Does anyone know how to simplify this expression? I know it has to do with properties of summation and special summation rules but I can’t quite figure it out.
Answers
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Explanation:
Let's expand this out up to say n = 6 terms
I'll write the summation in such a way that each (1/(k+1) - 1/k) grouping will get its own row. Each row has the same value of k.
If we plug in k = 1 through k = 6, but not evaluate just yet, we will have this:
1/(1+1) - 1/1
1/(2+1) - 1/2
1/(3+1) - 1/3
1/(4+1) - 1/4
1/(5+1) - 1/5
1/(6+1) - 1/6
All I've done so far is replace k with 1 through 6. Again, each row represents a different k value. Each row has the general format 1/(k+1)-1/k.
Let's simplify everything in the first column.
So the (1+1) turns into 2, the (2+1) turns into 3, and so on.
Doing that leads to...
1/2 - 1/1
1/3 - 1/2
1/4 - 1/3
1/5 - 1/4
1/6 - 1/5
1/7 - 1/6
Then note how we have the cancellations shown in the diagram below. The color coding shows how the terms pair up to cancel out. By "cancel out", I mean specifically that the fractions add up to 0. Eg: 1/2 + (-1/2) = 0.
Nearly everything cancels out. The only things left after the dust settles is the -1/1 in the first row and the 1/7 in the bottom row. This evaluates to 1/7-1/1 = 1/7 - 7/7 = (1-7)/1 = -6/7
If you were to try this with n = 7, then you should find that again nearly everything cancels but the fractions -1/1 and 1/8. Then 1/8 - 1/1 = -7/8.
For n = 8, you should end up with 1/9 - 1/1 = -8/9.
You can probably see the pattern.
A conjecture is that the answer is -n/(n+1). The general proof of this isn't too tricky. Simply follow the same ideas as mentioned above. Expand out a few terms and see how things cancel out. You'll find that everything cancels except for the -1/1 in the first row and the 1/(n+1) just one term before the very end. Then we can say:
1/(n+1) - 1/1
1/(n+1) - (n+1)/(n+1)
(1-(n+1))/(n+1)
(1-n-1)/(n+1)
-n/(n+1)
If we tried n = 6, then we find,
-n/(n+1) = -6/(6+1) = -6/7
which was the result we found earlier when we added the first n = 6 terms of this series. Trying out n = 7 should lead to -n/(n+1) = -7/8, and so on.
Related Questions
Use the given information to find the minimum sample size required to estimate an unknown population mean μ.
How many students must be randomly selected to estimate the mean weekly earnings of students at one college? We want 95% confidence that the sample mean is within $4 of the population mean, and the population standard deviation is known to be $25.
Answers
Answer:
151 students must be selected.
Step-by-step explanation:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1 - 0.95}{2} = 0.025[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1 - \alpha[/tex].
That is z with a pvalue of [tex]1 - 0.025 = 0.975[/tex], so Z = 1.96.
Now, find the margin of error M as such
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
In which [tex]\sigma[/tex] is the standard deviation of the population and n is the size of the sample.
The population standard deviation is known to be $25.
This means that [tex]\sigma = 25[/tex]
How many students must be randomly selected to estimate the mean weekly earnings of students at one college? Sample mean within $4 of the population mean
This is n for which M = 4. So
[tex]M = z\frac{\sigma}{\sqrt{n}}[/tex]
[tex]4 = 1.96\frac{25}{\sqrt{n}}[/tex]
[tex]4\sqrt{n} = 1.96*25[/tex]
[tex]\sqrt{n} = \frac{1.96*25}{4}[/tex]
[tex](\sqrt{n})^2 = (\frac{1.96*25}{4})^2[/tex]
[tex]n = 150.1[/tex]
Rounding up:
151 students must be selected.
What is a,b,c and d please help me
Answers
Step-by-step explanation:
Edit:
a is 18
b is 36
c is 18
d is 18 times sqr root of 3
Answers:
a = 18b = 36c = 18d = 18*sqrt(3)sqrt is shorthand for "square root"
=====================================================
Explanation:
The smaller triangle on the left is a 45-45-90 triangle (aka isosceles right triangle). One special property of such triangles is that the legs are the same length, meaning a = c. Another special feature is that the hypotenuse is exactly sqrt(2) times that of the leg.
In other words, if x is the leg length, then x*sqrt(2) is the length of the hypotenuse. That means mean a = 18 in order for the hypotenuse to be 18*sqrt(2). This then leads to c = 18 as well.
