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Answers
\(\omega\) is a 2020th root of unity, so \(\omega^{2020}=1\) and we have a kind of reflection identity of \(\omega^{2020-n} = \omega^{-n}\) for \(n\in\Bbb Z\).
Let's evaluate the product first. Denote it by \(P_k\). We split the product where the \(j=k\) factor would belong, and pull out powers of \(\omega^k\).
\(\displaystyle P_k = \prod_{j=1, j\neq k} (\omega^k - \omega^j) \\\\ ~~~~ = \prod_{a=1}^{k-1} (\omega^k - \omega^a) \prod_{b=k+1}^{2019} (\omega^k - \omega^b) \\\\ ~~~~ = (\omega^k)^{k-1} \prod_{a=1}^{k-1} (1-\omega^{a-k}) \cdot (\omega^k)^{2019-k} \prod_{b=k+1}^{2019} (1 - \omega^{b-k}) \\\\ ~~~~ = (\omega^k)^{2018} \prod_{a=1}^{k-1} (1 - \omega^{-a}) \prod_{b=1}^{2019-k} (1 - \omega^b) \\\\ ~~~~ = \omega^{-2k} \prod_{a=1}^{k-1} (1-\omega^{-a}) \prod_{b=1}^{2019-k} (1-\omega^b)\)
Now introduce some factors to "complete" the \(b\)-product and have it contain 2019 factors.
\(\displaystyle P_k = \omega^{-2k} \prod_{a=1}^{k-1} (1-\omega^{-a}) \frac{\displaystyle \prod_{b=1}^{2019} (1 - \omega^b)}{\displaystyle \prod_{b=2020-k}^{2019} (1 - \omega^b)}\)
It's relatively straightforward to show that if \(\zeta\) is an \(n\)-th root of unity, then
\(\displaystyle \sum_{m=1}^{n-1} (1-\zeta^m) = n\)
which gives
\(\displaystyle P_k = 2020 \omega^{-2k} \frac{\displaystyle \prod_{a=1}^{k-1} (1-\omega^{-a})}{\displaystyle \prod_{b=2020-k}^{2019} (1 - \omega^b)}\)
Shifting the index in the denominator and again using the reflection property eliminates all but one factor.
\(\displaystyle P_k = 2020 \omega^{-2k} \frac{\displaystyle \prod_{a=1}^{k-1} (1-\omega^{-a})}{\displaystyle \prod_{b=1}^k (1 - \omega^{-b})} \\\\ ~~~~ = \frac{2020 \omega^{-2k}}{1 - \omega^{-k}}\)
Now evaluate the sum. We can exploit symmetry. Split the sum at the 1010th term, so that
\(\displaystyle - \sum_{k=1}^{2019} P_k = -2020 \left(\sum_{k=1}^{1009} \frac{\omega^{-2k}}{1 - \omega^{-k}} + \frac{\omega^{-2020}}{1-\omega^{-1010}} + \sum_{k=1011}^{2019} \frac{\omega^{-2k}}{1-\omega^{-k}}\right)\)
The middle terms reduces to 1/2. Shifting the index in the second sum, we can condense it to
\(\displaystyle -\sum_{k=1}^{2019} P_k = -2020 \left(\frac12 + \sum_{k=1}^{1009} \left(\frac{\omega^{-2k}}{1-\omega^{-k}} + \frac{\omega^k}{1-\omega^k}\right)\right)\)
Join the fractions.
\(\displaystyle \frac{\omega^{-2k}}{1-\omega^{-k}} + \frac{\omega^k}{1-\omega^k} = 1 - \frac{2-\omega^{2k}-\omega^{-2k}}{2-\omega^k-\omega^{-k}} = -(1+\omega^k + \omega^{-k})\)
The remaining sums are trivial.
\(-\displaystyle \sum_{k=1}^{2019} P_k = -2020 \left(\frac12 - 1009 - \frac{1-\omega^{1010}}{1-\omega} - \frac{1-(-\omega)^{1010}}{1+\omega}\right) \\\\ ~~~~ = 2020\cdot1009 - 1010 \\\\ ~~~~ = (2000+20)(1000+9) - 1000 - 10 \\\\ ~~~~ = 2\cdot1000^2 + 37\cdot1000 + 170\)
Taking this last result mod 1000, we find the last 3 digits to be 170.
