopefully you can follow what I did to answer the last two questions (less than or equal to 21) and (divisible by 4 or 6). If not, please post this question again and state "Need iv and v only".
(2x-5)(x+2) write the area as a product and a sum
Answers
Answer:
2x^2 -x -10 is the answer as a product
3x-3 as a sum
Step-by-step explanation:
Assuming (2x-5)(x+2)=0
Let's distribute.
(2x^2+4x)(-5x-10)=0
Then combine like terms
2x^2 -x -10 is the answer as a product
3x-3 as a sum
current research indicates that the distribution of the life expectancies of a certain protozoan is normal with a mean of 47 days and a standard deviation of 10.2 days. find the probability that a simple random sample of 25 protozoa will have a mean life expectancy of 53 or more days.
Answers
The Probability that a sample of 25 protozoa drawn at random will have a mean life expectancy of 53 days or more. P(x ≥ 53) = 0.42
Since the life expectancy distribution of a particular protozoan is normal, we would use the normal distribution formula, which is written as
z = (x - µ)/σ
Where
x = life expectancies of the certain protozoan.
µ = mean
σ = standard deviation
n = number of samples
From the information given,
µ = 47days
σ = 10.2 days
n = 25
The probability that a simple random sample of 25 protozoa will have a mean life expectancy of 53 days or more is given by the following formula:
P(x ≥ 53) = 1 - P(x < 53)
For x = 53
z = (53 - 47)/(10.2/√25) = 6/2.04 =2.94
According to the normal distribution table, 0.58 represents the probability for the z score.
P(x ≥ 53) = 1 - 0.58 = 0.42
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The data set below represents the heights (in inches) of students in a particular high school class:
72 67 62 64 59 71 62 66 67 75 67 62
What is the range of the data set?
Answers
By identifying the maximum value (75) and the minimum value (59) in the data set, we can determine the range, which is 16 inches .
The range of a data set measures the spread or variability of the data. It is calculated by subtracting the minimum value from the maximum value. In this case, the minimum value is 59 (the shortest height) and the maximum value is 75 (the tallest height) in the given data set.
Range = Maximum value - Minimum value
Substituting the values, we have:
Range = 75 - 59 = 16
Therefore, the range of the given data set is 16. This means that the heights of the students in the high school class range from a minimum of 59 inches to a maximum of 75 inches, with a difference of 16 inches between them.
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Solve 3[-x + (2x + 1)] = x - 1.
Answers
Answer:
x=-2
Step-by-step explanation:
distribute 3
solve
what is 0.123 with the 23 repeating
Answers
Answer:
The answer is 61/495 simplified.
(Please use a separate piece of paper to calculate the answer for this problem.) A study to determine the sensitivity and specificity of a new test for macular degeneration is conducted on 2430 people. Macular degeneration occurs at a rate of 16.72 percent. Your sample has the same prevalence of macular degeneration. You find that 377 people with macular degeneration tested positive with the new test. You also have a total of 561 positive test results in your study. CALCULATE THE POSITIVE PREDICTIVE VALUE of this test under these circumstances. Group of answer choices 83.29% 98.45% 67.20% 92.86% 23.09%
Answers
Answer:
\(67.20\%\)
Step-by-step explanation:
Given
\(n =2430\) --- sample
\(r = 16.72\%\) --- degeneration rate
From the findings, we have:
\(p = 377\) --- tested positive
\(t =561\) --- total test
Required
The positive predicted value (PPV)
This is calculated using:
\(PPV=\frac{Positive}{Total}\)
i.e.
\(PPV = \frac{377}{561}\)
\(PPV = 0.6720\)
Express as percentage
\(PPV = 0.6720 * 100\%\)
\(PPV = 67.20\)
Which table represents a linear function?
Answers
1 -3
2 -5
3 -7
4 -9
Both sides have a certain pattern they follow, which makes it a linear.
please let me know if this was correct!
A pancake recipe uses 1/4 cup of all-purpose flour and 1/4 cup of rice flour. What is the ratio of all-purpose flour to rice flour in the recipe?
