Consider the following theorem. Main Theorem. Assume that n is any positive integer. Then 7 Ik = n(n+1) 2 k%3D1 (a) Illustrate the main theorem for the values n =1,2,3, 4, 5, 6 and 7. value the sum k n(n+1) 2 Does the IM k=1 of n theorem hold? 1 2 3 4 5 6 7 (b) State the Induction Step Theorem for n = 64, and prove it. Use a direct proof.
(c) State the Induction Step Theorem for n = 583, and prove it. Using direct proof
(d) Assume that N is any positive integer. State the Induction Step Theorem for the value n = N, and prove it.
(e) Explain in English in a few sentences in your own words why the proof of the induction step in d) combined with the verification of several base cases in a) complete the proof of the Main Theorem.
This is all one question based on the same main theorem
Answers
(a) The main theorem states that the sum of terms in the given sequence satisfies the equation n(n+1)/2 for various values of n.
(b) The induction step theorem for n = 64 states that if the main theorem holds for n = k, it also holds for n = k + 1.
(c) The induction step theorem for n = 583 states that if the main theorem holds for n = k, it also holds for n = k + 1.
(d) The induction step theorem for n = N states that if the main theorem holds for n = k, it also holds for n = k + 1.
(e) The combination of the induction step in (d) and verification of base cases in (a) completes the proof of the main theorem for all positive integers.
(a) The main theorem states that for any positive integer n, the sum of the terms k=1 to n of the sequence n(n+1)/2 is equal to n(n+1)/2.
Illustrating the main theorem for different values of n:
For n = 1: The sum is 1(1+1)/2 = 1, which holds true.
For n = 2: The sum is 2(2+1)/2 = 3, which holds true.
For n = 3: The sum is 3(3+1)/2 = 6, which holds true.
For n = 4: The sum is 4(4+1)/2 = 10, which holds true.
For n = 5: The sum is 5(5+1)/2 = 15, which holds true.
For n = 6: The sum is 6(6+1)/2 = 21, which holds true.
For n = 7: The sum is 7(7+1)/2 = 28, which holds true.
(b) The Induction Step Theorem for n = 64 states that if the main theorem holds for n = k, then it also holds for n = k + 1.
To prove it, assume that the main theorem holds for n = k, i.e., the sum of k terms is k(k+1)/2.
Then, we need to show that the sum of (k+1) terms is (k+1)((k+1)+1)/2 = (k+1)(k+2)/2.
By adding the (k+1)th term to the sum of k terms, we get (k(k+1)/2) + (k+1) = (k+1)(k+2)/2, which is the desired result.
(c) The Induction Step Theorem for n = 583 states that if the main theorem holds for n = k, then it also holds for n = k + 1.
To prove it, assume that the main theorem holds for n = k, i.e., the sum of k terms is k(k+1)/2.
Then, we need to show that the sum of (k+1) terms is (k+1)((k+1)+1)/2 = (k+1)(k+2)/2.
By adding the (k+1)th term to the sum of k terms, we get (k(k+1)/2) + (k+1) = (k+1)(k+2)/2, which is the desired result.
(d) The Induction Step Theorem for n = N states that if the main theorem holds for n = k, then it also holds for n = k + 1.
To prove it, assume that the main theorem holds for n = k, i.e., the sum of k terms is k(k+1)/2.
Then, we need to show that the sum of (k+1) terms is (k+1)((k+1)+1)/2 = (k+1)(k+2)/2.
By adding the (k+1)th term to the sum of k terms, we get (k(k+1)/2) + (k+1) = (k+1)(k+2)/2, which is the desired result.
(e) The proof of the induction step in (d) combined with the verification of several base cases in (a) completes the proof of the Main Theorem because it establishes that the theorem holds for all positive integers. The induction step shows that if the theorem holds for any positive integer, it also holds for the next integer. By verifying the base cases, we ensure that the theorem holds for the initial integers. Therefore, by the principle of mathematical induction, the theorem holds for all positive integers.
