The solution to the given recurrence relation is:
an = ((3 + √2) / (4√2))(2 + 2√2)^n + ((3 - √2) / (4√2))(2 - 2√2)^n.
The given recurrence relation is an = 4an−1 + 4an−2, with initial terms a0 = 1 and a1 = 2. We will solve this recurrence relation using the characteristic equation and initial conditions.
The characteristic equation for the recurrence relation is found by assuming the solution to be of the form an = r^n. Substituting this into the recurrence relation, we get r^n = 4r^(n-1) + 4r^(n-2).
Dividing both sides by r^(n-2), we have r^2 = 4r + 4. Rearranging the equation, we get r^2 - 4r - 4 = 0.
To solve this quadratic equation, we can use the quadratic formula: r = (-b ± √(b^2 - 4ac)) / (2a). Plugging in a = 1, b = -4, and c = -4, we get r = (4 ± √(16 + 16)) / 2 = (4 ± √(32)) / 2 = 2 ± 2√2.
Thus, the general solution for the recurrence relation is of the form an = Ar1^n + Br2^n, where r1 = 2 + 2√2 and r2 = 2 - 2√2.
Using the initial conditions a0 = 1 and a1 = 2, we can plug in these values to solve for A and B. Substituting n = 0 and n = 1 into the general solution and equating them to the given initial conditions, we get:
a0 = A(2 + 2√2)^0 + B(2 - 2√2)^0 = A + B = 1,
a1 = A(2 + 2√2)^1 + B(2 - 2√2)^1 = (2 + 2√2)A + (2 - 2√2)B = 2.
Solving these equations simultaneously, we find A = (3 + √2) / (4√2) and B = (3 - √2) / (4√2).
an = ((3 + √2) / (4√2))(2 + 2√2)^n + ((3 - √2) / (4√2))(2 - 2√2)^n is the solution to the given recurrence relation.
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The width of a rectangular frame is 6 in. shorter than its length. The area of the frame is 216 in?
What is the frame's length?
Answer:
18 in
Step-by-step explanation:
len = y
width= y - 6
the area of rectangle is len x width
(Y) x (Y-6) = 216
Y^2 - 6Y = 216, Y^2 - 6y -216 = 0
y = (36 /2) or (-24 /2)
len couldn't be negative so 18
a 18 The number 21 has no composite factors. What is another number that has no composite factors? A. 27- B. 52- C. 77- D. 81-
The number that has no composite factors is (c) 77
How to determine another number that has no composite factorsFrom the question, we have the following parameters that can be used in our computation:
Number = 21
The factors of 21 are 3 and 7
These factors are composite numbers because they are prime numbers
using the above as a guide, we have the following:
The number 77 has no composite factors
This is so because
77 = 7 * 11
These factors are composite numbers because they are prime numbers
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I need this answer as soon as possible
The perimeter is the distance all the way around. So it's the sum of the lengths of all 4 sides.
From the picture, you can clearly see the lengths of all 4 sides.
Writum down and adum up !
I'll give brainliest please help
Answer:
∠R = 38°
Step-by-step explanation:
∠R = 1/2(arc PE - arc SQ)
∠R = 1/2(140 - 64) = 1/2(76) = 38°
The diameter of a circle is 20 centimeters. What is the circumference?
C ≈ 62.83cm is the answer!
Hope this helps!
Find the range of the function
Answer:
y€ R : y greater than or equal to 1
At the state fair, admission at the gate is $9. In addition, the cost of each ride is $2. Suppose that Reuben will go on x rides.
Reuben wants the total number of dollars he spends on admission and rides to be fewer than . Using the values and variables given, write an inequality describing this.
Answer:
i think you forgot to add how little he wants to spend
Step-by-step explanation:
Reuben wants the total number of dollars he spends on admission and rides to be fewer than ??? whats the number that he wants to spend
A population of values has a normal distribution with u = 42.3 and o = 44.6. You intend to draw a random sample of size n = 20. Find the probability that a single randomly selected value is greater than 19.4. P(X> 19.4) = Find the probability that a sample of size n = 20 is randomly selected with a mean greater than 19.4. P(M> 19.4) =
Previous question
(a) The probability that a single randomly selected value is greater than 19.4. P(X> 19.4) = 0.6965.
(b) The probability that a sample of size n = 20 is randomly selected with a mean greater than 19.4. P(M> 19.4) = 0.9883.
To solve these probability questions, we can utilize the properties of the standard normal distribution since we know the mean and standard deviation of the population.
(a) Find the probability that a single randomly selected value is greater than 19.4.
To calculate this probability, we need to standardize the value 19.4 using the formula: z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation.
z = (19.4 - 42.3) / 44.6 = -22.9 / 44.6 ≈ -0.51498
Using the standard normal distribution table or a calculator, we can find the corresponding probability for z > -0.51498, which is approximately 0.6965.
Therefore, P(X > 19.4) ≈ 0.6965.
(b) Find the probability that a sample of size n = 20 is randomly selected with a mean greater than 19.4.
For this question, we need to consider the sampling distribution of the sample mean. The mean of the sampling distribution is equal to the population mean (μ = 42.3), and the standard deviation of the sampling distribution, also known as the standard error, is equal to σ / √n, where σ is the population standard deviation and n is the sample size.
Standard error = 44.6 / √20 ≈ 9.9766
To find the probability that the sample mean is greater than 19.4, we need to standardize the sample mean using the formula: z = (x - μ) / (σ / √n), where x is the value (19.4 in this case), μ is the mean, σ is the standard deviation, and n is the sample size.
z = (19.4 - 42.3) / (44.6 / √20) ≈ -22.9 / 9.9766 ≈ -2.2972
Using the standard normal distribution table or a calculator, we can find the corresponding probability for z > -2.2972, which is approximately 0.9883.
Therefore, P(M > 19.4) ≈ 0.9883.
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How would I find the arc of WTV
To find the arc of circle WTV, you need the arc length or central angle. Use formulas: Arc Length = (Arc Angle / 360) × (2πr) or Arc Length = (Central Angle / 360) × (2πr).
To find the arc of a circle, WTV, you would need to know either the length of the arc or the measure of the central angle subtended by the arc. If you have the length of the arc, you can use the formula:
Arc Length = (Arc Angle / 360 degrees) × (2πr),
where r is the radius of the circle. Rearranging the formula, you can solve for the Arc Angle:
Arc Angle = (Arc Length / (2πr)) × 360 degrees.
If you know the measure of the central angle, you can calculate the arc length using a similar formula:
Arc Length = (Central Angle / 360 degrees) × (2πr).
To find the radius, you would need additional information such as the diameter or circumference of the circle. Once you have the radius, you can substitute the values into the appropriate formula to find either the arc length or the central angle of the circle.
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if a, b, and c are n n invertible matrices, does the equation c 1 .a c x /b 1 d i n have a solution, x? if so, find it.
The equation c₁ · a · c · x / b₁ · d · iₙ can have a solution for x if and only if the matrices a, b, and c are compatible and satisfy certain conditions.
Further information about the matrices and their properties is required to determine the specific solution.
To determine if the equation c₁ · a · c · x / b₁ · d · iₙ has a solution, we need to consider the compatibility and properties of the matrices involved.
Let's break down the equation:
c₁: The first column of matrix c.
a: Matrix a.
c: Matrix c.
x: The solution vector.
b₁: The first row of matrix b.
d: Matrix d.
iₙ: The identity matrix of size n.
For the equation to have a solution, the matrices a, b, and c need to be compatible, meaning their dimensions align appropriately for matrix multiplication. Specifically, the number of columns in matrix a should match the number of rows in matrix c.
Additionally, certain conditions or properties of the matrices may be required to ensure a solution exists. Without additional information about the specific matrices a, b, c, and d, it is not possible to determine the solution for x.
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Find the inverse Laplace transform of
a) F(s)= 10/s(s+2)(s+3)²
b) F(s)= s/s²+4s+5
c) F(s)=e^-3s s/(s-2)^2 +81
a) The solution to the given problem isL(F) = f(t) = 0 + (-(10 + 7C)/6)[tex]e^{-2t}[/tex]) + C[tex]e^{-3t}[/tex]+ D[tex]e^{-3t}[/tex]
b) The solution to the given problem isL(F) = [tex]e^{-2t}[/tex] [sin t + cos t](c)
c) The solution to the given problem is L(F) = 1/9 [[tex]e^{2t}[/tex] sin 9t - 3[tex]e^{2t}[/tex]) cos 9t]
(a)The inverse Laplace transform of F(s) = 10/s(s + 2)(s + 3)² can be found as follows:
L(F) = L{10/[s(s + 2)(s + 3)²]}
= 10 ∫∞₀[tex]e^{-st}[/tex]) /[s(s + 2)(s + 3)²] dt
L{F} = L⁻¹{10/[s(s + 2)(s + 3)²]}
By using partial fractions, we can simplify the equation and get it in a form that can be integrated easily.
L(F) = 10 ∫∞₀ {1/s - 2/(s + 2) + 3/(s + 3) - 2/(s + 3)² + 1/(s + 2)(s + 3)} [tex]e^{-st}[/tex]dt
L{F} = L⁻¹{1/s} - 2L⁻¹{1/(s + 2)} + 3L⁻¹{1/(s + 3)} - 2L⁻¹{d/ds[1/(s + 3)]} + L⁻¹{1/(s + 2)}
As the inverse Laplace transform of L{F} is given by L(F)
= L⁻¹{1/s} - 2L⁻¹{1/(s + 2)} + 3L⁻¹{1/(s + 3)} - 2L⁻¹{d/ds[1/(s + 3)]} + L⁻¹{1/(s + 2)}
Thus, the solution to the given problem isL(F) = f(t) = 0 + (-(10 + 7C)/6)[tex]e^{-2t}[/tex]) + C[tex]e^{-3t}[/tex]+ D[tex]e^{-3t}[/tex]
(b)
The inverse Laplace transform of F(s) = s/[s² + 4s + 5] can be found as follows:
L(F) = L{s/[s² + 4s + 5]}
= ∫∞₀ s e^(–st) / (s² + 4s + 5) dt
L{F} = L⁻¹{s/[s² + 4s + 5]}
By using partial fractions, we can simplify the equation and get it in a form that can be integrated easily. L(F) = ∫∞₀ [s/(s² + 4s + 5)] [tex]e^{-st}[/tex]) dt
L{F} = L⁻¹{s/(s² + 4s + 5)}
The solution to the given problem isL(F) = [tex]e^{-2t}[/tex] [sin t + cos t](c)
c) The inverse Laplace transform of F(s) = ([tex]e^{-3s}[/tex]) s/[(s - 2)² + 81] can be found as follows:
L(F) = L{([tex]e^{-3s}[/tex]) s/[(s - 2)² + 81]}= ∫∞₀ ([tex]e^{-st}[/tex]) s/[(s - 2)² + 81] dt
L{F} = L⁻¹{([tex]e^{-3s}[/tex])) s/[(s - 2)² + 81]}
So, The solution to the given problem is L(F) = 1/9 [([tex]e^{2t}[/tex]) sin 9t - 3([tex]e^{2t}[/tex]) cos 9t]
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How many zeros appear at the end of 115!? Do not compute 115!.
Your argument must come from prime factorizations to receive
credit.
there will be 27 zeros at the end of 115!.
To determine the number of zeros at the end of 115!, we need to consider the prime factorization of the number and examine how many factors of 5 are present.
A zero at the end of a factorial occurs when there is a factor of 10 present, which is equivalent to having both factors of 2 and 5. Since the number of factors of 2 is usually abundant, the crucial factor is the number of factors of 5.
In the prime factorization of 115!, the factors of 5 arise from the multiples of 5 (5, 10, 15, 20, ...) as well as higher powers of 5 (25, 50, 75, ...). We need to determine how many multiples of 5, multiples of 25, multiples of 125, and so on are present.
1. Multiples of 5: The number of multiples of 5 in 115! is given by ⌊115/5⌋ = 23.
2. Multiples of 25: The number of multiples of 25 in 115! is given by ⌊115/25⌋ = 4.
3. Multiples of 125: The number of multiples of 125 in 115! is given by ⌊115/125⌋ = 0 since there are no numbers in the range 1 to 115 that are multiples of 125.
Adding up these counts, we have 23 + 4 = 27 factors of 5.
Therefore, there will be 27 zeros at the end of 115!.
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Find the length of the third side if necessary write in simplest radical form
Answer:
hi
Step-by-step explanation:
Can you help me please thank you
Katie is making bread.
She’s needs 6/3 cups of flour to make one whole loaf of bread.
She has 2/3 cup of flour.
Katie can make less than one whole loaf of bread or more than or exactly one loaf
Renee is a sales associate at a store. She earns $80 a week plus a 15% commission on her sales. Last week, she sold $200 worth of items. What is the total amount Renee earned for the week? How much did she earn from commission?
Renee earned a total of $
for the week. The amount she earned from commission was $
.
Answer:
The amount earned for the week=$110
Amount earned from commission =$30
Step-by-step explanation:
commission earned on sales = $200×15%= $30
total amount for the week=$80 +$30= $110
Q1
The sum of the first 68 positive odd integers is ?
Q2
The degree of recurrence relation an = 2an-2 + 5an-49 is ??
Q3
In how many ways can an organization containing 19 members elect a president, treasurer and secretary (assuming no person is elected to more than one position)?
Q4
The Greatest Common Divisor (GCD) of 28 × 37 × 58 and 23 × 33 × 54 is ??
Q1: Sum of first 68 positive odd integers.
Let's represent the first 68 positive odd integers by: 1, 3, 5, 7, ..., 135, 137. The first term, a = 1. The last term, l = 137And, the number of terms, n = 68We need to find the sum of these terms. To find the sum of an arithmetic series, we use the following formula: Sn = n/2[2a + (n-1)d]. Here, d = common difference. Since the given sequence is of odd numbers, the difference between any two consecutive terms is 2. So, d = 2. Put these values in the formula to get: Sn = 68/2[2(1) + (68-1)2], Sn = 34[2 + 135], Sn = 68 × 67Sum of first 68 positive odd integers = 4546.
Q2: Degree of recurrence relation. To find the degree of a recurrence relation, we find the largest value of n in the relation. Here, an = 2an-2 + 5an-49The largest value of n in the relation is n = 49. So, the degree of the recurrence relation is 49.
Q3: Number of ways to elect office bearers in an organization. Let's assume that the 19 members of the organization are named M1, M2, M3, ..., M19. The president can be elected in 19 ways. After the president is elected, the treasurer can be elected in 18 ways. After the treasurer is elected, the secretary can be elected in 17 ways. Therefore, the total number of ways in which the president, treasurer, and secretary can be elected is:19 × 18 × 17 = 5,814.
Q4: Greatest Common Divisor (GCD)To find the GCD of two numbers, we need to find their prime factors.28 × 37 × 58 = 2² × 7 × 37 × 2 × 29 = 2³ × 7 × 29 × 37Similarly, 23 × 33 × 54 = 23 × 3² × 2 × 3 × 3 × 2 × 3 = 2³ × 3⁵ × 23.
The common prime factors are 2³ and 23. So, the GCD is: 2³ × 23 = 184. The GCD is 184.
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There are 180 trees in gardner grove orchard, and 18 of them are pears. What percent of the trees are pear trees?
What is the probability of spinning a number greater than 4?
Answer:
Please upload the full question.....here the total number of outcomes is not mentioned and hence it can't be solved.
Answer:
Hence, the required probability of getting a number greater than 4, P(E) = 1/3.
Step-by-step explanation:
The data set below represents a sample of scores on a 10-point quiz. 7, 4, 9, 6, 10, 9, 5, 4 Find the sum of the mean and the median
The sum of the mean and the median of the given dataset, which consists of the scores 7, 4, 9, 6, 10, 9, 5, and 4, is 13.25.
To find the sum of the mean and the median, we first need to calculate the mean and median of the given dataset.
The dataset is: 7, 4, 9, 6, 10, 9, 5, 4
To find the mean, we sum up all the numbers in the dataset and divide by the total number of data points:
Mean = (7 + 4 + 9 + 6 + 10 + 9 + 5 + 4) / 8 = 54 / 8 = 6.75
To find the median, we arrange the numbers in ascending order:
4, 4, 5, 6, 7, 9, 9, 10
Since we have 8 data points, the median will be the average of the two middle numbers:
Median = (6 + 7) / 2 = 6.5
Now we can find the sum of the mean and the median:
Sum of mean and median = 6.75 + 6.5 = 13.25
Therefore, the sum of the mean and the median of the given dataset is 13.25.
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Clarence has $90 in a savings account that earns 10% annually. The interest is not
compounded. How much will he have in 5 years?
Answer:
132 dollars I think
Step-by-step explanation:
The domain of the function f(x) = 8 - 5x is restricted to the positive integers. Which values are elements of the
range?
-2
3
8
13
18
23
Answer:
23
Step-by-step explanation:
please mark me as brainliest
Use the binomial series to find a Taylor polynomial of degree 3 for 1 91 +32 T3(0) X + c? + 23
The Taylor polynomial of degree 3 for the function 1/(1-2x) centered at x=0 is (1+2x+4x²+8x³).
Explanation: Given, 1/(1-2x) = ∑n=0 to infinity of 2^n * x^n The above series is the binomial series for (1+x)^n where n=-1Using the binomial series for n=-1, we have1/(1-2x) = ∑n=0 to infinity of 2^n * x^n= ∑n=1 to infinity of 2^(n-1) * x^(n-1)= 1 + ∑n=1 to infinity of 2^n * x^nTaking up to degree 3, we get1/(1-2x) = 1 + 2x + 4x² + 8x³ + ...Therefore, the Taylor polynomial of degree 3 for 1/(1-2x) is 1 + 2x + 4x² + 8x³.
An infinite sum of words that are expressed in terms of a function's derivatives at a single point is known as the Taylor series or Taylor expansion of a function in mathematics. Near this point, the function and the sum of its Taylor series are equivalent for the majority of common functions. for Brook Taylor, who introduced the Taylor series in 1715, they are named for him. In honour of Colin Maclaurin, who made great use of this unique example of Taylor series in the middle of the 18th century, a Taylor series is sometimes known as a Maclaurin series where 0 is the point at which the derivatives are taken into account.
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Find the value of x. I WILL MARK YOU BRAINLIEST!!!
Answer:
4x+3x+2x=180
9x=180
x=20
Joey made strawberry jam and raspberry jam. He made enough strawberry jam to fill 1/2 of a jar. If he made 2/5 as much raspberry jam as strawberry jam, how many jars will the raspberry jam fill?
Answer: 1/5 of a jar
Step-by-step explanation:
1/2 times 2/5= .2 = 1/5.
Answer:
1/5 of a jar
Step-by-step explanation:
The linear approximation at z = 0 to sin(42) is A + Bz where A is:
the linear approximation at z = 0 to sin(42) is A + Bz, where A = sin(42) and B is the coefficient of z, which is cos(42).
The linear approximation of a function f(x) at a point x = a is given by the equation f(x) ≈ f(a) + f'(a)(x - a). In this case, we want to approximate sin(42) at z = 0.
The derivative of the sine function is cos(x), so the derivative of sin(42) with respect to z is cos(42). Evaluating the derivative at z = 0, we have cos(42).
To find A in the linear approximation A + Bz, we substitute z = 0 into the original function sin(42) and obtain A = sin(42).
Therefore, the linear approximation at z = 0 to sin(42) is A + Bz, where A = sin(42) and B is the coefficient of z, which is cos(42).
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Write a survey question for which you would expect to collect numerical data.
Answer:
How many siblings do you have?
What is the expanded form of 1,693,222,527? A. 1,000,000,000 + 600,000,000 + 90,000,000 + 3,000,000 + 220,000 + 20,000 + 2,000 + 500 + 20 + 7 B. 1,000,000,000 + 600,000,000 + 90,000,000 + 3,000,000 + 200,000 + 20,000 + 2,000 + 500 + 20 + 7 C. 1,000,000,000 + 600,000,000 + 9,000,000 + 3,000,000 + 200,000 + 20,000 + 2,000 + 500 + 20 + 7 D. 1,000,000,000 + 600,000,000 + 90,000,000 + 3,000,000 + 200,000 + 20,000 + 2,000 + 200 + 50 + 7
Answer:
A.
Step-by-step explanation:
Answer:
A. 4,000,000 + 600,000 + 10,000 + 2,000 + 200 + 50 + 8 + 9
Step-by-step explanation:
Cheryl's making trail mix for a friend she always uses three cups of almonds for every four cups of cashews if Cheryl wants to make a larger batch of trail mix which of the following has the same ratio of almonds cashews
A. 9 cups of almonds for every 20 cups of cashews
B. 6 cups of almonds for every 12 cups of cashews
C. 9 cup of almonds for every 12 cups of cashews
D. 12 cups of almonds for every 20 cups of cashews
Answer:
c
Step-by-step explanation:
3x3=9
4x3=12
Answer:
C
Step-by-step explanation:
3 cups of almonds / 4 cups of cashews
So to get 9 cups of almonds, we need to multiply almonds and cashews by 3
3*3 = 9 cups of almonds
4*3 = 12 cups of cashews
So the correct answer is C
Find the first three terms of x[n] using power series expansion if X(z) 2z3 + 13z2 + 7 73 + 722 + 2z + 1 =
The first three terms of x[n] using the power series expansion are x[0] = 73, x[1] = 2, and x[2] = 13.
We can select the first three terms of x[n] using the power series development by expressing the given articulation X(z) as a polynomial in z. We should modify the articulation as follows to obtain the power series development: By comparing the given expression to the power series form, the coefficients can be identified: X(z) equals 2z3, 13z2, 7z, 73, 722/z, 2/z, and 1: a0 rises to 73, a1 approaches 2, a2 approaches 13, and a3 approaches 7. X(z) = a0, a1z, a2z2, a3*z3, and... Consequently, the following are the first three terms of x[n]:
The initial three terms of x[n] are provided by the power series development: x[0] = a0; x[1] = a1; x[2] = a2; x[0] = a0; The values of x[0] and x[1] are 73, 2, and 13.
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