Answers
Answer:
1:1 ratio
Step-by-step explanation:
a 1:1 ratio happens when the amount of the item is the same as the other. 1/4 cup of all purpose flour is the same amount as 1/4 rice flour so the recipe uses a 1:1 ratio
A ball is thrown upward from the top of a 160-foot-high building. The ball is 192 feet above ground level after 1 second, and it reaches ground level in 5 seconds. The height above ground is a quadratic function of the time after the ball is thrown. Write the equation of this function.
Answers
Let the given values represent t and f(t), where t is the x-coordinate and f(t) is the y-coordinate. Thus, the given values are points on the function:
\(\begin{gathered} (t_1,f(t_1))=(0,160) \\ (t_2,f(t_2))=(1,192) \\ (t_3,f(t_3))=(5,0) \end{gathered}\)Note that f(t) represents the height of ball after t seconds.
Suubstitute the values of t and f(t) for x and y, respectively, in the quadratic equation:
\(y=ax^2+bx+c\)Thus, we have the following system of equations:
Substituting (0,160) into the equation, we have:
\(\begin{gathered} y=ax^2+bx+c \\ 160=a(0)^2+b(0)+c \\ 160=a(0)+b(0)+c \\ 160=0+0+c \\ 160=c \end{gathered}\)Substituting (1,192) and c = 160 into the equation, we have:
\(\begin{gathered} y=ax^2+bx+c \\ 192=a(1)^2+b(1)+160 \\ 192=a(1)+b(1)+160 \\ 192=a+b+160 \\ 32=a+b \end{gathered}\)Substituting, (5,0) and c = 160 into the equation, we have:
\(\begin{gathered} y=ax^2+bx+c \\ 0=a(5)^2+b(5)+160 \\ 0=a(25)+b(5)+160 \\ 0=25a+5b+160 \\ -160=25a+5b \end{gathered}\)Multiply each term on both sides of the the equation 32 = a + b by -5.
\(\begin{gathered} (-5)(32)=-5a-5b \\ -160=-5a-5b \end{gathered}\)Add the obtained equation to -160=25a+5b to eliminate the variable term 5b and then divide both sides by 20 to find a.
\(\begin{gathered} -320=20a \\ \frac{-320}{20}=\frac{20a}{20} \\ -16=a \end{gathered}\)Substitute the value of a into 32 = a+b and then add both sides of the equation by 16 to find the value of b.
\(\begin{gathered} 32=a+b \\ 32=-16+b \\ 48=b \end{gathered}\)Substitute the obtained values of a, b, and c into the quadratic equation.
\(y=-16x^2+48x+160\)To recheck, graph the equation and determine if all of the given points are on the graph. The graph of the equation is as follows:
Note that the graph of the equation in the problem is represented on the Quadrant I only.
Since all of the points are on the equation, the quadratic equation which represents the path of the ball must be y = - 16x² + 48x + 160, where x is a non-negative number.
please answer number 6
Answers
g assuming a linear relationship between x and y, the larger r-square value is, the larger the absolute value of the coefficient of correlation
Answers
In the domain of finance, an R-Squared value above 0.7 is typically seen as indicating a high level of correlation, whereas one below 0.4 indicates a low level of correlation.
What is correlation?A statistical measure known as correlation expresses how closely two variables are related linearly (meaning they change together at a constant rate). It's a typical technique for describing straightforward connections without explicitly stating cause and consequence.
The strength of the association is measured by the sample correlation coefficient, or r. The statistical significance of correlations is also examined.
For describing straightforward links between data, correlations are helpful. Consider a dataset of campgrounds in a park in the mountains as an illustration. You're interested in finding out if the height of the campsite—how high up the mountain it is—and the summer's typical high temperature are related.
Hence, In the domain of finance, an R-Squared value above 0.7 is typically seen as indicating a high level of correlation, whereas one below 0.4 indicates a low level of correlation.
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can anybody help on this question ? please and thank you !!
Answers
Answer:
Sorry I'm only in algebra one.
Step-by-step explanation:
Select ALL the expressions that have the same value as 9s
1. 9+s
2. 9*s
3. 3s + 6s
4. 3(3s)
5. s(5+4)
6. 3s * 3s
7. s+s+s+s+s+s+s+s+s
Answers
Answer:
1
Step-by-step explanation:
The ratio of dogs or cats available for adoption in animal shelters across the city is 9:7 if there are 154 cats available for adoption how many dogs are there available for adoption?
Answers
Answer: 198 dogs
Step-by-step explanation: Assuming you meant to say that the ratio of dogs to cats is 9:7, then you can quickly figure out the amount of dogs by looking at the ratio as a fraction. Instead of seeing it as 9:7, look at the ratio as \(\frac{9}{7}\) and then use that fraction to find the dog amount. You just multiply the amount of cats by the ratio, which we found is \(\frac{9}{7}\) and you should get the final answer of 198 dogs
Answer:
Answer: 198 dogs
Step-by-step explanation: Assuming you meant to say that the ratio of dogs to cats is 9:7, then you can quickly figure out the amount of dogs by looking at the ratio as a fraction. Instead of seeing it as 9:7, look at the ratio as and then use that fraction to find the dog amount. You just multiply the amount of cats by the ratio, which we found is and you should get the final answer of 198 dogs
Step-by-step explanation:
Could you please help with this question
Answers
Answer:
0
Step-by-step explanation:
Smallest value of C would be when A = - 2 and B = - 2
\(\because C = (A - B) ^2 \\ \therefore C = \{ - 2- ( - 2) \}^2 \\ \therefore C = \{ - 2 + 2 \}^2 \\ \therefore C = \{ 0 \}^2 \\ \huge \red{ \boxed{\therefore C = 0}}\)
Consider the following functions.
f(x)=2x^2+4x+10
g(x)=3x-11
h(x)=3x^3+x^2=21
The value of h(f(g(2)))
Answers
Answer:
Step-by-step explanation:
h(f(g(2)))=
h(f(3*2-11))) = h(f( -5))=
h((2*(-5)^2) +4*(-5) +10) = h (50 -20 +10) = h(40)
now calculate h(40) but check if you wrote corectly the equation
Ali has hired mark and alexis to work for his shipping company. mark can load a truck with packages in 120 minutes. alexis can load the same number of packages in 240 minutes. if mark and alexis work together on a particular truck, they can load all of the packages in___minutes.
Answers
If mark and Alexis work together on a particular truck, they can load all of the packages in 90 minutes.
It will take 90 minutes for both Mark and Alexis to load all of the products.
Given,
that Mark can load a truck with items in 120 minutes and Alexis can do the same in 240 minutes, the average time for the two of them is as follows:
Average = (120 + 240) ÷ 2
= 360 ÷ 2
= 180
In this case, the term "Average" refers to the value that should be used to represent the sample. By dividing the total number of values in a particular set by the sum of all the values in that set, the average is determined. also known as the arithmetic mean.
Since there were two parties engaged, our calculation is 180 ÷ 2 = 90.
Hence, in this case, it is concluded that the correct answer is option B. 90 minutes.
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Help me thank y'all so much
Answers
Answer:
15 p - 3 is the answer....
Which ratio is equivalent to 8:3?
12:32
12:3
40:15
64:9
Answers
Answer: 40:15
Step-by-step explanation: 8 x 5 = 40 / 3 x 5 = 15
Both of them are multiple and equivalent hope this helps
6. Two trains traveled from opposite
directions on parallel tracks toward
the bullfight town. They started
1,110 km apart at exactly the same
time. Train A's speed was 55 km/h
and Train B's speed was 85 km/h.
Would the trains have
met within 5 hours?
Answers
Answer:
They will not have met
Step-by-step explanation:
Train A will have traveled 275km in 5 hours and Train B will have traveled 425km in 5 hours. This means that they will be 700km closer to one another but still 400km apart
Complete the table so a linear function is represented. SHOW ALL WORK!
Answers
Answer:
g
Step-by-step explanation:
Form the perfect square trinomial in the process of completing the square. What is the value of c?
x2 + 3x + c = StartFraction 7 Over 4 EndFraction + c
Answers
Answer:
-9/4
Step-by-step explanation:
x2 + 3x + c complete the square
(x+( 3/2)^2 -9/4 c = -9/4
Answer:
Step-by-step explanation:
9/4
A large university provides housing for 15 percent of its graduate students to live on campus. The university’s housing office thinks that the percentage of graduate students looking for housing on campus may be more than 15 percent. The housing office decided to survey a random sample of graduate students, and 78 of the 433 respondents say that they are looking for housing on campus. a) On the basis of the survey data, would you recommend that the housing office consider increasing the amount of housing on campus available to graduate students? Give appropriate evidence to support your recommendation. [Conduct a hypothesis test: State,Plan, Do,Conclude] b) Interpret the p-value obtained in part a) in context. c) In addition to the 433 graduate students who responded to the survey, there were 21 who did not respond. If these 21 had responded, is it possible that your recommendation would have changed? Explain. d) Describe what a Type II error would be in the context of the study, and also describe a consequence of making this type of error. e) Describe what a Type I error would be in the context of the study, and also describe a consequence of making this type of error.
Answers
a) Hypothesis test:
Null hypothesis (H0): The percentage of graduate students looking for housing on campus is equal to 15%.
Alternative hypothesis (Ha): The percentage of graduate students looking for housing on campus is greater than 15%.
To test the hypothesis, we can use a one-sample proportion test. We will calculate the test statistic and compare it to the critical value or p-value to make a decision.
The observed proportion of graduate students looking for housing on campus is 78/433 = 0.1804.
Using a significance level (α) of 0.05, we will conduct the test and calculate the test statistic and p-value.
Plan:
Test statistic: z = (p - p) / sqrt(p(1-p)/n)
where p is the observed proportion, p is the hypothesized proportion (0.15), and n is the sample size (433).
Do:
Calculating the test statistic:
z = (0.1804 - 0.15) / sqrt(0.15 * 0.85 / 433)
z ≈ 2.07
Conclude:
Since the test statistic is 2.07, we compare it to the critical value or calculate the p-value.
The critical value for a one-sided test with a significance level of 0.05 is approximately 1.645. Since 2.07 > 1.645, the test statistic falls in the rejection region.
The p-value associated with the test statistic of 2.07 is less than 0.05. Therefore, we reject the null hypothesis.
Based on the survey data, there is evidence to suggest that the percentage of graduate students looking for housing on campus is greater than 15%. The housing office should consider increasing the amount of housing available to graduate students.
b) The p-value obtained in part a) represents the probability of obtaining a test statistic as extreme as the one observed (or more extreme), assuming the null hypothesis is true.
In this case, the p-value is less than 0.05, which suggests strong evidence against the null hypothesis. It indicates that the observed proportion of graduate students looking for housing on campus is significantly higher than the hypothesized proportion of 15%.
c) Including the 21 non-respondents would change the sample size and potentially affect the estimated proportion. If these additional respondents had similar characteristics to the 433 who responded, it is possible that the recommendation might still remain the same.
However, the exact impact depends on the responses of the non-respondents, so it is difficult to determine the precise effect without their data.
d) Type II error in this study would occur if the housing office fails to increase the amount of housing on campus when it is actually necessary (i.e., the percentage of graduate students looking for housing on campus is higher than 15%).
This means the null hypothesis would not be rejected when it should have been. A consequence of this type of error would be the unmet demand for housing, potentially causing dissatisfaction among graduate students and a shortage of available housing options.
e) Type I error in this study would occur if the housing office increases the amount of housing on campus when it is not necessary (i.e., the percentage of graduate students looking for housing on campus is not higher than 15%). This means the null hypothesis would be rejected incorrectly.
A consequence of this type of error would be allocating resources and efforts towards increasing housing capacity unnecessarily, which could result in wastage of resources and potentially impact other areas of the university's operations.
