It looks like you have to find the value of the sum,
[tex]C = \displaystyle \frac1{21\times22\times23} + \frac1{22\times23\times24} + \cdots + \frac1{200\times201\times202}[/tex]
so that the n-th term in the sum is
[tex]\dfrac1{(21+(n-1))\times(21+n)\times(21+(n+1))} = \dfrac1{(n+20)(n+21)(n+22)}[/tex]
for 1 ≤ n ≤ 180.
We can then write the sum as
[tex]\displaystyle C = \sum_{n=1}^{180} \frac1{(n+20)(n+21)(n+22)}[/tex]
Break up the summand into partial fractions:
[tex]\dfrac1{(n+20)(n+21)(n+22)} = \dfrac a{n+20} + \dfrac b{n+21} + \dfrac c{n+22}[/tex]
Combine the fractions into one with a common denominator and set the numerators equal to one another:
[tex]1 = a(n+21)(n+22) + b(n+20)(n+22) + c(n+20)(n+21)[/tex]
Expand the right side and collect terms with the same power of n :
[tex]1 = a(n^2+43n+462)+b(n^2+42n+440) + c(n^2+41n + 420) \\\\ 1 = (a+b+c)n^2 + (43a+42b+41c)n + 462a+440b+420c[/tex]
Then
a + b + c = 0
43a + 42b + 41c = 0
462a + 440b + 420c = 1
==> a = 1/2, b = -1, c = 1/2
Now our sum is
[tex]\displaystyle C = \sum_{n=1}^{180} \left(\frac1{2(n+20)}-\frac1{n+21}+\frac1{2(n+22)}\right)[/tex]
which is a telescoping sum. If we write out the first and last few terms, we have
C = 1/(2×21) - 1/22 + 1/(2×23)
… … + 1/(2×22) - 1/23 + 1/(2×24)
… … + 1/(2×23) - 1/24 + 1/(2×25)
… … + 1/(2×24) - 1/25 + 1/(2×26)
… … + … - … + …
… … + 1/(2×198) - 1/199 + 1/(2×200)
… … + 1/(2×199) - 1/200 + 1/(2×201)
… … + 1/(2×200) - 1/201 + 1/(2×202)
Notice the diagonal pattern of underlined and bolded terms that add up to zero (e.g. 1/(2×23) - 1/23 + 1/(2×23) = 1/23 - 1/23 = 0). So, like a telescope, the sum collapses down to a simple sum of just six terms,
C = 1/(2×21) - 1/22 + 1/(2×22) + 1/(2×201) - 1/201 + 1/(2×202)
which we simplify further to
C = 1/42 - 1/44 - 1/402 + 1/404
C = 1,115/1,042,118 ≈ 0.00106994
Which of the following is the discriminant of the polynomial below?
X2 +6X+8
A. 8
B. 6
C4
D. 26
Amber had 56 of a book of postage stamps. She used 26 of the book of postage stamps. Which fraction of the book of postage stamps does Amber have now?
An Step-by-step explanation: There are 56 stamps in total, and Amber used 26. So, she used 26/56. We need know which fraction has now. so she have 30/56 (becauase 56-26= 30). And by simplify, she have now 15/28 (if you simplify by 2)
Sam feels that students score better on math tests that take place on Fridays. The school provides Sam with the school math test average, but they cannot separate them by date taken. The school is too large for him to ask every student, so he randomly selects a sample of willing students' test scores after the next Friday's math test. He uses the method of hypothesis testing to compare the Friday scores against the true average of all scores. Frame the correct set of hypotheses for his research. Decide where the rejection region will lie.
Answer:
H0 : μ ≤ x
H1 : μ > x
The hypothesis :
The null hypothesis, H0 : μ ≤ x
The alternative hypothesis, uses the Friday score samples to compare the hypothesis ;
The alternative hypothesis, H1 : μ > 8
For then null, the mean scores is assumed to be the same, and has such mean of Friday's test score takes the value of the population.
Let the population mean value = x
The alternative hypothesis is the claim that the sampled mean gives a greater value than the population mean :
H1 : μ > x
Which ordered pair would form a proportional relationship with the point graphed below?
On a coordinate plane, point (60, negative 20) is plotted.
(–10, 30)
(30, –15)
(–30, 10)
(80, –30)
Answer: its not b or d that leaves a and c its
A AAnswer:
it might be c its def not a for i got it wrong when i chose a
Step-by-step explanation:
dont get mad if im wrong i.d.k if i am right or wrong so im gonna say c?
The formula p=nc-e gives the profit p when a number of items n are each sold at a cost c and expenses e are subtracted. If p = 3750, n = 3000, and e = 900, what is the value of c?
Answer:
1.55
Step-by-step explanation:
p =nc-e
p=3750, n=3000, e=900, c=?
3750 = 3000c - 900
3750 + 900 = 3000c
4650 = 3000c
3000c = 4650
c = 4650/3000
c = 1.55
The temperature of a city B (x) last week was lesser than or equal to - 9°C.
Represent this as an inequality. Does -8°C satisfy this inequality?
O xs-9, No.
O x2-9, No.
XS-9, Yes.
O x2-9, Yes.
