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Answers
[tex]\omega[/tex] is a 2020th root of unity, so [tex]\omega^{2020}=1[/tex] and we have a kind of reflection identity of [tex]\omega^{2020-n} = \omega^{-n}[/tex] for [tex]n\in\Bbb Z[/tex].
Let's evaluate the product first. Denote it by [tex]P_k[/tex]. We split the product where the [tex]j=k[/tex] factor would belong, and pull out powers of [tex]\omega^k[/tex].
[tex]\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)[/tex]
Now introduce some factors to "complete" the [tex]b[/tex]-product and have it contain 2019 factors.
[tex]\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)}[/tex]
It's relatively straightforward to show that if [tex]\zeta[/tex] is an [tex]n[/tex]-th root of unity, then
[tex]\displaystyle \sum_{m=1}^{n-1} (1-\zeta^m) = n[/tex]
which gives
[tex]\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)}[/tex]
Shifting the index in the denominator and again using the reflection property eliminates all but one factor.
[tex]\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}}[/tex]
Now evaluate the sum. We can exploit symmetry. Split the sum at the 1010th term, so that
[tex]\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)[/tex]
The middle terms reduces to 1/2. Shifting the index in the second sum, we can condense it to
[tex]\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)[/tex]
Join the fractions.
[tex]\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})[/tex]
The remaining sums are trivial.
[tex]-\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[/tex]
Taking this last result mod 1000, we find the last 3 digits to be 170.
Related Questions
Write 2 x 10 ^-5 in standard notation
Answers
Answer:
0.00,002
Step-by-step explanation:
Since the exponent is negative. The number written in standard form is much smaller than 2. How much smaller? We need to move the decimal 5 places to the left. When we do not see the decimal it is at the end of the number.
We do not see the decimal in 2, so we know it is to the right of the 2. If I move that 5 places to the left, I get. 0.00,002
Three times the sum of a number and two is equal to -9.
Answers
x=-5
98,615 kg of sugar was produced in the year 1993.This was 75069 kg more than the quantity of sugar
produced in 1992.What was the production in 1992 ?
Answers
173,684 was the production in 1992 in a linear equation .
What in mathematics is a linear equation?
There are only one or two variables in a linear equation. No variable can be multiplied by a number larger than one or used as the denominator of a fraction in a linear equation. All of the points fall on the same line when you identify the values that together make a linear equation true and plot those values on a coordinate grid.Ax+By=C is the usual form for two-variable linear equations. A standard form linear equation is, for instance, 2x+3y=5. When an equation is given in this format, finding both intercepts is rather simple (x and y).the production in 1992 = 75069 × 1 + 98,615
= 75069 + 98,615
= 173,684
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What is the standard unit of volume? What can be used to measure volume?
Answers
Answer:
The answer is Cubic Units (Thank me later)
Step-by-step explanation:
What the answer to (-9.5)(-2)
Answers
Answer:
19
Step-by-step explanation:
Solve for the variable. 4/5 = 20/?
A. x = 5
B. x = 16
C. x = 20
D. x = 25
Answers
Answer:
Its 25
Step-by-step explanation:
It is 25 as 4x5 is 20 so u have the denominator so u do the same for 5 , 5x5 is 25 so 4/5=20/25
describe the translation of the point to its image
Answers
Answer:
13 to the left and 13 up
Step-by-step explanation:
molly is making small bags of sweets for a childrens party she buys seven big bags of sweets each bag contains 47 sweets. how many sweets does molly have?
she wants to give each child 20 sweets in their bag, how many full bags can she make
Answers
16 full bags
Explanation:
7bags x 47sweets per bag = 329sweets
329sweets / 20sweets = 16 FULL bags
I will give brainliest to who ever helps
Answers
Answer:
Use Pythagorean theorem. [tex]\sqrt{52}[/tex]
Step-by-step explanation:
Tony plays running back for his football team. During a football game, Tony gained six yards on the first play. Then he lost eight yards on the second play.
Answers
Um... -2 net yards. Also, Tony should reconsider his position if he's losing 8 yards on one play.
evaluate the expression 4k^{5} -3k^{3} +k^{2} -k for k=2
Answers
Answer: 106
Step-by-step explanation:
To solve this problem, simply substitute k in the equation for 2 and evaluate:
[tex]4(2)^5 - 3(2)^3 +(2)^2 -2 = 106[/tex]
The graph of the function f is shown. The domain of f is [0, 50]. What is the range of g(x) = 4f(x)?
The graphs with the X-coordinate marks 0, 10, 20, 30, 40 and 50. The Y-coordinate mark 0, 0.5, 1, and 1.5. There is an absolute value function with vertex at (25, 1.75). It starts at origin and ends at (50, 0).
A. [0, 50]
B. [0, 70]
C. [0, 1.75]
D. [0, 7]
Answers
Based on the calculations using the given graph, the range of g(x) is equal to: D. [0, 7].
How to identify the range and domain of this graph?The vertical extent of a graph represents all range values and they are always read and written from smaller to larger numerical values, and from the bottom of the graph to the top.
Similarly, the horizontal extent of a graph represents all domain values and they are also read and written from smaller to larger numerical values, and from the left of the graph to the right.
By critically observing the graph of f(x) shown, we can infer and logically deduce the following:
Domain = (0, 50) i.e the domain is all real numbers.
Range = (0, 1.75).
What is the range of g(x) = 4f(x)?At the minimum f(x) = 0, we have:
g(x) = 4 × 0
g(x) = 0.
At the maximum f(x) = 1.75, we have:
g(x) = 4 × 1.75
g(x) = 7.
Therefore, the range of g(x) is equal to [0, 7].
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Anita, Bhanu and Cathy were born on the same year. Anita's birthday is 3 days before Bhanu's birthday and 2 days after Cathy's birthday. Bhanu was born on 2nd March 2008. On what day was Cathy born?
a. 25th February 2008
b. 26th February 2008
c. 28th February 2008
d. 4th March 2008
Will mark the best answer with explanation as Brainliest
Answers
Jayme submitted the following work when asked to convert 34/5 to a mixed number
Answers
Answer:
jayme submitted the following work when asked to convert 34/5 into mixed fractions
6 4/
Step-by-step explanation:
Answer:
6 4/5
Step-by-step explanation:
A charge of $8.99 is automatically deducted from Ari's bank account each
month for a live TV streaming service. What is the change in Ari's bank
account after 6 months for this service? Lesson 1-3 7.NSO.2.2, 7.NSO.
Answers
Answer:
53.94
Step-by-step explanation:
Multiply 8.99 x 6 to get the answer $53.94
Francis, Georgie, and Harrieta go to the library to study every few days. Francis goes to the library every eighth day, Georgie goes every 18th day, and Harrieta goes every ninth day. In how many days will Francis, Georgie, and Harrieta all be at the library together?
Answers
Based on the days that Francis, Georgie, and Harrieta go to the library to study, the number of days till they will all be at the library together is 72 days.
How to find the day they'll be at the library together?The day that they'll be at the library together can be found by checking for the least common multiple of the days that Francis, Georgie, and Harrieta go to the library to study.
The days that they go to study are every 8, 18th, and 9th day.
You therefore need to find the least common multiple of 8, 9, and 18.
This number is 72.
This means that the number of days till Francis, Georgie, and Harrieta go to the library to study together will be in 72 days.
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Is this a nonlinear or linear?
Answers
The given data represent linear equation
A linear equation is represented by straight line. An equation between two variables that gives a straight line when plotted on a graph is called a linear equation.
as given in this data we can see that the value of y is uniformly increasing by 25 when we increase x by 1. and this pattern follows the whole data.
So, when we represent this data on x and y graph, we will obtain a straight line graph
Final answer, the given data will represent a linear equation graph
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Is the y-axis on the vertical or horizontal line?
Answers
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Find an ordered pair (x, y) that is a solution to the equation 6x+y=7
Answers
6(1)+1=7
Hope this helps!!
A local gym is running a promotional offer of 40% off their annual gym memberships.
The annual membership costs £340.
How much discount would you get off?
What is the cost of the membership after the discount?
Answers
The cost of the membership after a discount of $ 136 will be $ 204.
We are given that:
a local gym is giving a discount for the promotional offer it has to make.
Discount percentage = 40 %
Cost of the annual membership = $ 340
Discount = 40 % of $ 340
Discount = 40 / 100 × $ 340
Discount = 0.4 × $ 340
Discount = $ 136
So, the cost of annual membership after discount will be:
Cost = $ 340 - $ 136
Cost = $ 204
Therefore, the cost of the membership after a discount of $ 136 will be $ 204.
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he radius of an iron atom has been calculated to be about 0.00000000014 m. What is this length in scientific notation?
Answers
Answer: 1.4 × 10^-10 m
Calculator
Concert R charges an admission fee of $10.00 and $6.25 per souvenir brochure. Concert S charges an admission fee of $13.00 and $5.75 per souvenir brochure.
How many souvenir brochures must be purchased in order for the total cost at Concert R and Concert S to be the same?
6 souvenir brochure
10 souvenir brochure
12 souvenir brochure
18 souvenir brochure
Answers
The number of souvenir brochures that must be purchased in order for the total cost at Concert R and Concert S to be the same is 6 souvenir brochure
When does the number of souvenir brochures for both concerts become the same?
In order for the total cost of souvenir brochures to be the same in for both concerts, their total costs must be the same
Concert R:
let x be the number of souvenir brochures for both concerts
total cost=10.00+6.25x
Concert S:
let x be the number of souvenir brochures for both concerts
total cost=13.00+5.75x
10.00+6.25x=13.00+5.75x
6.25x-5.75x=13.00-10.00
0.50x=3.00
x=3.00/0.50
x=6
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How can you tell if a relation has symmetry?
Answers
Answer:
they r two angles day sum up to 180°
but de complementary angles sum up to 90°
Step-by-step explanation:
i hope it helped ;)
Ching Weng's budget and expenses for personal items for January are
below. How much more or less did she spend than she budgeted?
Newspapers, gifts
A$39.75 less
B$39.75 more
C $44.75 more
D $44.75 less
Answers
Ching Weng spend 39.75 less then she budgeted on the personal items for the month of January.
We have Ching Weng's budget and expenses for personal items.
We have to determine how much more or less did she spend than she budgeted.
What is budget ?A budget is an estimation of revenue and expenses over a specified future period of time.
According to the question -
The total amount kept by Ching Weng as budget = $95 + $15 + $20 = $130
The total expenses of Ching Weng = $60.75 + $7.85 + $21.65 = $90.25
Since, expenses are less than budget, she spend less than she budgeted. Thee amount saved by her is =
Total budget - total expenses = 130 - 90.25 = $39.75
Hence, Option A is the correct alternative.
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Solve the equation 8 + dw = m for the variable w.
Answers
Answer:
hi there is no dear sir please
Answer:
w= (m-8)/d
Step-by-step explanation:
8+dw=m
dw=m-8
w= (m-8)/d
point(s) possible
Acme Movers charges $210 plus $20 per hour to move household goods across town. Hank's Movers charges $55
per hour. For what lengths of time does it cost less to hire Hank's Movers?
Answers
For less than 6 hours, Hank's Movers charge less.
For what lengths of time does it cost less to hire Hank's Movers?Acme movers charge $210 plus $20 per hour, so if they work for x hours they charge:
A(x) = $210 + $20*x
Hank's movers only charge $55 per hour, so here the linear equation is:
H(x) = $55*x
They charge the same amount when:
A(x) = H(x)
$210 + $20*x = $55*x
Now we can solve that to get:
$210 + $20*x = $55*x
$210 = $55*x - $20*x = $35*x
$210/$35 = x = 6
So, for 6 hours, both companies charge the same amount.
For less than 6 hours, Hank's Movers charge less.
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The point b is the midpoint of ac find the location of a
Answers
Answer:
A = -23
Step-by-step explanation:
If B is directly between A and C, and C is 9 points above B, then A must be 9 points below B.
-14 - 9 = -23
AC = 3x + 3
AB= -1 + 2x
B is between A and C. Find x
None of the other answers are correct
O-8
07
O 12
0 1
BC = 11
Answers
The value of the x is 7 ; By considering the linesegment AC of algebraic length 3x+3,AB of algebraic length -1+2x and BC of length 11 units.
Given, A linesegment AC of algebraic length 3x+3,AB of algebraic length -1+2x and BC of length 11 units, B is the midpoint of the linesegment AC and asked to find the value of x
AC=AB+BC {as B is the midpoint of the linesegment AC}
⇒3x+3=-1+2x+11
⇒x=7
Therefore,The value of the x is 7 ; By considering the linesegment AC of algebraic length 3x+3,AB of algebraic length -1+2x and BC of length 11 units.
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PLEASE HELP!!!!! PLEASE!!!!
I need all of parts 1, 2, and 3
Please don't do number 6 :)
Answers
Part I
The domain of the graph is represented by 0 ≤ x ≤ 12In terms of the problem the domain means the time in hours taken to notice change in temperature in degrees Celsius.Part II
The range of the graph is represented by -2 ≤ x ≤ 4In terms of the problem the range means the temperature in degrees Celsius noted during the time spent, the time is in hoursPart III
The function is decreasing from 0 to 3 hours, this is the point in the x axis that is mathematically represented as 0 ≤ x ≤ 3.This decreasing interval means that it is getting cold, since the temperature is dropping and temperature drop implies getting cold.What temperature?This a term that represents the amount of how hot or cold a substance is the unit used to measure temperature as regards the question is degrees Celsius.
The range of the graph is temperature while the domain is the time
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*Insert cricket noises
cause no one is anwering my questions anymore... #lonely
Answers
Answer:
1.01 × [tex]10^{-5}[/tex]
2.7 × [tex]10^{-5}[/tex]
1.01 × [tex]10^{-4}[/tex]
2.7 × [tex]10^{5}[/tex]
Step-by-step explanation:
The negative exponent automatically makes it smaller than the positive exponent. The more negative the exponent is, the less the value it will be. However, since 1.01 < 2.7, then it is less.
All work and answers are provided in the attached screenshot! :)