Related Questions
A painter leans an 18 foot ladder against the house so that the base of the ladder is 4 feet away from the wall. Approximately how far up the side of the house does the ladder reach?
Answers
Answer: 17.5ft
Step-by-step explanation:
Use Pythagorean theorem
A^2 + B^2 = c^2 where c is the hypotenuse
The ladder leaning against the wall is the hypotenuse.
4^2 + b^2 = 18^2
16 + b^2 = 324
324-16 = b^2
308 = b^2
B = 17.5ft
Help me with this please
Answers
Answer:
178 maybe tell its right or wronv
Solve for r 1+3r>10 please I need this solved aspap
Answers
The solution to the inequality 1 + 3r > 10 is r > 3.
This means that any value of r greater than 3 will satisfy the inequality.
To solve the inequality 1 + 3r > 10, we need to isolate the variable r on one side of the inequality sign.
Let's begin by subtracting 1 from both sides of the inequality:
1 + 3r - 1 > 10 - 1
This simplifies to:
3r > 9
Next, we can divide both sides of the inequality by 3 to solve for r:
(3r)/3 > 9/3
This simplifies to:
r > 3
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pls help me with this problem :)
The areas of two similar triangles are 9 cm² and 49 cm². One of the sides of the first
triangle is 2 cm. What is the length of the corresponding side of the other triangle?
Answers
Answer:
I believe it should be rounded to 11, if not decimal would be 10.88
Step-by-step explanation:
49 divided by 9 equals 5.44 which is the amount times the 49cm^2 triangle is compared to the 9cm^2. So then the side is 2cm, multiply by the amount times bigger the other triangle is which would be 10.88. Hopefully I'm not wrong:)
Answer:
The answer is 4 and 2/3 cm²
Step-by-step explanation:
Take the square root of 49 and 9 and you'll get 7 and 3 since the scale factor is the large area over the small area.
The large area would be √49 cm²
The small area would be √9 cm²
7 x cm
------ = ----------, cross multiply
3 2 cm
7 × 2 = 3 × x
14 = 3x
x = 14/3, x = 4 and 2/3
Which equation can be used to find the volume of this solid?
V = 3 times 5 times 4
V = 3 + 5 + 4
V = 5 times 4
V = 3 times 5
Answers
Answer:
You would use V=3 times 5 times 4
First you would multipy 5 times 4 then when you get that answer you would multiply the 3.
EASY MATH POINTS!
Answer from the screenshot.
Answers
Answer:
That answer is -3
Step-by-step explanation:
The way you do this is Rise/Run
so you would do y2-y1/x2-x1
In this case it's 10-1/-4-(-1)
Simplified is 9/-3 which is -3
hope this helped
Joseph is 33 years old. Five years ago, He was twice as old as Ann. How old will Ann be in 5 years time?
Answers
Answer:
19 years oldStep-by-step explanation:
\(Joseph = 33 \:years \:old\\ \\Let \: ann's \:age be x\\\\33-5 = 2x\\\\28 = 2x\\\\Divide \:both \:sides \:of \:the \:equation \: by \:2\\\\\frac{2x}{2} = \frac{28}{2} \\\\x = 14\\\\Ann's \:present \:age \:= 14\\\\In \:5 \:years \:time ; \\\\14+5 = 19\\\)
Miss Lianto mowed 2 7 of her lawn. Her son mowed 1 3 of it. Who mowed most of the lawn? How much of the lawn still needs to be mowed?
HURRY!!!!!!!!!!!!!!!!!!!!!!!!!
Answers
8/21 of the lawn still needs to be mowed.
To compare fractions, we need a common denominator. The least common multiple of 7 and 3 is 21.
Miss Lianto's fraction: (2/7) x (3/3) = 6/21
Son's fraction: (1/3) x (7/7) = 7/21
To calculate how much of the lawn still needs to be mowed, we can subtract the combined fractions mowed from the total:
Combined fractions mowed: 6/21 + 7/21 = 13/21
So, Remaining fraction: 1 - 13/21 = 8/21
Therefore, 8/21 of the lawn still needs to be mowed.
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Edwin sells jars of jam for $1.90 each. Determine how many jars of jam Edwin needs to sell to break even if the variable cost per jar is $1.10 and fixed expenses are $35,700.00 per year.
Answers
Edwin needs to sell 44,625 jars of jam to break even.