--------------
The smaller triangle on the right side has a = 18 as the shorter leg, opposite the smallest angle (30 degrees). The hypotenuse b is twice that of the short leg, and this works because we're dealing with a 30-60-90 triangle. Therefore b = 2a = 2*18 = 36.
The longer leg d is equal to sqrt(3) times the short leg, ie,
d = a*sqrt(3)
d = 18*sqrt(3)
You could use the pythagorean theorem to solve for d like so:
a^2 + d^2 = b^2
18^2 + d^2 = 36^2
d^2 = 972
d = sqrt(972)
d = 18*sqrt(3)
I skipped a bunch of steps, but you get the idea.
What is the surface area of the sphere below?
5
2.5
A. 1257 units2
B. 1007 units2
C. 12573 units2
O
D. cannot be determined
Answers
Here's the solution :
In a sphere,
[tex] \boxed{ \mathsf{surface \: \: area = 4\pi {r}^{2} }}[/tex]
now, let's solve :
[tex]s = 4 \times \pi \times 5 \times 5[/tex][tex]s = 100\pi[/tex]hence, correct answer is B. 100π unit²
The surface area of the sphere is 100 π unit².
The correct option is B.
What is Surface area?The area is the area occupied by a two-dimensional flat surface. It has a square unit of measurement. The surface area of a three-dimensional object is the space taken up by its outer surface. Square units are used to measure it as well.
We have,
Radius of Sphere= 5 unit
Now, the formula for Surface area of Sphere is
= 4πr²
= 4 (3.14) (5)(5)
= 4 x 3.14 x 25
= 314
= 10 π unit²
Thus, the Surface area is 10 π unit².
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Help me pleaseeee thank you
Answers
For each day is an increase of 21
Answer:
D
Step-by-step explanation:
Time is directly proportional to the Cost. And if you use the calculator you'll still get the answer by substituting the time into 21t
Please help!!!!!!!!!!!
Answers
Answer: 4 X's on 2 1/2, 3 X's on 3, 2 X's on 3 1/2, 1 X on 4
Step-by-step explanation:
1/6 divided by 1/2 as a reduced fraction
Answers
Answer:
1/3
Step-by-step explanation:
1/3
Step-by-step explanation:
1/6 ÷ 1/2
Copy dot flip
1/6 * 2/1
Multiply
2/6
Reduce
1/3
Answer it plssss........
Answers
Answer:
A
Step-by-step explanation:
Because x has the degree one and it’s figure at the numerator
A circle has radius 6 cm. Suppose an arc on the circle has length 31 cm. Answer the
following:
a. The fraction of the whole circle represented by this arc is
b. The measure of the central angle defined by the arc is
degrees.
C. The area of the sector defined by the arc is
cm2. (Express answer
in terms of r or pi).
Answers
Answer:
[tex]Fraction = \frac{31}{38}[/tex]
[tex]\theta = 296.18^\circ[/tex]
[tex]Area = 29.62 \pi[/tex]
Step-by-step explanation:
Given
[tex]r = 6[/tex]--- radius
[tex]L = 31[/tex] --- arc length
Solving (a): Fraction represented by the arc
First, calculate the circumference (C)
[tex]C = 2\pi r[/tex]
So, we have:
[tex]C = 2 * 3.14 * 6[/tex]
[tex]C = 38[/tex] --- approximated
So, the fraction represented by the arc is:
[tex]Fraction = \frac{L}{C}[/tex]
[tex]Fraction = \frac{31}{38}[/tex]
Solving (b): Measure of the center angle
Using arc length formula, we have:
[tex]L = \frac{\theta}{360} * 2\pi r[/tex]
This gives
[tex]31 = \frac{\theta}{360} * 2 * 3.14 * 6[/tex]
[tex]31 = \frac{\theta}{360} * 37.68[/tex]
Make [tex]\theta[/tex] the subject
[tex]\theta = \frac{360 * 31}{37.68}[/tex]
[tex]\theta = 296.18^\circ[/tex]
Solving (c): The area of the sector
This is calculated as:
[tex]Area = \frac{\theta}{360} * \pi r^2[/tex]
So, we have:
[tex]Area = \frac{296.18}{360} * \pi * 6^2[/tex]
[tex]Area = \frac{296.18}{360} * \pi * 36[/tex]
Divide 360 and 36
[tex]Area = \frac{296.18}{10} * \pi[/tex]
[tex]Area = 29.618 * \pi\\[/tex]
[tex]Area = 29.62 \pi[/tex] --- approximated
Picture Below
(i have to make this 20 characters so I'm doing this now)
Answers
Answer:
H.
Step-by-step explanation:
16x^2 is the big square area and the -10x comes from subtracting the smaller area from the big one
Answer:
H. is the answer
hope this helps
have a good day :)
Step-by-step explanation:
Researchers surveyed 250 college students at one time. They were asked if they were taking any engineering classes and whether or not they felt depressed. If 75 students reported taking at least one engineering class and 15 of those students said they were depressed, what is the incidence of depression in this group
Answers
Answer:
the incidence of the depression is 1 ÷ 5
Step-by-step explanation:
Given that
There is 250 college students that are surveyed
In the case when 75 students reported for minimum ne engineering class
And, out of this 15 students would be depressed
We need to find out the incidence of the depression
So
= 15 ÷ 75
= 1 ÷ 5
Hence, the incidence of the depression is 1 ÷ 5
Please can somebody help me
Answers
Answer:
Rectangle: bh or l · w
Circle: pi(r)²
Which of the following statements about the mapping is not true?
Answers
Answer:
I'm pretty sure the answer is A
Step-by-step explanation:
range can have multiple domain but domain can't have multiple ranges. So you can have (1,3) , (2,3) , (4,3)
but you can NOT HAVE (1,1) , (1,2) , (1,6)
the x value can not repeat in order for it to be a function due to the vertical line test thing
Find the value of x.
Answers
Answer:
x=17 plz the brainliest!
Step-by-step explanation:
3x+39=90
3x+39-39=90-39
3x=51
\frac{3x}{3}=\frac{51}{3}
x=17
The focus of a parabola is (2, 5) and the directrix is y = 1. What is the equation of the parabola?
Answers
Answer:
y = 0.125x² - 0.5x + 3.5
Hope that this helps!
Answer:
y = 0.125x² - 0.5x + 3.5
Step-by-step explanation:
I dont know if its correct hope it helped
Suppose that the function y = f(x) is decreasing on the interval ( 5, 7). What can be said about the graph of y= -f(x)
Answers
The graph of y = -f(x) is increasing on the interval ( 5, 7) if the function y = f(x) is decreasing on the interval ( 5, 7).
What is a function?It is defined as a special type of relationship, and they have a predefined domain and range according to the function every value in the domain is related to exactly one value in the range.
It is given that:
Suppose that the function y = f(x) is decreasing on the interval ( 5, 7).
The function:
y = -f(x)
Let f(x) be:
f(x) = -aˣ
Here 'a' is the real number a > 0
f(x) is decreasing on the interval ( 5, 7).
Applying transformation:
y = -f(x)
y = aˣ
Function y is increasing on the interval ( 5, 7).
Thus, the graph of y = -f(x) is increasing on the interval ( 5, 7) if the function y = f(x) is decreasing on the interval ( 5, 7).
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y = 3/5x + 5
y = -3/5x + 5
y = 3/5x - 5
y = -3/5x - 5
Answers
Answer:
y= 3/5x+5
Step-by-step explanation:
The y intercept is 5 and the graph is going upwards so it is a postive slope. If you look at at the graph from the y intercept point to point B you can see it goes up by 3 and right by 5 so the slope is 3/5. This makes the equation y=3/5x+5
HELP ME WITH THIS PLS! MATH :)
Answers
Answer:
(-7,5)
Step-by-step explanation:
the question was cut off, but I'm assuming that you want to find the location of Denver. Looking at the coordinate plane, Denver, would sit at (-7,5) as you find what x and y values they meet up at. Hopefully this answers your question
If we are looking for the location of Dallas (that is all I see), the coordinates would be as follows.
(2,-4)
Hope this helps.
. Given the significance level 0.025, the F-value for the degrees of freedom, df = (7,3) is A. 5.89. B. 14.62. C. 8.45. D. 27.67.
Answers
Answer:
B. 14.62
Step-by-step explanation:
Given that
The significance level is 0.025
We need to find out the degree of freedom df(7,3)
So for this please see the attached table
And, the calculation is given below;
As we can see that f-value = 14.62
df1=7
and
df2=3
Therefore the second option is correct
The same is relevant and considered
Please look at the image
Answers
Answer:
m<E = 60 deg
Step-by-step explanation:
The sum of the measures of the angles of a triangle is 180 deg.
m<E + m<F + m<G = 180
m<E + 75 + 45 = 180
m<E + 120 = 180
m<E = 60
Answer:
60 is the ans
Step-by-step explanation:
let angle E be x
then add all sides of triangle and subtract it from 180 you will get 60 degree as ans
a) Area:
b) Perimeter:
13 cm
add speaker notes
Answers
Answer:
area: 169
perimeter: 52
Find the product of (√2 + √5) (√7 - √5)
PLs help( a guy answered to this image (linked below) said I have to find that to find the answer) pls solve for me! THx :))))
Answers
Answer:
[tex](\sqrt 2 + \sqrt 5)(\sqrt 7 - \sqrt 5) = \sqrt{14} - \sqrt{10} +\sqrt{35} - 5[/tex]
Step-by-step explanation:
Given
[tex](\sqrt 2 + \sqrt 5)(\sqrt 7 - \sqrt 5)[/tex]
Required
Solve
We have:
[tex](\sqrt 2 + \sqrt 5)(\sqrt 7 - \sqrt 5)[/tex]
Open bracket
[tex](\sqrt 2 + \sqrt 5)(\sqrt 7 - \sqrt 5) = \sqrt{2*7} - \sqrt{2*5} +\sqrt{5*7} - \sqrt{5* 5}[/tex]
[tex](\sqrt 2 + \sqrt 5)(\sqrt 7 - \sqrt 5) = \sqrt{14} - \sqrt{10} +\sqrt{35} - 5[/tex]
Please help me with this, its due today
Answers
180-135=45
(x+2)=45
x=43
Answer:
43 is answer
Step-by-step explanation:
65+x+2+70=180
137+x=180
x=43
The apartments in Greg's building are named using the letters A,B,C, and D and the digits 1 through 9. How many apartments are there in Greg's building if each apartment is named by a single letter followed by a single digits
Answers
Answer:
36
Step-by-step explanation:
If there are 9 apartnments for everyletter 9 times the four letters is 36
9 * 4 = 36
The total number of apartments if The apartments in Greg's building are named using the letters A, B, C, and D, and the digits 1 through 9, are 36.
What is the combination?A combination is a choice of items from a group of different items, where the order of the choices is irrelevant.
Given:
The apartments in Greg's building are named using the letters A, B, C, and D and the digits 1 through 9,
Calculate the total number of apartments as shown below,
The number of apartments = total number of digits × total number of letters,
The number of apartments = (1, 2, 3, 4, 5, 6, 7, 8, 9) × (A, B, C, D)
The number of apartments = 9 × 4
The number of apartments = 36
Thus, The total number of apartments is 36.
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5th grade math. correct answer will be marked brainliest
Answers
Answer:
C
Step-by-step explanation:
9
Reduce to its lowest term.
36/100
Answers
Answer:
9/25
Step-by-step explanation:
Find a polynomial f(x) of degree 3 that has the following zeros. -1 (multiplicity 2), 8 Leave your answer in factored form
Answers
Answer:
Ganancia neta de la ciencia y dibujos de los gobernante en la vida de las ciudadanos y ciudadanas que ah implementado la Municipalidad de tu distrito para enfrentar la pandemia de mi corazón del gobierno estás a cargo del gobierno estás a cargo
Step-by-step explanation:
Ganancia neta de la ciencia y dibujos de los gobernante en la vida de las ciudadanos y ciudadanas que ah implementado la Municipalidad de tu distrito para enfrentar la pandemia de mi corazón del gobierno estás a cargoIs the function shown in the graph one-to-one?
Yes, there exists one input for every output.
No, there exists more than one input for at least one output.
Yes, there exists one output for every input.
No, there exists more than one output for at least one input.
Answers
could someone help please ??
Answers
Answer:
i think it's 0.66666666666 or 1.5
Answer:
The quotient is 1.5.
Step-by-step explanation:
If m < JKN = 28° and m < KLM = 103°, find m < JML.
Answers
Answer:
152
Step-by-step explanation:
The required measure of the angle ∠JML is given as 98°.
What is quadrilateral?A quadrilateral is a four-sided shape as given in the picture in question.
Here,
∠KLM = ∠MJK = 103°
∠ JKL = 2×28 = 56°
The required measure of the ∠JML is given as,
A quadrilateral has 4 angles. The sum of its interior angles is 360 degrees. So the sum of the angle of the kite is equal to 360°
∠KLM + ∠ JKL + ∠MJK + ∠ JML = 360
103 + 56 + 103 + ∠JML = 360
∠JML = 98°
Thus, the required measure of the angle ∠JML is given as 98°.
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18) In a class of 50 boys 3/5of the boys got marks above 60 in Mathematics. How
many boys got above 60 marks?
Answers
Answer:
thx for the points mwah
Step-by-step explanation: