Answers
Answer: Convergent; converges to 9/5 = 1.8
======================================================
Explanation:
The expression 8/(n^2-1) is the same as 8/( (n+1)(n-1) )
Use partial fraction decomposition to get
8/( (n+1)(n-1) ) = A/(n+1) + B/(n-1)
8 = (n+1)(n-1)*( A/(n+1) + B/(n-1) )
8 = A(n-1) + B(n+1)
8 = An-A + Bn+B
0n + 8 = (A+B)n + (-A+B)
We can see that
[tex]\begin{cases}A+B = 0\\-A+B = 8\end{cases}[/tex]
Use elimination to get 2B = 8 which leads to B = 4
Plug B = 4 into either equation of that system to solve for A getting A = -4
Therefore,
8/( (n+1)(n-1) ) = A/(n+1) + B/(n-1)
8/( (n+1)(n-1) ) = -4/(n+1) + 4/(n-1)
--------------------------------------------
Why do all this? Because it'll be helpful to set up the telescoping sum
Let's plug in a few values of n. Let's say from n = 5 to n = 10
If we do so, then we have...
-4/(5+1)+4/(5-1) = -4/6 + 4/4
-4/(6+1)+4/(6-1) = -4/7 + 4/5
-4/(7+1)+4/(7-1) = -4/8 + 4/6
-4/(8+1)+4/(8-1) = -4/9 + 4/7
-4/(9+1)+4/(9-1) = -4/10 + 4/8
-4/(10+1)+4/(10-1) = -4/11 + 4/9
I'm not evaluating or simplifying further because we'll have cancellations as shown in the diagram below. The left panel is what you see before cancelling happens, and the right hand panel is what happens after the terms pair up and cancel (since those pairs add to 0).
Note the color coding so you can see what terms pair up and cancel. The only stuff left is the 4/4 and the 4/5
So,
4/4 + 4/5 = 1 + 4/5 = 5/5+4/5 = 9/5 = 1.8
is what the infinite sum converges to. You can use a tool like wolfram alpha to confirm this answer.
Related Questions
4. Suppose that you are offered the following "deal." You roll a die. If you roll a six, you win $10. If you roll a four or five, you win $5. If you roll a one, two, or three, you pay $6. a. What are you ultimately interested in here (the value of the roll or the money you win)? b. In words, define the Random Variable X. c. List the values that X may take on. d. Construct a PDF. e. Over the long run of playing this game, what are your expected average winnings per game? f. Based on numerical values, should you take the deal? Explain your decision in complete sentences.
Answers
Answer:
Explained below.
Step-by-step explanation:
The rules of the deal are as follows:
If you roll a six, you win $10. If you roll a four or five, you win $5. If you roll a one, two, or three, you pay $6.(a)
The correct option is: "the money you win".
(b)
The random variable X denotes the amount won or lost.
(c)
The value of X are as follows:
X = {-$6, $5 and $10}
(d)
The probability function of X is as follows:
P (X = 10) = 1/6
P (X = 5) = 1/3
P (X = -6) = 1/2
(e)
Compute the expected value as follows:
[tex]E(X)=\sum x\times P(X=x)\\[/tex]
[tex]=(10\times\frac{1}{6})+(5\times\frac{1}{3})+(-6\times\frac{1}{2})\\\\=0.333333\\\\\approx $0.33[/tex]
Thus, the average winnings per game is $0.33.
(f)
Since the expected value of the game is a positive amount, the deal will not lead to me paying up.
Thus, you take the deal.
What is the remainder when 18,049 is divided by 14? (Input numeric values only)
Answers
Answer:
18049/14= 1289 with a remainder of 3
Answer:
3
Step-by-step explanation:
Which number line can be used to find the distance between (4, -1) and (8,-1)?
Answers
Answer:
the first one
The number line which can be used to find the distance between (4, -1) and (8,-1) is 1
What is the distance between two points ( p,q) and (x,y)?The shortest distance (length of the straight line segment's length connecting both given points) between points ( p,q) and (x,y) is:
[tex]D = √[(x-p)² + (y-q)²] D = \sqrt{(x-p)^2 + (y-q)^2} \: \rm units.[/tex]
Given the points;
(x1, y1) = (4, -1)
(x2, y2) = (8, -1)
Here, we have the same values of y for both points.
This means that the line that contains both points is a horizontal line.