To determine how many jars of jam Edwin needs to sell to break even, we'll calculate the breakeven point using the following formula:
Breakeven Point = Fixed Expenses / (Selling Price per Unit - Variable Cost per Unit)
Given information:
Selling Price per Unit (SP) = $1.90
Variable Cost per Unit (VC) = $1.10
Fixed Expenses = $35,700.00 per year
Plugging in the values into the formula:
Breakeven Point = $35,700 / ($1.90 - $1.10)
Breakeven Point = $35,700 / $0.80
Breakeven Point = 44,625 jars
Therefore, Edwin needs to sell 44,625 jars of jam to break even.
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A pet store has 10 puppies, including 4 poodles, 3 terriers and 3 retrievers. If Rebecca and Aaron in that order each select one puppy at random with replacement (they may both select the same one) find the probability that they both select a poodle.
Answers
Help me please. I kinda need it right now.!
Answers
Step-by-step explanation:
Given
\(5y - 4 = 126(corresponding \: angles) \\ 5y = 126 + 4 \\ 5y = 130 \\ y = 130 \div 5 \\ = 26\)
Also given
\(126 + (6z + 6) = 180(angles \: on \: a \: straight \: line) \\ 132 + 6z = 180 \\ 6z = 18 0 - 132 \\ 6z = 48 \\ z = 48 \div 6 \\ = 8\)
Lastly,
\(x = 180 - 126 \\ = 54(angles \: on \: the \: same \: straight \: line \: and \: corresponding \: angles)\)
Answer:
x = 126
y = 26
z = 8
Step-by-step explanation:
6z + 6 + 126 = 180
6z + 132 = 180
6z = 180 - 132
6z = 48
z = 8
6z + 6 + 5y - 4 = 180
6(8) + 6 + 5y - 4 = 180
48 + 2 + 5y = 180
50 + 5y = 180
5y = 180 - 50
5y = 130
y = 26
x = 126 (Because of alternate exterior angles thm)
Let A and B be n-by-n matrices in R^n, and let c be a real number. Which of the following statements about trace is not necessarily true? a. tr(A+B) = tr(A) + tr(B) b. tr(AB) = tr(BA) c. tr(AB) = tr(A)tr(B) d. tr(CA) = ctr(A)
Answers
Answer:
c. tr(AB) = tr(A)tr(B)
Step-by-step explanation:
The trace of a matrix is only valid for a square matrix, that is a n by n matrix. The trace of a matrix is the sum of all its diagonal elements. The following properties of trace holds for a matrix A and B with size n by n and a real number c.
i) The trace sum of two matrix is equal to the sum of their individual traces. That is:
tr(A + B) = tr(A) + tr(B)
ii) The trace of the product of a scalar and a matrix is the same as the product of the scalar and the trace of the product, that is:
tr(cA) = ctr(A)
iii) The trace of a transpose of a matrix is equal to the trace of the matrix, that is:
\(tr(A^T)=tr(A)\)
iv) The trace of a product of matrix is given as:
tr(AB) = tr(BA)
Answer:
c. tr(AB) = tr(A)tr(B)
(b) Find the sum of first twenty terms of the following arithmetic progressions A.P
(i) -4,-1, 2,...
(ii) 2, 3, 5,...
(iii) -2, -5, -8, -11,...
Answers
Answer:
-4*20+3*190=490
2*20+190=230
-2*20-3*190=-610
what is 67% of 89.5?
Answers
Answer:
59.96
Step-by-step explanation:
when 67% if 89.5 is expressed in numbers,
\(\frac{67}{100}*89.5\\ \frac{5996.5}{100}\\ =59.96\)
Claire's mother is 4 years more than twice claire's age the sum of their ages is 58 claire's age is 16,18, or 40 years and her mothers age is 18,40, or 42 years?
Answers
Answer: Claire is 18 and her mother is 40
Step-by-step explanation:
Read the following prompt and type your response in the space provided.
Describe the relationship between the probability of an event and its complement.
If the probability of an event is 0.95, what is the probability of its complement?
Answers
The probability of the complement of the event is 0.05.
The complement of an event is the probability of that event not occurring. The relationship between the probability of an event and its complement is that they always add up to 1.
Therefore, if the probability of an event occurring is p, then the probability of its complement not occurring is 1-p.
In the case where the probability of an event is 0.95, the probability of its complement not occurring would be 1-0.95 = 0.05. This means that the probability of the complement of the event is 0.05.
This concept is important in probability theory and is used to calculate the probabilities of events and their complements. It allows us to consider all possible outcomes of a given situation and calculate the likelihood of each of them.
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1. What is the surface area of the polyhedron? Explain your reasoning.
Answers
Answer:
24
Step-by-step explanation:
i counted the square me good at counting
How many 20kobo make up #20
Answers
Answer:
100
Step-by-step explanation:
#20 naira - 20 * 100
= 2000kobo
2000/20
100
QED✅✅
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Solve 2,341 ÷ 68 = ________
32 r 42
34 r 29
34 r 42
32 r 29
Answers
Answer:
it's third one
Step-by-step explanation:
make it braintliest please
Hank made payments of $219 per month at the end of each month for 30 years to purchase a piece of property. He promptly sold it for $195,258. What interest rate, compounded monthly, would he need to earn on an ordinary annuity for a comparable rate of return?
Answers
To achieve a comparable rate of return, Hank would need to earn an interest rate of approximately 0.86% per month, compounded monthly on his ordinary annuity.
To find the interest rate, compounded monthly, that Hank would need to earn on an ordinary annuity for a comparable rate of return, we can use the present value formula for an ordinary annuity.
First, let's calculate the present value of Hank's payments. He made payments of $219 per month for 30 years, so the total payments amount to $219 * 12 * 30 = $78840.
Now, we need to find the interest rate that would make this present value equal to the selling price of the property, which is $195,258.
Using the formula for the present value of an ordinary annuity, we have:
PV = P * (1 - (1+r)\(^{(-n)})\)/r,
where PV is the present value, P is the payment per period, r is the interest rate per period, and n is the number of periods.
Plugging in the values we have, we get:
$78840 = $219 * (1 - (1+r)\({(-360)}\))/r.
Solving this equation for r, we find that Hank would need to earn an interest rate of approximately 0.86% per month, compounded monthly, in order to have a comparable rate of return.
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A rectangular field is four times as long as it is wide. If the perimeter of the field is 560 yards, what are the field’s dimensions?
Answers
Answer:
Width=56 yards
length=224 yards
Step-by-step explanation:
let:
Width=x
length= x*4= 4x
perimeter = 2*x + 2*4x
working
\((x \times 2) + (4x \times 2) = 560\)
\(2x + 8x = 560\)
\(10x = 560\)
\( \frac{10x}{10} = \frac{560}{10} \)
\(x = 56\)
\(4x = 56 \times 4 = 224\)
width =56 yards
length = 224 yards
Polynomials
(7n^4-14 - 5n^3) (7-8n^3 + 11n^4)
Answers
Answer:
\(\boxed{22n^8-16n^7-140n^4+112n^3-98}\)
Step-by-step explanation:
\((7n^4-14 - 5n^3) (7-8n^3 + 11n^4)\)
According to the distributive property, each term multiplies each of the terms in the other equation. Therefore, we can do the following:
\(\begin{aligned}\Rightarrow &7n^4 (7-8n^3 + 11n^4)= 49n^4-56n^3 n^4+77n^4n^4 \\&=49n^4-56n^7+77n^8 \end{aligned}\)
\(\begin{aligned}\Rightarrow &-14 (7-8n^3 + 11n^4)\\& = -98+112n^3-154n^4 \end{aligned}\)
\(\begin{aligned}\Rightarrow &-5n^4 (7-8n^3 + 11n^4)= -35n^4+40n^3n^4-55n^4n^4 \\&=-35n^4+40n^7-55n^8 \end{aligned}\)
Next, we combine all the terms of the plynomial equation:
\(49n^4-56^7+77n^8-98+112n^3-154n^4-35n^4+40n^7-55n^8\)
common factor:
\(22n^8-16n^7-140n^4+112n^3-98\)
By this, we have solved the exercise.
\(\text{-B$\mathfrak{randon}$VN}\)
Suppose a life insurance company sells a
$280,000
1-year term life insurance policy to a
20-year-old
female for
$270.
According to the National Vital Statistics Report, 58(21), the probability that the female survives the year is
0.999544.
Compute and interpret the expected value of this policy to the insurance company.
Answers
Answer:
$142.32, profit on sale of the policy
Step-by-step explanation:
You want to know the expected value of a $280,000 life insurance policy sold for $270, if the probability the insured will live for the year is 0.999544.
CostThe insurance company expects to have to pay the $280,000 death benefit for 0.000456 of the policies issued. That means their expected payout on any one policy is ...
0.000456 × $280,000 = $127.68
ProfitThe company gets a premium of $270 for the policy, so the expected value of the policy to the company is ...
$270 -127.68 = $142.32
The expected value of the policy to the company is $142.32.
This represents its profit from sale of the policy.
__
Additional comment
Of course, the company has expenses related to the policy, perhaps including a commission to the agent selling it, and expenses related to handling claims. That is to say that not all of the difference between the premium and the average death benefit is actually profit. It is what might be called "contribution margin."
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Find the perimeter of a rectangular garden that has a width of 4x−6 and a length of 2x+4.
Answers
Answer:
perimwter = 2(4x-6 + 2x+4) = 2 (6x-2) = 12x-4
Which of the following is a step in simplifying the expression x multiplied by y to the power of 3 over x to the power of negative 4 multiplied by y to the power of 4, the whole to the power of negative 2.?
Answers
Answer:
-3
Step-by-step explanation:
There are 3 red jelly beans, 5 blue jelly beans, 2 orange jelly beans, and and 5 yellow jelly beans in a bag. Another bag has 1 pink jelly bean, 7 purple jelly beans, and 2 green jelly beans. What is the probability of randomly selecting a blue jelly bean from the first bag and then randomly selecting a green jelly bean from the second bag?
1/15
3/10
7/25
1/4
Answers
The probability of randomly selecting a blue jelly bean from the first bag and then randomly selecting a green jelly bean from the second bag is 1/15 (option a).
Firstly, we need to determine the total number of jelly beans in both bags.
The first bag contains 15 jelly beans (3+5+2+5) and the second bag contains 10 jelly beans (1+7+2).
Therefore, the total number of jelly beans in both bags is 25.
Next, we need to determine the probability of randomly selecting a blue jelly bean from the first bag.
Since there are 5 blue jelly beans out of a total of 15 jelly beans in the first bag, the probability of selecting a blue jelly bean is 5/15 or 1/3.
After selecting a blue jelly bean from the first bag, we move on to the second bag to select a green jelly bean.
Since there are 2 green jelly beans out of a total of 10 jelly beans in the second bag, the probability of selecting a green jelly bean is 2/10 or 1/5.
To determine the probability of both events occurring, we use the multiplication rule of probability.
Therefore, the probability of randomly selecting a blue jelly bean from the first bag and then randomly selecting a green jelly bean from the second bag is (1/3) x (1/5) = 1/15.
Hence, the answer is option (a) 1/15.
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Can someone tell me what they're asking?
Answers
Which the answer is x^20
Xx
Which equation models the rational function shown in
the graph?
2(x+2)
X-2
O f(x) =
O f(x) = x-2
x+2
2(x-2)
x+2
Of(x)=.
O f(x) = x+2
X-2
Answers
At x=2, the function in the graph is approaches to infinity. This is satisfied by equation 1 that models the rational function, hence option 1 is the correct answer.
What is equation?A mathematical definition of an equation is a claim that two expressions are equal when they are joined by the equals sign ("="). For illustration, 2x - 5 = 13.
Two expressions are combined in an equation using an equal symbol ("="). The "left-hand side" and "right-hand side" of the equation are the two expressions on either side of the equals sign. Typically, we consider an equation's right side to be zero.
From given graph we can see that
At x=2, the function in the graph is approaches to infinity.
It means function is not defined at x=2
We know that a rational function is undefined when denominator is equal to zero.
For equation 1:
f(x)= 2(x+2)/(x-2)
Equating the denominator:
x-2=0
x=2
It means function is not defined at x=2
The y-intercept is: y= -2
The function is passing through the point (1,-6).
When we substitute x=1
Hence, option 1 is true.
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Is (7,-6) a solution to this system of equations?
y = 1/7x+ 7
X = 7
yes
no
Answers
If you want to get is
Y=mx+b
Y=1/7x+7
Y=1/7*7+7 Since x=7
Y=1+7 Compute 1/7*7
Y=8
If the food bill is $38.92, then what would the tip be if you want to leave a tip of 15%?
Answers
Answer:
$5.84
Step-by-step explanation: