The given statement lim n tends to infinity s_n doesn't exist by showing (s_n) is not a cauchy sequence.
To show that the sequence (s_n) does not converge, we need to demonstrate that it is not a Cauchy sequence.
A sequence is said to be a Cauchy sequence if, for any positive epsilon (ε), there exists an integer N such that for all m, n > N, |s_n - s_m| < ε.
Let's analyze the sequence (s_n) step by step:
s_1 = 4
s_2 = s_1 + (-1)^2 * 2/5 = 4 + 2/5 = 4.4
s_3 = s_2 + (-1)^3 * 3/7 = 4.4 - 3/7 = 4.057
s_4 = s_3 + (-1)^4 * 4/9 = 4.057 + 4/9 = 4.507
s_5 = s_4 + (-1)^5 * 5/11 = 4.507 - 5/11 = 4.052
Continuing this pattern, we can observe that the terms of the sequence (s_n) oscillate and do not converge to a specific value. As n tends to infinity, the sequence does not approach a single value. Therefore, the limit of (s_n) does not exist.
To show that (s_n) is not a Cauchy sequence, we need to find an epsilon (ε) such that for any integer N, there exist m, n > N for which |s_n - s_m| ≥ ε.
Let's choose ε = 0.1. For any N, we can find m and n such that |s_n - s_m| ≥ 0.1. For example, we can choose n = N + 2 and m = N + 1. In this case:
|s_n - s_m| = |s_{N+2} - s_{N+1}| = |s_{N+2} - (s_{N+1} + ( - 1)^{N+1} * (N+1)/(2(N+1) + 1))| = |s_{N+2} - s_{N+1} + (-1)^{N+1} * (N+1)/(2(N+1) + 1))|
Since the terms of the sequence oscillate and do not converge, for any choice of N, we can always find m and n such that |s_n - s_m| ≥ ε. Therefore, (s_n) is not a Cauchy sequence.
In conclusion, we have shown that the sequence (s_n) does not converge and is not a Cauchy sequence.
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please help!
if a trail runner zig-zags 15m to the west and then 15m to the east:
what is the runner's total distance?
what is the runner's total displacement?
Answer:
Total distance = 30m
Total displacement = 0m
Step-by-step explanation:
a) Distance is a scalar quantity that does not specify direction
Hence if a trail runner zig-zags 15m to the west and then 15m to the east, the total distance will be the sum of the distance travelled both to the west and east.
total distance = distance through west + distance through east
total distance = 15m + 15m
total distance = 30m
Hence the runner's total distance is 30m
b) Displacement is a vector quantity and it specifies direction. Hence we will take into consideration the direction taken
Since East direction is positive, the displacement through east is +15m
Since West direction is negative, the displacement through west is -15m
Total displacement = +15 - 15
Total displacement = 0m
Hence the runner's total displacement is 0m
Triangle ABC is an isosceles triangle where angles angle A and B are congruent of angle B measures 35 what is the measurement of angle c
Answer:
m<C = 110
Step-by-step explanation:
<a = <b
<b = 35 so,
<a = 35
<c = x
<a + <b + <c = 180
35 + 35 + x = 180
70 + x = 180
x = 180 - 70
x = 110
Find the missing side of this right triangle 8 16
Answer:
[tex]\boxed {\boxed {\sf b= \sqrt{192}}}[/tex]
Step-by-step explanation:
Since this is a right triangle, we can use Pythagorean Theorem to find the sides.
[tex]a^2+b^2=c^2[/tex]
where a and b are the legs and c is the hypotenuse.
In this triangle, 8 and x are the legs because they form the right angle. 16 is the hypotenuse because it is opposite the right angle.
[tex]a=8 \\c=16[/tex]
Substitute the values into the formula.
[tex]8^2+b^2=16^2[/tex]
Solve the exponents.
8² = 8*8= 64 16²=16*16=256[tex]64+b^2=256[/tex]
Subtract 64 from both sides to isolate the variable.
[tex]64-64+b^2= 256-64 \\b^2=192[/tex]
Take the square root of both sides.
[tex]\sqrt {b^2}=\sqrt{192}\\ b=\sqrt{192}[/tex]
The missing side is equal to √192
If a linear function has the points (3,-1) and (-3,0) on its graph, what is the rate of change of the function? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The rate of change is (Type an integer or a simplified fraction.) B. There is no solution.
The rate of change of the function is -1/6
How to determine the rate of change of the functionFrom the question, we have the following parameters that can be used in our computation:
(3,-1) and (-3,0)
The rate of change of the function is calculated as
Rate = Change in y/Change in x
Using the above as a guide, we have the following:
Rate = (0 + 1)/(-3 - 3)
Evaluate
Rate = -1/6
Hence, the rate of change of the function is -1/6
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What is the midpoint of EF with endpoints E (−9,7) and F(0,−4)?
Answer:
midpoint = -4.5, 1.5
Step-by-step explanation:
x1 = -9, y1 = 7, x2=0 , y2=-4
midpoint = (x1 +x2)/2 , (y1 + y2)/2
= (-9 + 0)/2 , (7 + -4)/2
=-9/2 , 3/2
midpoint = -4.5, 1.5
You add 11/4 tablespoons of olive oil for every 3/2 tablespoons of balsamic vinegar to make 34 tablespoons of bread dip. How much olive oil and balsamic vinegar do you use?
You use ___ tablespoons of olive oil and
___ tablespoons of balsamic vinegar.
Answer:
The amount of olive oil and balsamic vinegar to use are;
You use 22 tablespoons of olive oil and 12 tablespoons of balsamic vinegar
Step-by-step explanation:
From the question, we have;
The number of tablespoons of olive oil added for every 3/2 tablespoon of balsamic vinegar = 11/4 tablespoons
Therefore, in the mixture, the ratio of olive oil to balsamic vinegar = 11/4:3/2
Let 'x', and 'y' represent the the number of tablespoons of olive oil and balsamic vinegar, in the mixture 34 tablespoons of bread dip, we can write;
x + y = 34 tablespoons
x/y = (11/4)/(3/2) = 11/6
∴ x = 11/6 × y
∴ x + y = 11/6 × y + y = 34
11/6 × y + y = 17·y/6 = 34
17·y = 34 × 6 = 204
y = 204/17 = 12
The number of tablespoons of balsamic vinegar in the bead dip, y = 12 tablespoons of balsamic vinegar
x = 11/6 × y = 11/6 × 12 = 22
The number of tablespoons of olive oil in the bead dip, x = 22 tablespoons of olive oil.
The driveshaft lifespan is measured in operating hours before failure, and the distribution of lifespans is known to follow a Gamma distribution. A sample of 50 driveshafts is carefully tracked until failure and their lifespans are measured to have sample mean of 1274 hours, and a standard deviation of 108 hours.
(a)Assuming a normal distribution with the mean and standard deviation of the sample, estimate the proportion of driveshafts that would fail prior to reaching 1200 operating hours.
(b)Using the central limit theorem and corresponding normality assumption, calculate a 95% confi- dence interval for the true mean lifespan.
a. The estimated proportion of driveshafts that would fail prior to reaching 1200 operating hours is approximately 0.246, or 24.6%.
b. The 95% confidence interval for the true mean lifespan is approximately (1244.79, 1303.21) hours.
How to calculate the value(a) First, we need to standardize the value 1200 using the sample mean and standard deviation. We calculate the z-score as follows:
z = (x - μ) / σ,
z = (1200 - 1274) / 108
≈ -0.685
Using a standard normal distribution table or calculator, we find that the proportion is approximately 0.246.
(b) The formula for the confidence interval is:
CI = x ± z * (σ / √n),
CI = 1274 ± 1.96 * (108 / √50)
CI = 1274 ± 1.96 * (108 / √50)
CI ≈ 1274 ± 1.96 * 15.297
The 95% confidence interval for the true mean lifespan is approximately (1244.79, 1303.21) hours.
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A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, X, is found to be 100, and the sample standard deviation found to be 8.
Construct a 98% confidence interval about µ, if the sample size n, is 20.
Lower bound ______ Upper bound _______
(Round to one decimal place as needed.)
As per the confidence interval Lower Bound is 94.874 and the Upper Bound is 105.126
Sample Mean = 100
Sample Standard Deviation = 8
Sample Size = 20
Calculating the confidence interval -
Confidence Interval = Sample Mean ± (Critical Value) x (Standard Deviation / √(Sample Size))
Substituting the values
= 100 ± (2.860) x (8 / √20)
= 100 ± 2.860 x (8 / 4.472)
= 100 ± 2.860 x 1.789
= 100 ± 5.126
Calculating the lower bound -
Lower Bound
= 100 - 5.126
= 94.874
Calculating the upper bound -
Upper Bound
= 100 + 5.126
= 105.126
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what is the area of the rectangular pool below whose dimension are 2 x +1 and units x-1 units
Answer:
(2x+1)(x-1) = 2x^2 - x - 1
Step-by-step explanation:
Answer:
2x^2 - x - 1
Step-by-step explanation:
Lets rewrite it to the area formula
(2x+1)(x-1)
Simplify-(distributive property of multiplication)
(2x*x) + (2x*-1) + (1*x) + (1*-1) =
Simplify
2x^2 + (-2x) + x + (-1)
Simplify
2x^2 - x - 1
The answer is 2x^2 - x - 1
If my answer is wrong, pls correct me
If you like my answer and my explanation, mark me as brainliest!
-Chetan K
HELP ASAP PLZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
Answer:
a. The price per ticket is 4 bucks, and b. you get 1/4 of a ticket for a dollar.
1. the penguin walks 75 feet in 45 seconds, and 2. it takes 27 seconds to walk 45 feet.
Step-by-step explanation:
Tickets:
16 dollars divided by 4 tickets is 4 dollars, so 4 dollars = 1 ticket. therefore, 1 dollar gives you 1/4 of a ticket.
Penguins
formula: X = feet, Y = seconds
X*0.6=Y
10 feet*0.6=6 seconds
(?) feet*0.6= 45 seconds
75 feet*0.6=45 seconds
same thing, but backwards this time
45 feet*0.6=(?) seconds
45 feet*0.6= 27 seconds
Hey can I have brainly please I am really close to leveling up
thx
Answer:
Step-by-step explanation:
A salt shaker company produces salt shakers of varying heights. They create
a table showing how many salt granules each size salt shaker holds. Does this
data represent a proportional relationship? Click on the correct response, then
answer the question below.
Height of Salt Shaker
Number of Salt Granules
Answer:
Proportional and h.200=g
Step-by-step explanation:
Solve Systems of Equations Algebraically.
y=x-8
y=5x
Answer:
(- 2, - 10 )
Step-by-step explanation:
Given the 2 equations
y = x - 8 → (1)
y = 5x → (2)
Substitute y = 5x into (1)
5x = x - 8 ( subtract x from both sides )
4x = - 8 ( divide both sides by 4 )
x = - 2
Substitute x = - 2 into either of the 2 equations for y
Substituting into (1)
y = - 2 - 8 = - 10
solution is (- 2, - 10 )
identify the domain and range of the inverse of f(x) = 0.5x.
The domain and range of the inverse of the function f(x) = 0.5x need to be determined. The domain and range of the inverse function g(x) = 2x are both all real numbers.
To find the domain and range of the inverse of a function, we can swap the roles of x and y in the original function and then solve for y.
The original function is f(x) = 0.5x. Swapping x and y, we get x = 0.5y. Solving this equation for y, we multiply both sides by 2, giving us y = 2x.
The domain of the inverse function, denoted as g(x), is the set of all possible x-values. In this case, the domain of g(x) is the same as the range of the original function f(x). The range of f(x) is all real numbers, so the domain of g(x) is also all real numbers.
Similarly, the range of the inverse function is the set of all possible y-values. In this case, the range of g(x) is the same as the domain of the original function f(x). The domain of f(x) is all real numbers, so the range of g(x) is also all real numbers.
Therefore, the domain and range of the inverse function g(x) = 2x are both all real numbers.
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how do I write an equation for a quadratic function represented by a table ?
Answer:
Hello There!!
Step-by-step explanation:
Firstly,select three ordered pairs from the table. Secondly,substitute the first pair of values in to the form of the quadratic equation: f(x) = ax^2 + bx + c and then solve for a.
hope this helps,have a great day!!
~Pinky~
You take out a loan for $478 at an interest rate of 8% compounded annually. What is the
total amount that you will have paid at the end of 4 years?
Answer:
650.3
Step-by-step explanation:
Answer:
the total payment will be 91.43 and the total interest pay will be 13.43.
Step-by-step explanation:
Tbh I dont get what you mean that much, but I think you mean 48 months of payment times the monthly payment, soooo.
13.43x 48= 644.64
644.64 after four years. I really really hope this helps you. Have a good day
Y= -3/2x +3 is the same line as x + y =
Write the system x' = e²tx - 8ty + 7 sin(t), y' = 2 tan(t) y + 7x - 2 cos(t) in the form Use prime notation for derivatives and write x and ï', etc., instead of ï(t), x'(t), or dr. 1-1 dt = P60 [*] + F(0). P(t)
The system in the requested form, using prime notation for derivatives and ï notation for variables, is ï' = e^2tï - 8tï' + 7sin(t) and ï'' = 2tan(t)ï' + 7ï - 2cos(t).
The system can be written in the form:
x' = e^2tx - 8ty + 7sin(t)
y' = 2tan(t)y + 7x - 2cos(t)
Using prime notation for derivatives, the system becomes:
x' = e^2tx - 8ty + 7sin(t)
y' = 2tan(t)y + 7x - 2cos(t)
Furthermore, we can write x and y using ï' notation as:
x = ï
y = ï'
Substituting these expressions into the system, we have:
ï' = e^2tï - 8tï' + 7sin(t)
ï'' = 2tan(t)ï' + 7ï - 2cos(t)
In summary, the system in the requested form is:
ï' = e^2tï - 8tï' + 7sin(t)
ï'' = 2tan(t)ï' + 7ï - 2cos(t)
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Twenty-five wooden beams were ordered for a construction project. The sample mean and the sample standard deviation were measured: xbar = 260 cm, s = 2 cm. Calculated confidence interval for the mean is [259.32;260.68].
Which confidence level was chosen? assume distribution to be normal.
The answer tο the questiοn "Which cοnfidence level was chοsen?" is that it cannοt be determined based οn the given infοrmatiοn.
What is Statistical Inference?Statistical inference refers tο the prοcess οf drawing cοnclusiοns οr making predictiοns abοut a pοpulatiοn based οn sample data. It invοlves using statistical methοds and techniques tο analyze the data and make inferences abοut the underlying pοpulatiοn parameters. Statistical inference allοws researchers tο generalize their findings frοm a sample tο the entire pοpulatiοn and make infοrmed decisiοns οr predictiοns.
Twenty-five wοοden beams were οrdered fοr a cοnstructiοn prοject. The sample means and the sample standard deviatiοn was measured:[tex]\bar{x} = 260 \, \text{cm}$, $s = 2 \, \text{cm}$.[/tex] The calculated confidence interval for the mean is [tex][259.32; 260.68]$.[/tex]
To determine the confidence level chosen, we need to consider the properties of a confidence interval. In this case, the confidence interval is given as[tex]$[259.32; 260.68]$[/tex], which represents an interval estimate for the population mean.
A confidence interval is typically written as [tex]\bar{x} \pm E$,[/tex]where [tex]$\bar{x}$[/tex] is the sample mean and E is the margin οf errοr. The margin οf errοr depends οn the sample size, standard deviatiοn, and the desired level οf cοnfidence.
In the given infοrmatiοn, the sample mean [tex]$\bar{x}$[/tex] is 260 cm, and the sample standard deviatiοn s is 2 cm. Hοwever, the level οf cοnfidence is nοt directly prοvided.
Tο determine the cοnfidence level chοsen, we need additiοnal infοrmatiοn abοut the calculatiοn οr the specific cοnfidence level used. Withοut this infοrmatiοn, we cannοt determine the exact cοnfidence level that cοrrespοnds tο the given cοnfidence interval.
Therefοre, the answer tο the questiοn "Which cοnfidence level was chοsen?" is that it cannοt be determined based οn the given infοrmatiοn.
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Find the confidence interval specified. Assume that the population is normally distributed. The principal randomly selected six students to take an aptitude test. Their scores were: 84.1 79.8 79.6 81.3 88.1 81.0 Determine a 90% confidence interval for the mean score for all students.
The 90% confidence interval for the mean score for all students is [79.35, 86.75].
What is the range within which the mean score for all students lies with 90% confidence?A 90% confidence interval was calculated to estimate the mean score for all students based on a sample of six students' scores. The sample mean was found to be 82.83, and the standard deviation was 3.67.
Using a t-distribution with five degrees of freedom (n-1), the critical value for a 90% confidence level was determined to be 2.571. By applying the formula for confidence intervals, the lower bound of the interval was calculated as 82.83 - (2.571 * (3.67/√6)) = 79.35, and the upper bound was calculated as 82.83 + (2.571 * (3.67/√6)) = 86.75.
This means that we can be 90% confident that the mean score for all students falls within the range of 79.35 to 86.75.
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HELP ASAP!!! select all the descriptions that apply to the transformation!
Answer: A and B
Step-by-step explanation:
The shape is moving from right to left and down
How am I supposed to do this?
Quickly please I only have a few minutes
Answer:
1=4
5=8
2=5
Step-by-step explanation:
Answer:
1, 5, 2 is the x values.
Step-by-step explanation:
you just subtract 3 from the y's
The measures of the angles of a triangle are shown in the figure below. Solve for x.
Answer:
26 degrees
Step-by-step explanation:
180 - 48 = 132 - 106 = 26
this one should work
Answer: x = 106°
Step-by-step explanation:
106° + 48° + x° = 180°
154 + x = 180
x = 106°
given the rectangular coordinates (-8,15), which of the following polar coordinate pairs represents the same point in radians?
a(17,2.65)
b(-17,-1.08)
c(12.7,2.65)
d(-12.7,-0.49)
Answer:
1. (x,y)=
2. (3, 480°), (3,-240°), (-3, -60°)
3. (-17, -1.08)
4. on the second circle from the center and 153° clockwise from the horizontal axis
Step-by-step explanation:
took quiz...100%
The weather reporter says that there is a 12% chance that it will be moderately windy tomorrow.
What is the probability that it will not be windy? & Will tomorrow be a good day to fly a kite? Explain.
(______Be specific ya numptys______)
The probability that it will not be windy is 88% or 0.88.
No, tomorrow be not a good day to fly a kite.
What is Probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.
The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.
The degree to which something is likely to happen is basically what probability means
The weather reporter says that there is a 12% chance that it will be moderately windy tomorrow.
Now, the probability that it will not be windy
= 100 - 12
= 88%
= 0.88
No, the lite totally depends on the wind but the next day the probability of no wind is 88%.
This probability is high probability.
So, there is huge probability that the kite will not fly.
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if h(x) = 3(x + 6), find h(-1).
a.3
b.9
c.15
d.21
Find value(s) of k so that the linear system is consistent? (Enter your answers as a comma-separated list.) 8x1-7x2 = 2 12x1 + kx2 =-1
There are no values of k that make the linear system consistent.
To determine the values of k for which the linear system is consistent, we need to check if the system of equations has a unique solution, infinitely many solutions, or no solution.
The given system of equations is:
8x1 - 7x2 = 2
12x1 + kx2 = -1
We can solve this system using various methods, such as substitution or elimination. Let's use the elimination method to simplify the system.
To eliminate x2, we can multiply equation 1 by k:
8kx1 - 7kx2 = 2k
Now, we can subtract equation 2 from equation 3:
(8k - 12)x1 + (-7k - k)x2 = (2k + 1)
Simplifying equation 4, we have:
(8k - 12)x1 - 8kx2 = 2k + 1
Now, for the system to be consistent, the coefficient of x1 and the coefficient of x2 in the resulting equation should be the same.
Comparing the coefficients, we get:
8k - 12 = 0 (coefficient of x1)
-8k = -1 (coefficient of x2)
Solving the first equation:
8k - 12 = 0
8k = 12
k = 12/8
k = 3/2
Now, substitute the value of k in the second equation to check if it is satisfied:
-8k = -1
-8(3/2) = -1
-12 = -1
The equation -12 = -1 is false. Therefore, there are no values of k for which the linear system is consistent.
In conclusion, there are no values of k that make the given linear system consistent.
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evaluate the exponent expression for a = –2 and b = 3. question 15 options: a) –9∕8 b) –2∕5 c) –6 d) 3
The correct option is A) 9∕8, evaluating the exponent expression with a = -2 and b = 3, we find that the value is -8.
We are given the expression a^b, where a = -2 and b = 3. Substituting these values into the expression, we have (-2)^3.
To evaluate this expression, we raise -2 to the power of 3. When we raise a negative number to an odd power, the result will be a negative number.
So, (-2)^3 will yield a negative value.
Calculating (-2)^3, we multiply -2 by itself three times: (-2) × (-2) × (-2). This equals -8.
Therefore, the correct option is A) 9∕8 and the value of the exponent expression (-2)^3 is -8.
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a cell phone plan costs $28.90 per month for 400 minutes of talk time. it costs an additional $0.07 per minute for each minute over 400 minutes. to get e-mail access, it costs 10% of the price for 400 minutes of talk time. your bill, which includes e-mail, is the same each month for 8 months. the total cost for all months is $298.56. write and solve an equation to find the number of minutes of talk time you use each month.
Answer:
Equation = 298.56=6(0.07m+23.90+0.10(23.90))
Minutes = 335
Step-by-step explanation:
Okay, so first simplify to get 298.56=0.52m+172.08
Next subtract to get 0.52m=126.48
Then divide 0.52 to get 335
I hope this helps ! :)
A biologist has two brine solutions, one containing 7% salt and another containing 28% salt. How many milliliters of each solution should she mix to obtain 1 L of a solution that contains 19.6% salt?
Answer:
Each litters can be represented as; x and y respectively to the 7% and the 28%
x + y =1
7x + 28y =19.6
This a simultaneous equation and can either be solved by using elimination or substitution method
Using the substitution method
eqn1, x=1-y
putting into eqn2
7(1-y)+28y =19.6
7-7y+28y=19.6
21y=12.6
y=3/5
but x=1-y
so
x=1-3/5
x=2/5