Partial fractions:
[tex]\displaystyle \frac1{(n^2-1)^2} = \dfrac14 \left(\frac1{n+1} - \frac1{n-1} + \frac1{(n+1)^2} + \frac1{(n-1)^2}\right)[/tex]
Now evaluate the sums. The first two terms form a telescoping sum,
[tex]\displaystyle \sum_{n=2}^\infty \left(\frac1{n+1} - \frac1{n-1}\right) = \sum_{n=1}^\infty \left(\frac1{n+2} - \frac1n\right) \\\\ ~~~~~~~~ = \lim_{N\to\infty} \sum_{n=1}^N \left(\frac1{n+2} - \frac1n\right) \\\\ ~~~~~~~~ = \lim_{N\to\infty} \left(\frac1{N+2} + \frac1{N+1} - 1 - \frac12\right) \\\\ ~~~~~~~~= -\frac32[/tex]
The others are flavors of [tex]\zeta(2)[/tex].
[tex]\displaystyle \sum_{n=2}^\infty \frac1{(n+1)^2} = \sum_{n=3}^\infty \frac1{n^2} \\\\ ~~~~~~~~~~~~~~~~~ = \sum_{n=1}^\infty \frac1{n^2} - 1 - \frac1{2^2} \\\\ ~~~~~~~~~~~~~~~~~ = \frac{\pi^2}6 - \frac54[/tex]
[tex]\displaystyle \sum_{n=2}^\infty \frac1{(n-1)^2} = \sum_{n=1}^\infty \frac1{n^2} \\\\ ~~~~~~~~~~~~~~~~~ = \frac{\pi^2}6[/tex]
It follows that
[tex]\displaystyle \sum_{n=2}^\infty \frac1{(n^2-1)^2} = \frac14 \left(-\frac32 + \left(\frac{\pi^2}6 - \frac54\right) + \frac{\pi^2}6\right) = \boxed{\frac{\pi^2}{12} - \frac{11}{16}}[/tex]
A train is going on a journey. First it travels east for 50 kilometers. Then it turns north and travels for 22 kilometers. Next it goes west for 50 kilometers. Finally, it travels south for a certain number of kilometers. At the end of its journey it is back at its original location. How many kilometers did it travel in the southward direction?
Answer:
Answer would be 22 km at southward direction.
Answer = 22km
Step-by-step explanation:
As it started from the point O.
* Goes 50 to East direction
* Goes 22 to North direction
* Goes 50 to West direction
* And the it travels back to the starting point when it turn to south direction. It traveled 22 km to southward direction.
> Total kilometers the train traveled is 144 kilometers.
I hope you get this right.
write the equation y=-6+5 in function notation using f (x) to denote the function
The equation y = -6x + 5 when rewritten is f(x) = -6x + 5
How to denote the equation as a function?The equation is given as
y = -6 + 5
Rewrite the above equation properly
So, we have
y = -6x + 5
Express y as a function
So, we have
f (x) = -6x + 5
Hence, the equation y = -6x + 5 when rewritten is f(x) = -6x + 5
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One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.
Answer: 32, 46
Step-by-step explanation:
a=number 1. b=number 2
a=b-14
a+b+7=85
b-14+b+7=85
b+b-14+7=85
2b-7=85
2b=92
b=46
a=46-14
a=32
Answer:
Step-by-step explanation:
the equation to find this is
x+(x+14)+7=85
first minus the 7
X+(x+14)+7-7= 85-7
x+(x+14)=78
78÷2 = 39
42+36
The coordinate point (-5, 18) is a point on the graph of f. Write this point in function notation.
f(-5) = 13
f(-5) = 18
f(18) = -3.6
f(18) = -5
The function notation of the coordinate point where the coordinate point is given as (-5, 18) is f(-5) = 18
What are coordinate points?Coordinate points are mathematical statements that are used to represent the location of points on a coordinate plane
How to rewrite the function notation of the coordinate point?The coordinate point is given as
(-5, 18)
The above coordinate point can be represented as
(x, y) = (-5, 18)
The above coordinate point can further be rewritten as
(x, f(x)) = (-5, 18)
So, we have
x = -5
f(x) = 18
Substitute x = -5 in the equation f(x) = 18
f(-5) = 18
Hence, the function notation of the coordinate point where the coordinate point is given as (-5, 18) is f(-5) = 18
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Use the figure for Items 2 and 3. Lines PS and TR intersect at point Q.
Applying the vertical angles theorem and the linear pair theorem, the measure of angle PQV is calculated as: 53°.
How to Apply the Linear Pair Theorem?The linear pair theorem states that when two angles form a linear pair, the sum of both angles of the linear pair would be equal to 180 degrees.
What is the Vertical Angles Theorem?Two angles that are a vertical pair, have a measure that is equal to each other, based on the vertical angles theorem. That is, they are congruent angles.
m<RQS + m<PQT = 180 degrees [linear pair theorem]
Substitute
(2m + 4)° + (3m + 1)° = 180°
Solve of the value of m
2m + 4 + 3m + 1 = 180
Add like terms together
2m + 3m + 4 + 1 = 180
5m + 5 = 180
5m + 5 - 5 = 180 - 5
5m = 175
5m/5 = 175/5
m = 35
m<PQR = 3m + 1 [vertical angles theorem]
Plug in the value of m
m<PQR = 3(35) + 1
m<PQR = 105 + 1
m<PQR = 106°
m<PQV = 1/2(m<PQR)
m<PQV = 1/2(106)
m<PQV = 53°
Thus, applying the vertical angles theorem and the linear pair theorem, the measure of angle PQV is calculated as: 53°.
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Answer:
The answer is 74
Step-by-step explanation:
PLEASE HELP !?!????!?
Answer: Supplementary Angles
Step-by-step explanation:
Supplementary angles are a pair of angles whose sum is 180°. 45° and 135° add up to 180°. Therefore, the answer is supplementary angles.
Maya is using a coordinate plane to create a map of her yard drawn to scale. Each unit
on the coordinate plane represents 1 meter. Her yard forms a quadrilateral with vertices
A(5, -3), B(10, -3), C(12, -11), and D(7, -11). Maya would like to fence in her entire
yard. Fencing costs $200 per meter. How much will it cost Maya to fence in her yard?
Answer:
$5,298.48
Step-by-step explanation:
Given vertices:
A = (5, -3)B = (10, -3)C = (12, -11)D = (7, -11)As points A and B, and points C and D, have the same y-coordinates, AB is parallel to CD.
As there is a difference of 5 units between points A and B, and C and D, AB = CD.
As AB is parallel to CD and AB = CD ⇒ AD = BD and AD is parallel to BD.
Therefore, the quadrilateral ABCD is a parallelogram (see attachment).
To find the perimeter of ABCD, calculate the side lengths of AB and AD using the Distance Formula.
[tex]\boxed{\begin{minipage}{4.7 cm}\underline{Distance Formula}\\\\$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the two endpoints.\end{minipage}}[/tex]
[tex]\begin{aligned}AB & =\sqrt{(x_B-x_A)^2+(y_B-y_A)^2}\\& =\sqrt{(10-5)^2+(-3-(-3))^2}\\& =\sqrt{(5)^2+(0)^2}\\& =\sqrt{25}\\& =5\:\:\sf units\end{aligned}[/tex]
[tex]\begin{aligned}AD & =\sqrt{(x_D-x_A)^2+(y_D-y_A)^2}\\& =\sqrt{(7-5)^2+(-11-(-3))^2}\\& =\sqrt{(2)^2+(-8)^2}\\& =\sqrt{4+64}\\& =\sqrt{68}\:\:\sf units\end{aligned}[/tex]
The perimeter of a two-dimensional shape is the distance all the way around the outside.
[tex]\begin{aligned}\textsf{Perimeter of $ABCD$} & = AB + CD + AD + BC\\& = 2\:AB + 2\:AD\\& = 2 \cdot 5+2 \cdot \sqrt{68}\\& = 10+2 \sqrt{68}\\& = 10+2\sqrt{4 \cdot 17}\\& = 10+2\sqrt{4}\sqrt{17}\\& = 10+2 \cdot2\sqrt{17}\\& = 10+4\sqrt{17}\:\:\sf units\end{aligned}[/tex]
As 1 unit = 1 meter, the perimeter of the fence is (10 + 4√17) m.
If fencing costs $200 per meter:
[tex]\begin{aligned}\sf \implies Cost \:of\:fencing& = 200(10 + 4 \sqrt{17})\\& = 2000+800 \sqrt{17}\\& = 5298.4845\\ & = \$5298.48\end{aligned}[/tex]
Therefore, it will cost Maya $5,298.48 to fence her yard.
The point G(-5, -1) is reflected over the line y= -3. What are the coordinates of the resulting point, G’
The coordinates of the resulting point after the reflection are (-5, -5)
What are the coordinates of the resulting point?
When we do a reflection over a given line, the perpendicular distancee between the line and the point is invariant.
In this case, we want to reflect the point (-5, -1) over the line y = -3
The distance between the y-value of the point and the line is:
D = |-1 - (-3)| = | -1 + 3| = 2
The point is at 2 units (above) from the line, then after the reflection, the point will be 2 units below the line.
So the new y-value is:
y = -3 - D = -3 - 2 = -5
The coordinates of the resulting point are (-5, -5).
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Directions: For the following problems....
(a) determine the correct conclusion or further conclusion based on the given information and diagram. Your
conclusions should only be based on the given information, not what you believe the diagram may imply.
(b) fill in the appropriate statement/reason chart to organize your work. REMINDER: A reason can only be a
definition or definition reversed.
1. Given: LFHEZKHE
HINT: Your conclusion should NOT be the m2FHE =
m2KHE! There is a better conclusion to draw here...
Conclusion: FHE = KHE
ΔFHE = ΔKHE, as the ∠FHE = ∠KHE. When two triangles contain the same two sides and include the same Angle, the remaining sides, Angle, and area are likewise identical, which means they're congruent.
∠FHE ≅ ∠KHE is an Identical part.
L(FE) = L(KE), as two Congruent angles have equal sides
L( FH) = L(KH)
L(EH)=L(EH), Common part in a figure.
Congruent angles are two or more angles that are exactly the same. As a result, the lengths of these angles are identical.
hence, ΔFHE = ΔKHE, as the ∠FHE = ∠KHE. When two triangles contain the same two sides and include the same Angle, the remaining sides, Angle, and area are likewise identical, which means they're congruent.
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Can someone explain this pls
Answer:
Step-by-step explanation:
( g ° f )(x) = ( x + 7 )² = x² + 14x + 49
20 + 3(x² + 14x + 49) = - 34
20 + 3x² + 42x + 147 = - 34
3x² + 42x + 201 = 0 ⇔ x² + 14x + 67 = 0
D = 196 - 268 = - 72 < 0
There is no any roots in set of real numbers.
The values of x in set of complex numbers are:
[tex]x_{12}[/tex] = [tex]\frac{- 14 }{2}[/tex] ± [tex]\frac{6\sqrt{2}i }{2}[/tex]
[tex]x_{12}[/tex] = - 7 ± 3√2 i
The company made a partial payment of $600 cash on the equipment purchased in transaction d.
Answer: can u finish the question
Step-by-step explanation:
What are two different names for the plane shown
in the diagram below?
Answer:
Plane Y and CDB
Step-by-step explanation:
Plane Y and CDB
put these numbers in order least to greatest 56,209 5,609 55,209 562,009
Answer:
5,609
55,209
56,209
562,009
Find the odd one out 1.234,
123.4 hundredths,
1234 thousandths,
12.34,
123 hundredths and 4 thousandths
The decimal odd one out is: 123 hundredths and 4 thousandths.
What are decimals?The accepted method for representing both integer and non-integer numbers is the decimal numeral system. It is the expansion of the Hindu-Arabic numeral system to non-integer values.
Decimal notation is the term used to describe the method of representing numbers in the decimal system.
Given: 1.234 is a decimal.123.4 hundredths, 1234 thousandths and 12.34 are all different forms of the given decimal. But 123 hundredths and 4 thousandths is not.
This is due to the fact that in the other selections, it is written in decimal format or is finished, but in the fourth or final option, it is purposefully not written in decimals. It is hence the outlier.
Hence, The decimal odd one out is: 123 hundredths and 4 thousandths.
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HELP PLEASE!!!!!!! A zookeeper wrote down the number of fish each of the ten dolphins in the aquarium ate per day. His results were as follows: 18, 15, 25, 17, 13, 15, 23, 27, 30, and 27. Find the mean absolute deviation of this data set. What does the mean absolute deviation tell you about the dolphins at the aquarium?
The mean absolute deviation is 5.4.
The mean absolute deviation tells us that the variance of each data set is 5.4.
What is the mean absolute deviation?The mean absolute deviation is a measure of variance that determines the average distance between each data point in the dataset and its mean.
The mean absolute deviation = 1/n ∑ l x - m(x) l
The first step is to determine the mean of the data set: (18 + 15 + 25 + 17 + 13 + 15 + 23 + 27 + 30 + 27) / 10
= 210 / 10
= 21
The second step is to determine the absolute difference between the mean and each data point. Then add the result together.
|(18 - 21) + (15 - 21)+ (25- 21) + (17- 21) + (13- 21) + (15- 21) + (23- 21) + (27- 21) + (30- 21) + (27- 21)| = 54
Now, divide the sum by the total numbers in the data set : 54 / 10 = 5.4
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The length of a rectangle is twice the width. The area of the rectangle is 94 square units. Notice that you can divide the rectangle into two squares with equal area. How can you estimate the side length of each square? Estimate the length and width of the rectangle.
Answer: w=[tex]\sqrt{47}[/tex], l=2[tex]\sqrt{47}[/tex]
Step-by-step explanation:
A=area of rectangle
A=l*w
A=2w*w
94=2w*w
w*w=47
w=[tex]\sqrt{47}[/tex] ==> width of rectangle and side length of square
94=l*w
94=l*[tex]\sqrt{47}[/tex]
l=[tex]\frac{94}{\sqrt{47} }[/tex]
l=[tex]\frac{94\sqrt{47} }{47 }[/tex]
l=2[tex]\sqrt{47}[/tex]
How much it’s 27+ 0.185
Answer:
27.185
Step-by-step explanation:
Best answer is ready to give gift prize?
Step-by-step explanation:
but please where is the question I'm eager to answer and win the pizza
Construct a parallelogram PQRS such that PQ = 8.1 cm, QR = 6.9 cm and the diagonal QS = 10.4 cm. Measure and write down the length of the diagonal PR.
Answer:
10.4 it is the correct answer please mark me as a brainlist
Write an inequality that represents the graph. An inequality that represents the graph is -1 on a number line
a .
The inequality of the number line is x > -1
What is a number line?A number line is a vertical or horizontal line that has endpoints and it is used to represent numbers at a constant interval
From the number line in the question, we can see that:
How to determine the inequality?From the question, we have
Point = -1
Assume the following
The endpoint of the circle on the number line is an open circleThe arrow points to the rightAs a general rule, open circles implies that the number at that endpoint is not included part of the solution set.
This means that the inequality of the number line is x > -1
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ASAP Please help and thank you!
Charles, age 45, would like to determine how much life insurance to purchase using the human life value approach. He assumes his average annual earnings will remain at $40,000. Of this amount, $20,912 is available annually for the support of his family. Charles will generate this income for 24 more years and he believes that 5.37 percent is the appropriate interest (discount) rate. What is Charles' human life value? (Round your answer to 2 decimal places)
Using it's formula, it is found that Charles' human life value is of $26,951.39.
What is the human life value formula?The human life value formula(HLV) is given by the following multiplication:
HLV = Annual Expenses x Number of Years x Interest Rate.
For this problem, we have that:
$20,912 is available annually for the support of his family, hence the annual expenses are of $20,912.Charles will generate this income for 24 more years, hence the number of years is of 24.As stated in the problem, the interest rate is of 5.37%, hence, as a decimal, is of 0.0537.Hence, applying the formula, we have that Charles' human life value is found as follows:
HLV = 20912 x 24 x 0.0537
HLV = $26,951.39.
Charles' human life value is of $26,951.39.
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For the month of December, a checking account has a balance of $500 for 12 days, $1250 for 12 days, $1150 for 3 days, and $600 for 4 days. What is the average balance for December?
Answer:
$866.13
Step-by-step explanation:
to get an average, you add up all the factors then divide them by how many there are. I added 500 12 times to 1250 12 times to 1150 3 times and 600 4 times.
aka (500*12)+(1250*12)+1150+1150+1150+600+600+600
then i divided by 31 because 12+12+3+4=31. giving the average $866.13.
Bryan ran 100 meters in 14.5 seconds. How long did he take to run 5 meters?
Answer: 0.725 seconds
Step-by-step explanation:
5 meters*14.5 seconds/100 meters=
5*14.5/100=
72.5/100=0.725 seconds
2. Find and simplify the product of f and g *
given the functions f (x)
; g(x) = -²
=
x+2
(2 Points)
The product of the functions f(x) and g(x), where the functions f(x) = [tex]x+2[/tex] and g(x) = [tex]-x^{2}[/tex] is
As per the given question statement, we are supposed to find and simplify the product of the functions f(x) and g(x), where the functions f(x) = [tex]x+2[/tex]and g(x) = [tex]-x^{2}[/tex].
Let's assume [tex]m(x)[/tex] be the product of the functions [tex]f(x)[/tex] and [tex]g(x)[/tex]
So, [tex]m(x) = f(x) * g(x)\\m(x)= (x+2)*(-x^{2} )\\m(x)=-x^{3} -2x^{2}[/tex]
Therefore the product of the functions f(x) and g(x), where the functions f(x) = [tex]x+2[/tex]and g(x) = [tex]-x^{2}[/tex] is [tex]m(x)=-x^{3} -2x^{2}[/tex]
This question is a classic example of how to solve functions using different mathematical operations and using the concept of algebraic expressions.
Algebraic Expressions: An algebraic symbol or set of symbols that includes one or more integers, variables, and arithmetic operations.To learn more about algebraic expressions, click on the link given below:
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Evaluate 3x - 7 when x = 4
Answer:
5
Step-by-step explanation:
3x-7
=3×4-7
=12-7
=5..........
Put the following equation of a line into slope-intercept form, simplifying all fractions. 9x+6y=42
Answer:
4
Step-by-step explanation:
9x+6y=42
solution
Balancing Method :
subtract both sides by 6
9x-6y+6y=42-6=
9x=42-6=36
divide both by 6
9 ÷9=36÷9=4
77. What is the circumference of the larger circle
shown above in terms of x?
A. 10 π x
B. 10 π +x
C. 10 π +2x
D. 10π+ 2πx
[tex]\textit{circumference of a circle}\\\\ C=2\pi r ~~ \begin{cases} r=radius\\[-0.5em] \hrulefill\\ r=5+x \end{cases}\implies C=2\pi (5+x)\implies C=10\pi +2\pi x[/tex]
Select all the products that are equivalent to 1/6
A. −3/4⋅(−2/9)⋅1/2
B. −3/2⋅(−2/9)⋅1/2
C. −1/2⋅(−2/3)⋅1/2
D. 1/2⋅(−2/9)⋅1/2
E. 1/2⋅(−5/6)⋅(−2/5)
B, C, and E are the products equivalent to 1/6
We have been given five different products and we have to find out which of them are equivalent to 1/6. So,
(-3/4)*(-2/9)*(1/2) => 6/72 => 1/12(-3/2)*(-2/9)*(1/2) => 6/36 => 1/6(-1/2)*(-2/3)*(1/2) => 2/12 => 1/6(1/2)*(-2/9)*(1/2) => -2/36 => -1/18(1/2)*(-5/6)*(-2/5) => 10/60 => 1/6Here we observe that the product of B, C, and E after solving them is equivalent to 1/6 whereas, the product of A and D is 1/12 and -1/18 respectively.
Hence, We get that the product for the numbers
(-3/2)*(-2/9)*(1/2), (-1/2))*(-2/3)*(1/2), and (1/2)*(-5/6)*(-2/5)Which are Options B, C, and E equivalent to 1/6
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what is the answer to 17^2-286
The answer for the given expression, 17²-286 will be 3.
What is the equation?An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.
It is given that,the expression is,
17²-286
The expression is solved by the arrthmetic operation in which we do the addition of numbers, subtraction, multiplication, and division. It has a basic four operators that is +, -, ×, and ÷.
=17²-286
=289-286
=3
Thus, the answer for the given expression, 17²-286 will be 3.
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Solve for c.
a=11+4b−4c
Answer:
c = -a/4 + b + 11/4
Step-by-step explanation:
Solve for c:
a = -4 c + 4 b + 11
a = -4 c + 4 b + 11 is equivalent to -4 c + 4 b + 11 = a:
-4 c + 4 b + 11 = a
Subtract 4 b + 11 from both sides:
-4 c = a + (-4 b - 11)
Divide both sides by -4:
Answer: c = -a/4 + b + 11/4