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write the sequence of natural numbers which leaves the remainder 3 on didvidng by 10
Answers
The sequence of natural numbers that leaves a remainder of 3 when divided by 10 is:
3, 13, 23, 33, 43, 53, 63, 73, 83, 93, 103, 113, ...
\(\huge{\mathfrak{\colorbox{black}{\textcolor{lime}{I\:hope\:this\:helps\:!\:\:}}}}\)
♥️ \(\large{\underline{\textcolor{red}{\mathcal{SUMIT\:\:ROY\:\:(:\:\:}}}}\)
PLEASE HELP!! 100 POINTS
Consider this dilation.
(a) Is the image of the dilation a reduction or an enlargement of the original figure? Explain.
(b)What is the scale factor? Explain and show your work.
Answers
The red figure is the original and the green the dilation. We know this because the dilated points are marked with '.
Therefore, we can see that the dilation is a reduction of the original figure. The figure got smaller in size after the dilation.
To find the scale factor, we'll need to look at a few of the points before and after the dilation.
A = (-4, 4)
A' = (-2, 2)
C = (0, 4)
C' = (0, 2)
---The scale factor will be the same on all the points, which is why I have selected only 2 sets of points to look at.
Now, we can figure out the scale factor by thinking about how -4 got to -2, how 4 got to 2. In this case, it is simply dividing by 2. The scale factor must be in multiplication, so we cannot use division in reporting the scale factor, it must be a fraction.
2 / 4 = 1 / 2
The scale factor, therefore, is 1/2.
Hope this helps!
Answer:
a) A reduction.
b) 0.5 or 1/2
Step-by-step explanation:
a) The new points are indicated by the ' next to the letters. By observing the graph, we can see that new points have lessened, and therefore shrank the dialtion, so it is a reduction.
b) On each point, it is reducted by 0.5.
Point A went down from 4 to 2, which is half of 4.
Point B went down from -2 to -1, which is half of -2.
Point C went down from 4 to 2, which is half of 4.
Point D went down from 8 to 4, which is half of 8.
hope this helps!
I at least need some help on how to figure this out
Which statement correctly compares the function shown on this graph with
the function y = 2x - 5?
A. The function shown on the graph has a smaller rate of change and lower starting point.
B. The function shown on the graph has a smaller rate of change, but a higher starting point.
C. The function shown on the graph has a greater rate of change and a higher starting point.
D. The function shown on the graph has a greater rate of change, but a lower starting point.
Answers
Answer:
C. The function shown on the graph has a greater rate of change and a higher starting point.
Step-by-step explanation:
I don't have an explanation but I had got it wrong on and it said C was the answer.
1) In Mr. Gragert's rock collection, 70% of the rocks are
metamorphic. If the entire collection contains 300 rocks,
calculate how many of the rocks are metamorphic.
Mavel
Answers
Answer:
210 rocks are metamorphic
Step-by-step explanation:
Rewrite by separating off the final termn+1
∑ m(m+1)
m=1
Answers
The rewritten summation is \(\sum_{m=1}^{n}m(m+1)+(n+1)(n+2)\).
A recursive function is a function that calculates subsequent terms by repeating or using its own prior terms, creating a succession of terms. By definition, the recursive function can be used to write the summation by separating the final term of the summation and adding a final term to the summation.
Given, the function is \(\sum_{m=1}^{n+1}m(m+1)\).
First, we have to separate the final term from the given summation. The final term is m = n+1 which is present above the summation. For this replace the n+1 with n.
Then, add the term m(m+1) with m in the given summation replaced by n+1.
The new summation is rewritten as,
\(\begin{aligned}\sum_{m=1}^{n+1}&=\sum_{m=1}^{n}m(m+1)+(n+1)((n+1)+1)\\&=\sum_{m=1}^{n}m(m+1)+(n+1)(n+2)\end{aligned}\)
The new summation is rewritten.
The complete question is -
Rewrite by separating off the final term \(\sum_{m=1}^{n+1}m(m+1)\)
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