Exact solutions for divide-and-conquer recurrence relations. Expand the terms of each recurrence relation in order to obtain an exact solution for T(n). Your solution should include all the constants in the expression for T(n), and not just the asymptotic growth of the function T(n). You can assume that the value of n, the input to the function T, is a power of 3. That is, n = 3k for some integer k. (a) T(n) = 3T(n/3) + 5n T(1) = 5 (b) T(n) = 3T(n/3) + 5n² T(1) = 5 Solution At each level, expand the expression for T, using the recurrence relation. Start with T(n) at level 0. Replace T(n) by 5n² at level 0 and add three T(n/3) terms at level 1. Then replace each T(n/3) at level 1 with 5 (n/3)². For each term at level 1, add three T(n/9) terms at level 2. Continue with the expansion until level L, where n/3 = 1. There will be 3 terms at level L, each of value T(1). Use the initial value T(n) and replace each T(1) terms at level L with the number 5. There are a total of L+1 levels. Since n/3¹ = 1, then n = 3 and by the definition of logarithms, L = log3 n. The value of T(n) is the sum of all the terms at each level. At level j, there are 3³ terms, each with value 5 (n/3¹)². Note that at level L, there are 3 terms, each with value 5 = 5 (n/34)², because n/34 = 1. The total value of all the terms at levelj is 2 3 3¹.5. (+)* n² = 3¹.5. 3²j = 5n² = 5n² The sum of all the terms at all the levels is logą n T(n) = Σ 5n²( -Σ*5m² (+)². (1/3)logs n+1 1 (1/3) - 1 j=0 1- (1/3)(1/3)log, n 3 = 5n² (1-(1/3)) -157² (1-3) = 1- (1/3) n 2 3n = 5n². 5n².
Answers
In this case, we have two recurrence relations: T(n) = 3T(n/3) + 5n and T(n) = 3T(n/3) + 5n². By expanding the expressions at each level and replacing the recursive terms, we can derive the exact solution for T(n).
To obtain the exact solution for T(n), we start by expanding the expression for T(n) at level 0, using the given recurrence relation. We replace T(n) with the initial value of 5n² and add three terms of T(n/3) at level 1. We continue this expansion process, adding three terms at each subsequent level until we reach the final level, where n/3 = 1.
At each level, the number of terms is determined by 3 raised to the power of the level. The value of each term is 5 times the square of n divided by 3 raised to the power of the level. Finally, we sum up all the terms at each level to obtain the total value of T(n).
In the end, we use the property of logarithms to determine the number of levels, which is log3 n. By simplifying the expression, we arrive at the exact solution for T(n) as 5n² times the sum of a geometric series.
By following this expansion and simplification process, we can obtain the exact expression for T(n) in terms of n, including all the constants involved in the recurrence relation.
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Related Questions
Do ∠ABE and ∠DBC share a ray
Answers
Answer:
yes they do
Step-by-step explanation:
they share a ray
can anyone help ?\me with this algebra answer
Answers
Answer:
B) 2x-y=5
Step-by-step explanation:
Info we have:
The x-intercept is 2.5
The slope is 2/1=2
Therefore, our equation so far is y=2x+b
We plug in our x-intercept in and make y=0 to get b:
0=2(2.5)+b
0=5+b
-5=b
Therefore, the equation of the line is y=2x-5
Given our found equation, we can see which option works.
A is wrong because 2x+y=-5 -> y=-5-2x -> y=-2x-5
B is right because 2x-y=5 -> -y=5-2x -> y=-5+2x -> y=2x-5
C is wrong because x+2y=-5 -> 2y=-5-x -> y=-2.5-1/2x -> y=-1/2x-2.5
D is wrong because 2y-x=10 -> 2y=10+x -> y=5+1/2x -> y=1/2x+5
Solve the rational equation for x and state all x values that are excluded from the solution set. If there is more than one excluded value then separate them with a comma and do not include any spaces. If a value is not an integer then type it as a decimal rounded to the nearest hundredth. \frac{5}{x+1}+\frac{1}{x-3}=\frac{-6}{x^2-2x-3} Solving for x gives us x=AnswerThe value for x cannot equal Answer
Answers
Answer:
Solving for x gives us x = 1.33
The value for x cannot equal 3,-1
Explanation:
The initial expression is:
\(\frac{5}{x+1}+\frac{1}{x-3}=\frac{-6}{x^2-2x-3}\)The denominator of the right side can be factorized as (x - 3)(x + 1) because -3 and 1 multiply to -3 and sum to -2. So, we can rewrite the expression as:
\(\frac{5}{x+1}+\frac{1}{x-3}=\frac{-6}{(x-3)(x+1)}\)Now, we can sum the expression on the left side to get:
\(\frac{5(x-3)+(x+1)}{(x-3)(x+1)_{}}=\frac{-6}{(x-3)(x+1)}\)Both sides have a denominator of (x-3)(x+1), so we need to exclude the values where this denominator is equal to 0. Those values are:
x - 3 = 0
x - 3 + 3 = 0 + 3
x = 3
x + 1 = 0
x + 1 - 1 = 0 - 1
x = -1
So, the values of x cannot be equal to 3 and -1.
Then, if the denominators are the same, we can find the solution of the equation, making equal the numerators, so we need to solve:
\(5(x-3)+(x+1)=-6\)Solving for x, we get:
\(\begin{gathered} 5(x_{})+5(-3)+x+1=-6 \\ 5x-15+x+1=-6 \\ 6x-14=-6 \\ 6x-14+14=-6+14 \\ 6x=8 \\ \frac{6x}{6}=\frac{8}{6} \\ x=1.33 \end{gathered}\)Therefore, the answers are:
Solving for x gives us x = 1.33
The value for x cannot equal 3,-1
A rectangle has a perimeter of 36 centimeters. The length of the rectangle is five centimeters more than the width. What is the length of the rectangle?
Answers
Answer:
11.5 cm
Step-by-step explanation:
Set up a system of equations and simplify.
l = w + 5
2l + 2w = 36
Substitute w + 5 for l
2w + 10 + 2w = 36
4w + 10 = 36
4w = 26
w = 6.5
l = 6.5 + 5
l = 11.5
calcuate the perimeter of a triangle with sides of length 21 in, 67
in, and 55 in
Answers
The perimeter of a triangle with sides of length 21 in, 67in, and 55 in is 143 inches.
Perimeter of a triangle is defined as the sum of all the sides of the triangle. To find the perimeter of a triangle with sides of length 21 in, 67in, and 55 in, we can use the formula of perimeter of a triangle as P = a + b + c. Therefore, we can calculate the perimeter of a triangle by adding all three sides given in the question. The perimeter of a triangle is given as:
Perimeter, P = a + b + c= 21 + 67 + 55= 143 inches.
:In geometry, the perimeter of a shape is defined as the distance around the shape. When it comes to a triangle, it is simply the sum of all the sides of the triangle. In the given problem, we have been given the three sides of a triangle. To find the perimeter, we simply add up all the given sides.
We can use the formula for perimeter of a triangle to calculate the answer.
That is,P = a + b + cwhere P is the perimeter of the triangle, and a, b, c are the three sides of the triangle given in the problem.We can now substitute the values of a, b and c from the given problem as follows:
Perimeter, P = a + b + c= 21 + 67 + 55= 143 inches.
Therefore, the perimeter of a triangle with sides of length 21 in, 67in, and 55 in is 143 inches.
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Graph the line y = 3x +3 (on Desmos) and the line perpendicular to
it that passes through (-3, 2). Then write the equation of the
perpendicular line.
Answers
Does 2x - 10 equal to 50
Answers
Two times -10 is -20. Therefore, it is not equal to 50.
What is the volume of a cylinder with a height of 4xy^4 and a radius of 2x? (The formula fo
the volume of a cylinder is V = pie r^2 h.)
Answers
Answer:
volume: \(\sf 16\pi x^3y^4\)
given:
height: \(\sf 4xy^4\)radius: \(\sf 2x\)cylinder volume:
\(\hookrightarrow \sf \pi r^2h\)
\(\hookrightarrow \sf \pi (2x)^2(4xy^4)\)
\(\hookrightarrow \sf \pi (4x^2)(4xy^4)\)
\(\hookrightarrow \sf 16\pi x^3y^4\)
represent the following on number line -5
Answers
Answer:
It would be 5 spots to the left of 0 on the number line.
Step-by-step explanation:
Can anyone do this thanks
Answers
2) sept.
4) aug. &' sept
Not 100% though .
se upper and lower sums to approximate the area of the given region and given number of subintervals.
1. = from x=0 to x=1 with 4 subintervals.
2. = 4― from x=0 to x=2 with 4 subintervals.
3. = 1
from x=1 to x=2 with 5 subintervals.
Answers
The estimated areas are listed below:
The area of the function is approximately 0.806.The area of the function is approximately 2.088.The area of the function is approximately 0.696.How to approximate the area of a given region with a given number of subintervals
Right Riemann sum based on trapezoids offer a compact method that combines upper and lower sums, which is defined below:
\(A = \frac{b-a}{2\cdot n}\cdot \sum_{i = 0}^{n-1} [f(x_{i})+f(x_{i+1})]\) (1)
Where:
b - Upper bounda - Lower boundn - Number of subintervalsi - Subinterval indexIn this question we must apply right Riemann sums to estimate the areas of functions on given intervals. Now we proceed to solve each case below:
Case I (\(f(x) = \sqrt{x}\), \(a = 0\), \(b = 1\), \(n = 4\))\(A = \frac{1-0}{2\cdot (4)} \cdot [(0+\sqrt{0.5})+(\sqrt{0.5} + 1)+(1+\sqrt{1.5})+(\sqrt{1.5}+2)]\)
A ≈ 0.806
The area of the function is approximately 0.806. \(\blacksquare\)
Case II (\(f(x) = 4\cdot e^{-x}\), \(a = 0\), \(b = 2\), \(n = 4\))\(A = \frac{2-0}{2\cdot (4)}\cdot [(4+1.472)+(1.472+0.541)+(0.541+0.199)+(0.199-0.073)]\)
A ≈ 2.088
The area of the function is approximately 2.088. \(\blacksquare\)
Case III (\(f(x) = \frac{1}{x}\), \(a = 1\), \(b = 2\), \(n = 5\))\(A = \frac{2-1}{2\cdot (5)} \cdot [(1+0.833)+(0.833+0.714)+(0.714+0.625)+(0.625+0.556)+(0.556+0.5)]\)
A ≈ 0.696
The area of the function is approximately 0.696. \(\blacksquare\)
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Use the Distributive Property to expand −3(8x−4) .
Answers
Answer: -24x+12
Step-by-step explanation:
-3(8x-4)
-3(8x)+-3(-4)
-24x+12
Which of the following statements for a simple graph is correct?a) Every path is a trailb) Every trail is a pathc) Every trail is a path as well as every path is a traild) Path and trail have no relation
Answers
Statement (c) is correct for a simple graph. Every trail is a path, and every path is a trail.
In graph theory, a simple graph is an undirected graph with no loops or multiple edges between the same pair of vertices. A path in a graph is a sequence of vertices where each consecutive pair is connected by an edge. A trail in a graph is a path that allows for repeated vertices and edges.
Statement (a) is not correct because not every path is a trail. A path does not allow for repeated vertices or edges, whereas a trail does.
Statement (b) is not correct because not every trail is necessarily a path. A trail may contain repeated vertices or edges, but a path does not.
Statement (d) is not correct because paths and trails do have a relation. A trail is a more general concept that encompasses paths by allowing for repetition of vertices and edges.
Therefore, statement (c) is the correct statement. In a simple graph, every trail is a path, and every path is a trail.
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Answer Number 2 or number 6 pleasee
Answers
Problem 2
Rewrite the fraction 2/3 into 4/6. This is valid because we multiply top and bottom by 2.
2/3 = 4/6
Now compare the fractions for Wednesday and Tuesday in that order. We have 4/6 and 5/6. We can ignore the denominators because they effectively cancel out. The numerators will form the answer.
Answer is the ratio 4:5=====================================================
Problem 6
7% = 7/100 = 0.07
7% of 12,500 = 0.07*12500 = 875
Answer: 875 dollarsAre the ratios 9:3 and 6:2 equivalent?
Answers
Answer:
yes
Step-by-step explanation:
they both simplify to 3:1
Answer:
Yes
Step-by-step explanation:
To see whether two fractions are equivalent or not we need to simplify them
9:3 can be simplified to 3:1 as the biggest common divisor of 9 and 3 is 3
6:2 can be simplified to 3:1 since the biggest common divisor of this set is 2
as seen we can simplify both to 3:1 so yes, they are equivalent
how do I simplify when there are fractions on fractions
Answers
Answer: y/y+x
Step-by-step explanation: Reduce the fraction at the bottom (1 - x/y) (1 + x/y) = 1 + x/y = simplify and you get y/y+x
Eric has a riddle: "I am thinking of a fraction that is equivalent to2/5and the numerator is 18 less than the denominator."What is Eric's fraction? Explain how you know.
Answers
Eric's fraction is 8/20. We can solve the riddle with the help of algebraic equations.
Eric's fraction is 8/20. We can solve the riddle by using algebraic equations. We are looking for a fraction that is equivalent to 2/5. Therefore, the denominator of the equivalent fraction is 5. Let's call the numerator of the fraction y. The second piece of information we have is that the numerator is 18 less than the denominator. Therefore: y = 5 - 18 y = -13. However, we know that y has to be a positive number since it represents a numerator. So, this is not the answer. We can solve this by setting the denominator of the equivalent fraction to 20. Since we have multiplied the denominator by 4, we will also multiply the numerator by 4. This gives us an equivalent fraction of 8/20. Since we know that the numerator is 18 less than the denominator, 18 + 8 = 26. So the fraction is 8/20.
Therefore, Eric's fraction is 8/20.
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You have $30 to buy muffins. However, at the shop you can buy 1 muffin for $4. (a) Write an inequality from the above information. (b) Solve this inequality to find the number of muffins you can buy. (c) How much money will you have left over?
Answers
Answer:
Let m be the number of muffins.
a) 4m <= 30
b) m <= 7.5, so the maximum number of muffins to buy is 7.
c) $4 × 7 = $28, so $2 will be left over.
A 9 foot ladder is leaning against a wall. The distance from the base of the ladder to the base of the wall is 3.5 feet. If you were to lean a 11 foot ladder against the wall at the same angle from the ground, what will be the distance between the base of the ladder and the base of the wall? Round to the nearest hundredth.
Answers
The distance between the base of the 11-foot ladder and the base of the wall, rounded to the nearest hundredth, is approximately 4.28 feet.
We can use the idea of right triangles that are similar to each other to solve this issue. In similar triangles, the lengths of the sides that are corresponding to each other are equal.
Let's say that there is a distance of "x" feet between the base of the wall and the base of the 11-foot ladder.
Regarding the 9-foot ladder:
The hypotenuse measurement of the ladder is 9 feet, and the adjacent side measurement of the distance is 3.5 feet. For the 11-foot ladder:
The length of the stepping stool (hypotenuse) = 11 feet
The separation from the foundation of the stepping stool to the foundation of the wall (neighboring side) = 'x' feet
Utilizing the proportion of comparing sides of comparative triangles, we can set up the accompanying extent:
9 / 3.5 = 11 / x We can cross-multiply to solve for "x":
9x = 3.5 * 11 9x = 38.5 Now divide by 9 on both sides:
x = 38.5 / 9 x 4.28 indicates that the 11-foot ladder's base is approximately 4.28 feet away from the wall's base, rounded to the nearest hundredth.
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convert 0.025 days to seconds
Answers
Answer:
2,160 seconds
Step-by-step explanation:
Mrs. Attaway has 5 girls and 16 boys in her first-grade class. Three children are selected at random to participate in a PTA program. Find the probability that two are girls and one is a boy. (Round your answer to three decimal places.)
Answers
The probability that two children selected at random from Mrs. Attaway's first-grade class are girls and one is a boy is 0.120.
To find the probability that two children selected at random from Mrs. Attaway's class are girls and one is a boy, we need to calculate the number of favorable outcomes (selecting two girls and one boy) and divide it by the total number of possible outcomes.
The total number of children in the class is 5 girls + 16 boys = 21 children.
To calculate the probability, we need to determine the number of ways to select two girls from the five available girls and one boy from the 16 available boys. This can be done using combinations.
The number of ways to select two girls from five is given by the combination formula:
C(5, 2) = 5! / (2! * (5 - 2)!)
= 5! / (2! * 3!)
= (5 * 4) / (2 * 1)
= 10
Similarly, the number of ways to select one boy from 16 is given by the combination formula:
C(16, 1) = 16
The total number of favorable outcomes is the product of these two combinations: 10 * 16 = 160.
Now, let's calculate the total number of possible outcomes when selecting three children from the class:
C(21, 3) = 21! / (3! * (21 - 3)!)
= 21! / (3! * 18!)
= (21 * 20 * 19) / (3 * 2 * 1)
= 1330
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = favorable outcomes / total outcomes
= 160 / 1330
≈ 0.120 (rounded to three decimal places)
Therefore, the probability that two children selected at random from Mrs. Attaway's first-grade class are girls and one is a boy is 0.120.
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Solve for x using
multiplication.
Cross
-
3x - 3
5
=
2x - 2
4.
x = [?]
![Solve for x usingmultiplication.Cross-3x - 35=2x - 24.x = [?]](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/pHqKWz00HpBoI1nBmduqzU2RzzFeItTK.png)
Answers
Answer:
x = 1
Step-by-step explanation:
\(\frac{3x-3}{5}\) = \(\frac{2x-2}{4}\) ( cross- multiply )
4(3x - 3) = 5(2x - 2) ← distribute parenthesis on both sides
12x - 12 = 10x - 10 ( subtract 10x from both sides )
2x - 12 = - 10 ( add 12 to both sides )
2x = 2 ( divide both sides by 2 )
x = 1
I will mark you as the brainiest if you get this correct
Thx so much xxx
Answers
Answer:
A= 168 minutes or 2 hours and 48 minutes.
B= 3 tyres
Step-by-step explanation:
please help thnx...
hi
Answers
Answer:
-17/5
Step-by-step explanation:
Multiply ten to everything to get
6x-1= x+2-20
Simplify to 5x= -17
Solve to get x = -17/5
two sets --m and n-- are shown in the venn diagram below. which statement is false? a.) set m has 30 elements. b.) sets m and n have 4 common elements. c.) there are a total of 112 elements shown in the venn diagram. d.) set n has 22 elements.
Answers
There are a total of 112 elements shown in the Venn diagram is false statement.
Describe Venn diagram.Venn diagrams are used to represent sets, their connections, and set-related actions graphically. The Venn diagram uses circles to represent the relationships between sets and the region that they overlap. A Venn diagram is used in mathematics to represent the logical relationship between sets and their constituent members and aids in the solution of instances based on these sets. A Venn diagram is a type of diagram designed to show every relationship between various sets. Venn diagrams are used in mathematics to show the connection between two or more sets. This approach of articulating sets or specific situations aids in a better comprehension of the problem and makes it easier to draw conclusions. As a result, we frequently utilize circles to illustrate the connections between objects or limited groupings of objects.
Given
Two sets --m and n-- are shown in the Venn diagram
Following statement is false,
c.) there are a total of 112 elements shown in the Venn diagram.
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D
Question 1
a
What is the Least common denominator (LCD) of
1/6 and 1/9 and 1/18
Answers
Answer:
its gonna be 1/18
Step-by-step explanation:
I'm pretty good at math honestly tbis was easy
Do number 9 please thank you
Answers
Answer:
82cm
Step-by-step explanation:
ihope it helps thanks
Mia uses the formula πr2 to find the area of a circle. What is πr2 an example of?.
Answers
πr2 is an example of a (D) model.
What exactly is a model?A mathematical model is a system description that makes use of mathematical concepts and language.The process of developing a mathematical model is known as mathematical modeling.Natural sciences (physics, biology, earth science, chemistry), engineering disciplines (computer science, electrical engineering), and non-physical structures such as the social sciences all use mathematical models (such as economics, psychology, sociology, and political science).The use of statistical equations to solve problems in business or military operations is a large part of the field of operations research.Mathematical models are used in music, linguistics, and philosophy (for example, intensively in analytic philosophy).A model is, for example, the πr2 method for calculating the area of a circle.
Therefore, πr2 is an example of a (D) model.
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The complete question is given below:
Mia uses the formula πr2 to find the area of a circle. What is πr2 an example of?
A. a robot
B. a car
C. a telescope
D. a model
A visitor to the Grand Canyon hiked the South Kaibab Trail and the River Trail on one day. The next day, she hiked the Bright Angel Trail. How far did she hike the first day? How much farther did she hike the first day than the second day? How much longer was her total route than if she had hiked the North Kaibab Trail?
Answers
At the first day: the visitor hiked the South Kaibab Trail and the River Trail on one day
so, the length for the first day = 10.1 + 2.7 = 10.1 + 2.7 = 12.8 km
At the second day: the visitor hiked the Bright Angel Trail
so, the length of the second day = 12.6 km
We will answer the questions:
How much farther did she hike the first day than the second day?
it will be = 12.8 - 12.6 = 0.2 kilometers
How much longer was her total route than if she had hiked the North Kaibab Trail?
The total route = 12.8 + 12.6 = 25.4
if she had hiked the North Kaibab Trail, the difference will be = 25.4 - 22.9 =
= 2.5 kilometers
i need help i will get a zero if i dont
Answers
Answer:
You would multiply 7cm twice. and then multiuply 9 twice and then you add the two sums.
so 49 plus 81 is 130
so you have 130 cm.
Step-by-step explanation:
Hope this helps Im bad at math.