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Related Questions
what is the number of fluid ounces in 7 cups.
Answers
Answer:
56 ounces :)
Step-by-step explanation:
56 fluid ounces
Step-by-step explanation:
7 × 8 = 56
Hope this helps, have a good day!
Sindhu’s present age is thrice of Shilpa. If Shilpa’s age three years ago was x, then Sindhu’s present age is
3 (x – 3)
3x + 3
3x – 9
3(x + 3)
Answers
Answer:
I think "D"- 3(x + 3)
Step-by-step explanation:
as 3 years ago - x
so present age of shilpa - x +3
~~~~~~~~~~~~~
present age of sindhu - 3 (x+3)
i guess : )
Shilpa's age three years ago
Shilpa's age now = x + 3
Sindhu's age now = 3(x + 3)
When testing the differences between means, the _____ hypothesis suggests that population means are not equal.
Answers
Answer: Research
Step-by-step explanation:
A carpenter can make at most 20 tables and at most 30 chairs per day. Each table requires 3 hours of labour and each chair requires 2 hours of labour. The maximum total hours of disposal is 96. (A) give 3 inequalities that express the condition above. (B) graph and shade the common region that satisfies these. (C) find the maximum number of chairs and table the carpenter can make.
Answers
A) The inequalities that express the condition is 3x + 2y ≤ 96, x ≤ 20 and y ≤ 30
B) The graph of the inequality is illustrated below.
C) The carpenter can make at most 16 tables and 24 chairs per day.
A) The three inequalities that express the conditions above are:
3x + 2y ≤ 96, which represents the maximum amount of time the carpenter has available to work.
x ≤ 20, which represents the maximum number of tables the carpenter can make.
y ≤ 30, which represents the maximum number of chairs the carpenter can make.
In these inequalities, x represents the number of tables and y represents the number of chairs.
B) To graph these inequalities, we will first plot the boundary lines for each inequality. To plot the first inequality, 3x + 2y = 96, we can rearrange it to solve for y:
y ≤ (96 - 3x) / 2
Then, we can create a table of values for x and y:
x y
0 48
16 32
32 16
48 0
Using these values, we can plot the line and shade the region below it, as shown in the graph below.
C) To find the maximum number of chairs and tables the carpenter can make, we look for the corner point of the feasible region that is furthest from the origin. In this case, the point (16, 24) represents the maximum number of chairs and tables that the carpenter can make, as shown in the graph above.
Therefore, the carpenter can make at most 16 tables and 24 chairs per day.
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Which of these expression is equivalent to 10% . 10%?
A. 100'
B. 103
C. 10,000,000
D. 2010
E. 107
what expression is equivalent to
Answers
Please help ive been trying at this stupid question for 45 minutes now.
Answers
Answer:
25
Step-by-step explanation:
But I’m not sure tho
Hope you get it right
FAST PLS!
Find the equation of a line. (-2,4). the slope is -2.
Answers
Answer:
y = -2x
Step-by-step explanation:
Because you are given a point and a slope you use what is called the point-slope formula.
y-y₁=m(x-x₁)
\(y-4=-2(x--2) \\ y-4=-2x-4\\y=-2x-4+4\\y=-2x\)
I NEED HELP ASAP
The temperature of a heat-producing chemical reaction is directly proportional to the square of the time elapsed. The reaction started at 0 degrees and reached 64 degrees in 5 minutes. Estimate the time taken by the reaction to reach 36 degrees.
A. 1.06
B. 2
C. 3.75
D. 4
Answers
The time taken by the reaction to reach 36 degrees will be 3.75 minutes. Then the correct option is C.
What are ratio and proportion?A ratio is a group of sequentially ordered numbers a and b expressed as a/b, where b is never equal to zero. When two objects are equal, a statement is said to be proportional.
The temperature of an intensity delivering synthetic response is straightforwardly relative to the square of the time slipped by. The response began at 0 degrees and arrived at 64 degrees in a short time.
The proportionality is given as,
T ∝ t²
T = kt²
Where T is the temperature, t is the time, and k is the constant of proportionality.
The time taken by the reaction to reach 36 degrees will be given as,
36 / 64 = t² / 5²
0.5625 = t² / 25
t² = 14.0625
t = 3.75 minutes
The time taken by the response to arrive at 36 degrees will be 3.75 minutes. Then the right choice is C.
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Solve for variable.Show work
-3(3+6b)=36-3b
Answers
\(-3(3+6b)=36-3b\\-9-18b=36-3b\\-9-36=18b-3b\\-45=15b\\b=-3\)
Use Evcel fo find the probabinty that in 112 trials with probabinty
0.42
of success that there are 30 or fewer suce ses. Round to 6 decimaiplaces Type your answer hern
Answers
Answer:
0.5
Step-by-step explanation:
0.5=0.42 then the answer is 0.5
PLEASE HELP IMAGE IS BELOW!!
Answers
Answer:
Number of small boxes shipped = 7
Number of large boxes shipped = 15
Step-by-step explanation:
Let the number of small boxes = s
And the number of large boxes = l
Weight of small box = 25 pound
Weight of the large box = 50 pounds
Total weight of the shipment = 925 pounds
Therefore, equation for the weight of shipment will be,
25s + 50l = 925
s + 2l = 37 ----- (1)
Total number of boxes shipped = 22 boxes
Therefore, equation will be,
s + l = 22 ------(2)
Subtract equation (2) from equation (1)
(s + 2l) - (s + l) = 37 -22
l = 15
From equation (2)
s + 15 = 22
s = 7
Therefore, number of small boxes shipped = 7
Number of large boxes shipped = 15
what is the correct way to measure the zone of inhibition?
Answers
The correct way to measure the zone of inhibition is to place a known amount of an antimicrobial agent on a solid medium.
What is Area?Area is a quantity that measures the size of a two-dimensional surface or shape. It is expressed in square units such as square centimetres (cm2), square metres (m2) or square kilometres (km2). Area is used to describe the size of a garden, a house, a room, a city and much more. It can also be used to calculate the amount of material required for a project, such as paint, carpet or tiles. Knowing the area of a shape can help to calculate costs, and to make sure that enough materials are ordered.
The zone of inhibition is then measured as the area surrounding the antimicrobial agent where microbial growth has been inhibited. This area is typically measured in millimeters. To accurately measure the zone of inhibition, it is important to ensure that the size of the inoculum and the amount of antimicrobial agent used remain constant. Additionally, it is important to consider the type of organism being studied, as different organisms can have different levels of susceptibility to the antimicrobial agent. Finally, the medium used should be appropriate for the organism being studied, as different media can produce different results.
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This is 7th grade math: Equations and Inequalities word problems
The Total cost of a teapot and some teacups is $100. The teapot costs $40. Each teacup costs $10. Write an equation to find t, the total number of teacups bought.
Answers
Answer:
I'm not a hundred percent sure but I think 6 teacups were bought for 60$
A car wash how to make their so blast six days if they only have 1/9 of a gallon of soap how much should they use each day so it lasts six days
Answers
They should use 1/45 of a gallon of soap out of 1/9 each day, so it lasts six days.
What is division?The division in mathematics is one kind of operation. In this process, we split the expressions or numbers into the same number of parts.
Given:
They have 1/9 of a gallon of soap.
To find the amount of soap so that it can last six days:
Divide the total amount of soap by the total number of days.
That means,
= 1/9 ÷ 6
= 1/9 x 1/6
= 1/ 45
Therefore, 1/45 is the required value.
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Over the course of 3 days, Calvin spent 10 hours delivering newspapers. He spent the same amount of time each day delivering newspapers. Which of the following equations represents the amount of time Calvin spent delivering papers each day?
A: 3 ÷ 10
B: 10 ÷ 3 =
C:10 ÷ 7 =
D: 7 ÷ 10 =
Answers
Please help 60 points for a rapid answer-In the figure below which of the following is true in circle E?
Answers
Answer:
all 3 options are true : A, B, C
Step-by-step explanation:
warning : it has come to my attention that some testing systems have an incorrect answer stored as right answer for this problem.
they say that A and C are correct.
but I am going to show you that if A and C are correct, then also B must be correct.
therefore, my given answer above is the actual correct answer (no matter what the test systems say).
originally the information about the alignment of the point F in relation to point E was missing.
therefore, I considered both options :
1. F is on the same vertical line as E.
2. F is not on the same vertical line as E.
because of optical reasons (and the - incomplete - expected correct answers of A and C confirm that) I used the 1. assumption for the provided answer :
the vertical line of EF is like a mirror between the left and the right half of the picture.
A is mirrored across the vertical line resulting in B. and vice versa.
the same for C and D.
this leads to the effect that all 3 given congruence relationships are true.
if we consider assumption 2, none of the 3 answer options could be true.
but if the assumptions are true, then all 3 options have to be true.
now, for the "why" :
remember what congruence means :
both shapes, after turning and rotating, can be laid on top of each other, and nothing "sticks out", they are covering each other perfectly.
for that to be possible, both shapes must have the same basic structure (like number of sides and vertices), both shapes must have the same side lengths and also equally sized angles.
so, when EF is a mirror, then each side is an exact copy of the other, just left/right being turned.
therefore, yes absolutely, CAD is congruent with CBD. and ACB is congruent to ADB.
but do you notice something ?
both mentioned triangles on the left side contain the side AC, and both triangles in the right side contain the side BD.
now, if the triangles are congruent, that means that each of the 3 sides must have an equally long corresponding side in the other triangle.
therefore, AC must be equal to BD.
and that means that AC is congruent to BD.
because lines have no other congruent criteria - only the lengths must be identical.
Alguien me podría ayudar aunque sea en la 2°
se los agradecería mucho
Answers
Answer:
photomath
Step-by-step explanation:
es una app q te ayuda en ejercicios de matemática
Really need help on this
Answers
Answer:
The function is t(n) = 3n - 9
A retail establishment closes each evening. If the establishment is open from 9 am to 5 pm, the objective of a simulation might be to estimate some measure of the
quality of customer service over the period beginning at 9 am and ending when the last customer who entered before the doors closed 5 pm has been served.
What type of model is the above model according to the output analysis?
Answers
According to the output analysis, the model presented above is of Discrete-event simulation (DES) type.
What is Discrete-event simulation (DES)?
Discrete-event simulation (DES) is indeed a type of computer-based simulation where events occur at specific points in time. In DES, a system's behavior is modeled by specifying the sequence of events that occur during the system's lifespan.
In a discrete-event simulation, the state of the system changes only when events occur, rather than continuously evolving over time. Events represent significant occurrences or activities within the system, such as customer arrivals, service completions, or resource allocations. Each event is associated with a specific time or occurrence point, and the simulation progresses by processing events in chronological order.
DES is particularly useful for modeling complex systems where the timing of events and their interactions play a crucial role. It allows for capturing the dynamic nature of the system, simulating the flow of entities (such as customers, vehicles, or messages) through the system, and observing the effects of event sequencing and resource allocation on system performance.
By simulating the system and observing the events and their consequences, DES enables analysts to evaluate different scenarios, study the impact of various parameters, and optimize system design or operation. It provides insights into system performance measures, such as throughput, waiting times, resource utilization, or customer satisfaction.
Overall, discrete-event simulation is a powerful technique for modeling and analyzing complex systems with discrete changes, events, and interactions over time.
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The model above is known as a discrete-event simulation model according to the output analysis.
A discrete event simulation is a type of computer-based modeling where a series of discrete events in a system are simulated.
In such models, the system under consideration evolves through a series of discrete events that are instantaneous.
For example, in the retail establishment model, each customer arriving at the store would be considered a separate, discrete event.
The simulation would model the sequence of events that occur between the time the store opens at 9 am and when the last customer leaves before closing time at 5 pm.
The objective of the simulation would be to estimate some measure of the quality of customer service during this time period.
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Divide: 3/7 ÷ 1/2
A) 116
B) 67
C) 25
D) 314
Answers
The graph of a line passes through the two points below.
Which of these can be used to determine the slope of the line?
ㅇ
ㅇ
2-3
4-1
2+3
(2, 4); (3,-1)
2-3
4+1
401 20
Answers
An expression which can be used to determine the slope of the line include the following: A. \(\frac{-1 - 4}{3-2}\)
How to calculate or determine the slope of a line?In Mathematics and Geometry, the slope of any straight line can be determined by using the following mathematical equation;
Slope (m) = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Slope (m) = rise/run
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
By substituting the given data points into the formula for the slope of a line, we have the following;
Slope (m) = (y₂ - y₁)/(x₂ - x₁)
Slope (m) = (-1 - 4)/(3 - 2) ⇒ (required expression).
Slope (m) = -5/1
Slope (m) = -5.
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Complete Question:
The graph of a line passes through the two points below.
(2, 4); (3,-1)
Which of these can be used to determine the slope of the line?
A. \(\frac{-1 - 4}{3-2}\)
B. \(\frac{-1 + 4}{3+2}\)
C. \(\frac{4 - 1}{3-2}\)
D. \(\frac{-1 - 3}{4-2}\)
According to a certain central bank from 2000 to 2016 the average price of a new home in a certain region increased by %87 to $ 572 thousand. What was the average price of a new home in 2000?
Answers
X=572/1.87
X=$306
a long distance runner ran a race in 5 hours, averaging 6 mph for the first 3 hours and 5 mph for the last 2 hours. how far did she run?
Answers
She ran a total distance of 28 miles in 5 hours.
We can solve this problem by using the formula: distance = rate × time
First, we need to calculate the distance traveled during the first 3 hours at 6 mph. Using the formula, we have:
distance = rate × time
distance = 6 mph × 3 hours
distance = 18 miles
Next, we need to calculate the distance traveled during the last 2 hours at 5 mph. Using the same formula, we have:
distance = rate × time
distance = 5 mph × 2 hours
distance = 10 miles
Finally, we can add the distances traveled during the first 3 hours and the last 2 hours to get the total distance:
total distance = 18 miles + 10 miles
total distance = 28 miles
Therefore, the runner ran a total distance of 28 miles in 5 hours.
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1. What is the slope of the line joining the points (2,5) and (7,15).
Answers
To calculate the slope of a line you have to use the following formula
\(m=\frac{y_1-y_2}{x_1-x_2}\)Where
(x₁,y₁) are the coordinates of one point on the line
(x₂,y₂) are the coordinates of a second point on the line
For the points (2,5) and (7,15), the slope of a line that crosses them is:
\(m=\frac{15-5}{7-2}=\frac{10}{5}=2\)The slope of the line is m=2
In 1980, there were 1. 2 million elephants living in Africa. Because the natural grazing lands for the elephant are disappearing due to increased population and cultivation of the land, the number of elephants has decreased by about 6. 8% per year. In 1987, what was the population of elephants? Round up to the nearest elephant
Answers
There were approximately 586,800 elephants in Africa, rounded up to the nearest elephant.
In 1980, there were 1.2 million elephants in Africa. With a decrease of 6.8% per year, we need to calculate the population in 1987, which is 7 years later.
To find the population in 1987, we use the formula:
Final population = Initial population * (1 - decrease rate) ^ number of years
Final population = 1,200,000 * (1 - 0.068) ^ 7
Final population ≈ 1,200,000 * 0.489
Final population ≈ 586,800
In 1987, there were approximately 586,800 elephants in Africa, rounded up to the nearest elephant.
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Please HELP!! Need to finish before the end of the day!
Answers
Answer & Step-by-step explanation:
I have attached two ways you can find the number of green Jolos, by adding and/or subtracting from the other known values. Both ways will give the answer of C. 280 Jolos
What does the 150 in the equation represent?
What does the 9 in the equation represent?
Answers
Max rides his scooter toward Kim and then passes her at a constant speed. His distance in feet, d, from Kim t seconds after he started riding his scooter is given by d = |150 – 9t|. Then 150 in the equation represents the initial distance between Max and Kim. 150 is how far Max is from Kim before he starts and 9 in the equation represents the rate at which Max is closing in on Kim. 9 feet per second is Max's rate.
What is Equation?
Two expressions with equal sign is called as equation.
Max rides his scooter toward Kim and then passes her at a constant speed. His distance in feet, d, from Kim t seconds after he started riding his scooter is given by d = |150 – 9t|.
Then 150 in the equation represents the initial distance between Max and Kim. 150 is how far Max is from Kim before he starts and 9 in the equation represents the rate at which Max is closing in on Kim. 9 feet per second is Max's rate.
Therefore Then 150 in the equation represents the initial distance between Max and Kim. 150 is how far Max is from Kim before he starts and 9 in the equation represents the rate at which Max is closing in on Kim. 9 feet per second is Max's rate.
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What is the surface area of the square pyramid 17ft 9ft 17ft
Answers
Answer:
The surface area of the square pyramid with dimensions 17 ft x 9 ft x 17 ft is 595 square feet.
Step-by-step explanation:
To calculate the surface area of a square pyramid, we need to add the area of the base to the sum of the areas of the four triangular faces.
For a square pyramid with side length s and slant height l, the surface area can be calculated as follows:
Surface area = s^2 + 2sl
where s is the length of one side of the square base, and l is the slant height of the pyramid, which is the height of each of the four triangular faces.
In your case, the dimensions given are 17 ft for the base and 9 ft for the height, which is also the slant height. Therefore, we can calculate the surface area as:
Surface area = 17^2 + 2(17)(9)
Surface area = 289 + 306
Surface area = 595
A minivan is traveling at 80.5 kilometers per hour. Cargo is strapped to the roof at a height of 1.75 meters. The car hits a concrete barrier, and the cargo is ejected from the roof. Use the following two equations to determine how long it takes for the cargo to hit the ground and how far it travels in the horizontal direction. Let y represent height in meters, x represent horizontal distance in meters, and t represent time in seconds. Equation 1: y = -4.9t2 + 1.75 Equation 2: y = -0.0081x2 + 1.75
Answers
The time taken for the cargo to hit the ground will be 0.6 seconds and it travels in the horizontal direction for around 14.69 feet.
What is equation?The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign. The point-slope form, standard form, and slope-intercept form are the three main types of linear equations. The definition of an equation in algebra is a mathematical statement that demonstrates the equality of two mathematical expressions. For instance, the equation 3x + 5 = 14 consists of the two equations 3x + 5 and 14, which are separated by the 'equal' sign.
Here,
The calculations are attached in images.
To calculate t we will use that is y because at the moment of cargo will have a height 0, so we get,
= −4.9t² +1.75
-1.75=-4.9t²
t²=-1.75/-4.9
t²=√0.3571
=0.6 seconds
Now we will calculate x, then y = 0
because since the cargo will be on the ground it will no have height, so
0= -0.0081x²+1.75
-1.75 =-0.0081x²
x²=-1.75/-0.0081
x= √216.04 = 14.69 feet
It will take the cargo 0.6 seconds to hit the ground, and it will travel 14.69 feet in a horizontal direction.
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help me out please asap! Giving brainliest!!!
Answers
Answer: the answer is -32
Step-by-step explanation: multiply -2 by itself 5 separate times.
Answer:
-32
Step-by-step explanation:
-2×-2×-2×-2×-2=4×4×-2
=-32