To find the distance, apply the formula:
[tex]d= \sqrt{(x2-x1)^{2} + (y2-y1)^{2} }[/tex]
Since they have the same y values, the number line to represent the distance between both points must have the points 4 and 8;
Hence, we have;
[tex]d= \sqrt{(8-4)^{2}+(-1+1)^{2} }[/tex]
[tex]d= \sqrt{4^{2} + 0^{2} }[/tex]
d= 4
The distance between both points is 4;
Therefore, the number line that can be used to find the distance between both points is the number line in option A.
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v/3=1 Please help what is V
Answers
Answer:
3
Step-by-step explanation:
v=1*3
Answer:
3
Step-by-step explanation:
Multiply each side by 3 and you get 3
v=1*3=3
v=3
on Monday it rained 1 3/4 of an inch and on Tuesday it rained 2 2/3 of an inch. how many inches did it rain altogether?
Answers
Answer:
4 5/12
Step-by-step explanation:
Convert to mixed fraction 7/4 + 8/3
Find common denominator, 12
21/12 + 32/12
add numerators (21+32)/12 = 53/12
reduce fraction 4 5/12
determine whether the ordered pair is a solution of the linear equation. thanks in advance!!
Answers
Work:
EQ: y= -x + 4
x= -6
y= 10
y= -x+4
1. 10= -(-6)+4
2. 10= +6+4
3. 10= 10
so, my teacher barley taught us this yesterday and i'm still having trouble with some questions, could anyone please help? ( this is 7th grade math )
Answers
5.) x>20
11.)x≤−15
13.)x≥−21
2.)x≤−7
6.)x≤−45
10.)x≤10
12.)x≥−12
16.)x≤−4
Answer:
2. x ≤ -7
5. x > 36
6. x ≤ -40
10. x ≤ 10
11. x ≤ -15
12. x ≥ -12
13. x ≥ -21
16. x ≤ -4
Step-by-step explanation:
2. x ≤ -7 (You want to add 8 to both sides like any equation, but don't forget it is an inequality, not an equation!)
5. x > 36 (You need to multiply by 2, to get rid of the division. So multiply both sides by 2, that gets rid of the 2 on the one side, and you get 36 on the other side when you multiply 18 by 2).
6. x ≤ -40 (You need to multiply by -5. REMEMBER: When you multiply or divide by a negative, you switch the sign. So if it is a < then it would be a > and same with if you had ≤ then it would be ≥ instead).
10. x ≤ 10 (You need to divide by -2. And once again you switch the signs).
11. x ≤ -15 (Add 8, like you did in number 2).
12. x ≥ -12 (Multiply by 6. REMEMBER: Do NOT switch the signs if you the number you are dividing by is in the numerator, or if you multiply and the negative number is first, UNLESS you have 2 negatives.
13. x ≥ -21 (Multiply by 7).
16. x ≤ -4 (Divide by 10).
A house is purchased for $150,000 in 2002. The value of the house
depreciates at a rate of 7%. How much is the house worth in 2013? *
Growth:
Decay:
y=a(1+r)*
y= a(1-r)*
a = initial amount before measuring growth/decay
r = growth/decay rate (often a percent)
X = number of time intervals that have passed
Answers
9514 1404 393
Answer:
$67,516
Step-by-step explanation:
"Depreciates" means the value is getting smaller. The function that describes the situation is a decay function:
y= a(1-r)^x
Here, a=150000, r = 0.07, and x=11. The value is predicted to be ...
y = 150000(1 -0.07)^11 = 150000(0.93^11) = 67,515.53
The house is worth about $67,516 in 2013.
Urgent. Solve -3−4x≤21.
Answers
Answer: x => -6
Step-by-step explanation: All you gotta do is alternate the form
The solution for the given inequality is x ≥ -6.
What are inequalities in mathematics?A declaration of an order connection between two numbers or algebraic expressions, such as greater than, equal to, less than, less than, or equal to. Either questions or theorems can be used to express inequality problems, and both can be solved using methods similar to those used to solve equations. The triangle inequality, for instance, stipulates that the length of the remaining side of a triangle is more than or equal to the sum of the lengths of any two of its sides. Many of these inequalities, like the Cauchy-Schwarz inequality, are used by mathematical analysis to prove some of its most significant theorems.
Given an equation -3−4x ≤ 21.
Solve for x
-3−4x ≤ 21
add three to both sides
-4x ≤ 24
divide by -4 into both sides (signs will be changed)
x ≥ -6
Therefore solution for the given equation in the form of inequalities is
x ≥ -6.
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120 eggs from 20 chicken
Answers
Answer:
60 eggs from 1 chicken
Step-by-step explanation:
How do you figure out
k+43=118
Answers
Answer:
Step-by-step explanation:
k+43=118
k=118-43
k=75
Answer:
Your answer is: k= 75
Step-by-step explanation:
Hope this helped : )
Can -0.05 be a probability?
Answers
Answer:
Yes
Step-by-step explanation:
Answer:
The probability of an event can only be between 0 and 1 and can also be written as a percentage. ... If P ( A ) > P ( B ) P(A) > P(B) P(A)>P(B)P, left parenthesis, A, right parenthesis, is greater than, P, left parenthesis, B, right parenthesis, then event A has a higher chance of occurring than event B.
so that means yes
A number is doubled and then the product is increased by 6. This sum is then multiplied by -3, and the result is 54 greater than three times the opposite of the number. What is the number? *
Answers
Answer:
-24Step-by-step explanation:
Let the number be x
Then we have the equation
(2x + 6)*(-3) = -3x + 54 -6x - 18 = -3x + 546x - 3x = -18 - 543x = -72x = -72/3x = - 24let the number be "x" .
now according to the given question ;
(2x+6)×(-3)=-3x +54 -6x -18 = -3x + 54 6x - 3x = -18-543x = -72x = -72/3x = -24 The required answer is -24 .What does it mean to be a medium of exchange
Answers
Answer:
The medium is the middle of something. More like math thought. So what even the numbers are it will be the middle.
Step-by-step explanation:
Answer: Any item that can be traded for another item.
Step-by-step explanation: took the quiz
Yasmin had 3 bananas she shared them equally among the 4 people in her family write an expression that shows how much banana each person got
Answers
Answer:
3/4 of a banana
Step-by-step explanation:
3 divided by 4 = 3/4
Sara has 4 pints of juice she wants to pour a equal amount of juice into 5 cups using all the juice. Which equation show how to find the amount of juice, in pints, in each glass?
Answers
Answer:
what are you in 5th? cause I might have the answer
An equation shows the amount of juice, in pints, in each glass is x =4/5.
What is an equation?In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign =.
Sara has 4 pints of juice she wants to pour a equal amount of juice into 5 cups using all the juice.
Quantity of juice poured into cups be x.
Now, x= 4/5
Therefore, the equation is x= 4/5.
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can you please answer the following
Answers
Answer:
5. 4
6. ?
7. c- 8
8. b-13
9. ?
10. d. 12
Step-by-step explanation:
i hope this helps, sorry for the question marks :)
5. B. 4
6. sq. root of 2
7. C. 8
8. B. 13
9. B. 12x12=144
10. C. 11
Explanation.
5. A perfect square is a number with a perfect square root. of the four numbers, 4 is a perfect square.
6. An irrational number is a number that cannot be expressed as the ratio of two integers. The square root of two is an example of this.
7. A square root is a number times itself, but you do not know what number is being multiplied. By testing all the options, we find that 8x8 is indeed 64.
8. A square root is a number times itself, but you do not know what number is being multiplied. By testing all the options, we find that 13x13 is indeed 169.
9. We know that a square root is when a number is multiplied my itself. 12x12 is 144 therefore the square root of 144 is 12.
10. Because the square root of 140 does not have an simple answer, we can estimate. The approximate answer is 11.83 which can round up to 12 if that answer is available, select it. If not, the answer is 11.
Hope this helps, let me know if it’s correct so others can use it.
Good luck.
Which of the following is NOT true about the following equation. y= -4
Answers
Answer:
Answer is D.
Step-by-step explanation:
Which is not true?
So, lets assess the answers given to the question.
A. It is the equation of a horizontal line.
Yes, it is. Since there is only one y - value which is y = -4, we can confirm this is the y intercept, so therefore the x value is zero. But it's not the answer we're looking for.
B. It's slope is zero, yes the slope is zero. It only lists the y-intercept which confirms that the slope is 0.
C. Yes, I mean i can't really explain how. Just use desmos calculator, the graph is exactly perpendicular.
D. The line actually does cross the x-axis, -4 is the y intercept and the x value is 0, and zero is still a point. So D is the one which isn't correct making it the correct answer.
What is an equation for the line that passes through the coordinates (- 1, 2) and (7, 6) ?
Answers
Answer:
y = 1/2x + 2.5
Step-by-step explanation:
First we want to plug it into the [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]. This gets us 6-2/7-(-1) which solves to 4/8 which simplifies to 1/2. This gives us the slope. Now, when using slope intercept form y = mx + b, we can plug our newly found slope in for m and get y = 1/2x + b. To find b plug in any point on the line into the equation, I chose (7,6) because we are given that, and both numbers are positive. Now plug those in for x and y. 6 = 1/2(7). Solve, and get 2.5. are final equation is
y = 1/2x + 2.5
You would like to know whether forwards in a basketball league average the same (or different) numbers of points than the overall league average of 7.5 points. Drawing a random sample of 10 forwards from the league, their averages were: 10, 9, 6, 11, 13, 14, 9, 9, 9, and 10 points. If you were going to compare the obtained average of scoring by the sampled forwards with the population value of 7.5.
Required:
What is the critical t-value for such a test given an alpha level of 0.05?
Answers
Answer:
The critical t-value for such a test given an alpha level of 0.05 is 2.26
Step-by-step explanation:
Null hypothesis : [tex]H_0:\mu = 7.5[/tex]
Alternate hypothesis :[tex]H_a:\mu \neq 7.5[/tex]
Population mean = [tex]\mu = 7.5[/tex]
Data : 10, 9, 6, 11, 13, 14, 9, 9, 9, and 10
Mean = [tex]\bar{x}=\frac{\text{Sum of all observations}}{\text{no. of observations}}[/tex]
Mean =[tex]\frac{10+9+6+11+13+14+9+9+ 9+ 10}{10}[/tex]
Mean =10
Standard deviation : [tex]\sqrt{\frac{\sum(x-\bar{x})^2}{n}}[/tex]
Standard deviation :[tex]\sqrt{\frac{(10-10)^2+(9-10)^2+(6-10)^2+(11-10)^2+(13-10)^2+(14-10)^2+(9-10)^2+(9-10)^2+(9-10)^2+(10-10)^2}{10}}[/tex]
Standard deviation s :2.144
[tex]t = \frac{x-\mu}{\frac{s}{\sqrt{n}}}t = \frac{10-7.5}{\frac{2.144}{\sqrt{10}}}t=3.687[/tex]
Df = n-1 = 10-1 = 9
T critical =[tex]t_{(df,\alpha)}=t_(9,0.05)=2.26[/tex]
T calculated > T critical
So, We failed to accept null hypothesis
Hence the critical t-value for such a test given an alpha level of 0.05 is 2.26
There is was an article in a travel magazine that posed the question..."Why are flights so much cheaper in Europe than in the United States?" The article reported that transatlantic round-trip flights originating in the United States can be more expensive than the same flights originating in Europe. The data on several flights was gathered for 3 airline companies. What is the Average Price of a ticket originating in Chicago using United Airlines? Round your answer to two decimal places. NOTE: This data corresponds with worksheet tab, Q33, in the Exam 2 Excel file in Canvas. You will use this for questions Q33-39. United Air France British Chicago 1063 1314 1442 1299 1493 1413 995 1443 1063 1299 1577 1442 1406 1314 1575 London 1015 1374 1603 856 977 1531 802 1043 1079 850 1070 807 935 1428 870
Answers
Answer:
The average price of a ticket originating in Chicago using United Airlines is $1212.40.
Step-by-step explanation:
The data provided is as follows:
United Air France British
Chicago 1063 1314 1442
1299 1493 1413
995 1443 1063
1299 1577 1442
1406 1314 1575
London 1015 1374 1603
856 977 1531
802 1043 1079
850 1070 807
935 1428 870
The data for the price of a ticket originating in Chicago using United Airlines is:
Chicago = {1063, 1299, 995, 1299, 1406}
Compute the average price as follows:
[tex]\bar X=\frac{1}{n}\sum X[/tex]
[tex]=\frac{1}{5}\times [1063+1299+995+1299+1406]\\\\=\frac{6062}{5}\\\\=1212.40[/tex]
Thus, the average price of a ticket originating in Chicago using United Airlines is $1212.40.
Standard of living is the level at which
Answers
Answer:
hope it helped you
Step-by-step explanation:
Standard of living refers to the level of wealth, comfort, material goods, and necessities available to a certain socioeconomic class or geographic area. Quality of life, on the other hand, is a subjective term that can measure happiness.
Answer:
D( consumer enjoy desired goods
Step-by-step explanation:
just did the quiz
Let R be a relation on a collection of sets defined as follows,
R={(A,B)|A⊆B}
Then pick out the correct statement(s).
1) R is reflexive and transitive
2) R is symmetric
3) R is antisymmetric.
4) R is reflexive but not transitive
Answers
Answer:
Options 1) and 3) are correct.
Step-by-step explanation:
R={(A,B)|A⊆B}
Reflexive:
As A⊆A, [tex](A,A)[/tex]∈ R.
So, R is reflexive
Symmetric:
Let [tex](A,B)[/tex]∈ R. So, A⊆B
Take [tex]A=\{1,2\}\,,\,B=\{1,2,3,4\}[/tex]
Here, A⊆B but B⊄A
So, [tex](B,A)[/tex]∉ R
R is not symmetric
Transitive:
Let [tex](A,B)[/tex]∈ R and [tex](B,C)[/tex]∈ R
So, A⊆B and B⊆C.
Therefore, A⊆C
So,
[tex](A,C)[/tex]∈ R
Hence, R is transitive.
Option 1) is correct.
Antisymmetric:
Let (A,B)∈R and (B,A)∈R
So, A⊆B and B⊆A
Hence, A = B
So, R is antisymmetric
Option 3) is also correct.
A commuter train travels 31 kilometers in 27 minutes. What is its speed in per hour?
Answers
The speed of the train in kilometer per hour is 68.9km/hr.
What is speed?The rate of change of position of an object in any direction.
Formula for speed[tex]speed = \frac{distance}{time}[/tex]
According to the given question
Distance travelled by commuter train = 21km
Time taken by a train to travel 31 km = 27 minutes = [tex]\frac{27}{60} =0.45hour[/tex]
Therefore, the speed of the train in kilometer per hour is given by
[tex]speed = \frac{31}{0.45}[/tex]
⇒[tex]speed = 68.9km/hr[/tex]
Hence, the speed of the train in kilometer per hour is 68.9km/hr.
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The graph shows a function that models the value V (in millions of dollars) of a stock portfolio as a function of time t (in months) over a 18-month period.
Answers
Answer:
Option (A)
Step-by-step explanation:
Graph attached shows a function that models the value V of a stock portfolio as a function of time.
y-values on the graph shows the value of V and x-values represent the time.
Lowest point of the curve shows the minimum value of V and highest point of the curve shows the maximum value.
Maximum value of the function → 6.93 at t = 4
Minimum value of the function → 1.05 at t = 15
Therefore, Option (A) will be the answer.
what is the greatest number of penss that can be bought with $ 8.15 if each pen costs 85 cents?
Answers
Answer:
9
Step-by-step explanation:
x=pens
.85x=8.15
.85x/.85= 8.15/.85
x=9.58
so 9 whole pens can be bought
A box of cereal states that there are 87 calories in a 3/4 -cup serving. What is the unit rate for calories per cup? How many calories are there in 4 of the cereal?
Answers
Answer: The unit rate is
116 calorie(s) per cup
and
There are 464 calories in 4 cups of the cereal
A driver averaged 51 miles per hour and took 14 hours to travel between two cities. What is the distance between the two cities?
Answers
Answer:
714 miles
Step-by-step explanation:
if he drove 14 hours and in each hour he drove 51 miles then multiply
14*51=714
Find the inverse of the function f(x) = 2x - 4.
g(x) = 1/2x - 1/4
g(x) = 1/4x - 1/2
g(x) = 4x + 2
g(x) = 3x+2
Answers
Answer:
None of the options are correct, but I got g(x)=1/2x+2.
Step-by-step explanation:
You start with the equation. You first need to change f(x) to y, and switch the x and y values. After this, your equation would be x=2y-4. Then, you need to icolate the y value, and to do that you first cancel out the four by adding it to both sides, and now your equation should be x+4=2y. The second step in icolating the y value is dividing everything in the equation by two. Now that the y is icolated, you equation should be y=1/2x+2. You then change the y to be g(x), and that's how I got my answer, g(x)=1/2x+2. You should ask your teacher about this, becuase from my calculations, none of the given responses would be correct. Sorry if that doesn't help you at all but yeah
Answer:
D, g(x) = 1/2 x + 2
Step-by-step explanation:
The second answer IS ONE TO ONE
What is the domain of the relation graphed below?
Answers
Answer:
you need to show us the graph