Answer:
XS-9, yes#####################
Determine whether the sequence converges or diverges. If it converges, give the limit.
50, 25/2, 25/8, 25/32, ...
Answer:
The sequence converges to 0.
Step-by-step explanation:
Every new number is divided by 4 and will continue to grow smaller until it converges to 0.
if 2/5 of the sweets in a bag is 60, how
many sweets would represent 1/3
Answer:
50 Candy
Step-by-step explanation:
We know that 2/5 of the bag is 60 candy. By taking this information we can conclude that 1/5 of the bag is 30 candy. We know that 1/5 of the bag is 30, so 30 times 5 is 150 candy.
Now that we have found out that there is 150 candy total we can know divde 150 by 3 to find out how much candy is in 1/3 of the bag.
150/3 = 50 Candy
1/3 of the bag represent 50 candy.
Write an equation in point-slope form for the line through the given point with the given slope. (Need ASAP help)
Is it a, b, c or d?
Mary spends her Saturday mornings buying groceries, helping her children with homework, cooking, and
working on her weekend assignment. She does not always do these tasks in the same order. In how many
possible ways can she do her tasks?
Answer:
I think in 2 or 3 process. But I am not sure
If the sum of seven odd consecutive numbers is 231, find the middle number.
Answer:
33
Step-by-step explanation:
(1+x) + (3+x) + (5+x) + (7+x) + (9+x) + (11+x) + (13+x) = 231
7x + 49 = 231
7x = 231 - 49
7x = 182
x = 182/7
x = 26
To find the middle number;
7 + x
= 7+26
= 33
There are j cars at the Rockford dealership and k cars at the Robinson dealership. The Robinson dealership has 6 more than three times as many cars as the Crockford dealership. What is the value of k in terms of j?
Answer:
[tex]k=3j+6[/tex]
Step-by-step explanation:
We can setup an equation in terms of k and j:
[tex]k=3j+6[/tex]
We get this equation because there are 6 more than 3 times the Crockford cars in the Robinson dealership. We see that the equation already is the value of k in terms of k, so we have our answer.
Which one is the geometric sequence.
Answer:
C
Step-by-step explanation:
A geometric sequence has a common ratio r between consecutive terms
A
4 ÷ 2 = 2
6 ÷ 4 = 1.5
Thus not a geometric sequence
B
1 ÷ 1 = 1
2 ÷ 1 = 2
Thus not a geometric sequence
C
[tex]\frac{1}{2}[/tex] ÷ 1 = [tex]\frac{1}{2}[/tex]
[tex]\frac{1}{4}[/tex] ÷ [tex]\frac{1}{2}[/tex] = [tex]\frac{1}{2}[/tex]
[tex]\frac{1}{8}[/tex] ÷ [tex]\frac{1}{4}[/tex] = [tex]\frac{1}{2}[/tex]
D
4 ÷ 7 = [tex]\frac{4}{7}[/tex]
1 ÷ 4 = [tex]\frac{1}{4}[/tex]
Thus not a geometric sequence
Then the geometric sequence is C
How is adding and subtracting radical expressions similar to adding and subtracting variables ?????
Answer: To add or subtract radicals, the indices and what is inside the radical (called the radicand) must be exactly the same. If the indices and radicands are the same, then add or subtract the terms in front of each like radical. If the indices or radicands are not the same, then you can not add or subtract the radicals.
Answer:
combining like terms. if the radicand is the you can combine the radical. this is the same as combining variables as long as the variable is the same if that makes sense
Step-by-step explanation:
so like you can add 7a and 3a because they have the same variable a. you can combined 2 root 2 and 4 root 2 because the radicand is same <3
PLZZ answer this in a simple method
Answer:
hope this help you
of my handwriting is bad then sorry.
Answer:
have a great day
may god bless you a lot
What is -20/64 in simplest form?
Answer:
-5/16
Step-by-step explanation:
-20 /64
Divide the top and bottom by 4
-5/16
URGENT PLEASE HELP. In the triangles below, DF MN, DG MP, D P. Can you prove that DFG MNP? Explain your answer. (15 points)
Answer:
FDG=NMP
Step-by-step explanation:
You can see below that the bases DG and MP are equal, and one of the saides DF and NM are equal, and D=P, so the triangles are the same, you can also see the triangles are isosceles, so they are the same since NM=FD=FG=NP and FD=NM is shown in the picture
I need help with #36.
Answer:
1/81
Step-by-step explanation:
Solve for b.
7
b
12
b= ✓ [?]
Pythagorean Theorem: a2 + b2 = c2
Enter
Answer: [tex]b=\sqrt{95}[/tex]
Step-by-step explanation:
To solve for b, we want to use the Pythagorean Theorem as given.
b and 7 are the legs, and 12 is the hypotenuse.
[tex]7^2+b^2=12^2[/tex] [exponent]
[tex]49+b^2=144[/tex] [subtract both sides by 49]
[tex]b^2=95[/tex] [square root both sides]
[tex]b=\sqrt{95}[/tex]
Now we know [tex]b=\sqrt{95}[/tex].
2. If (x+1,Y+2)=(3,4), find the value of x and y
Answer:
x = 2
y = 2
Step-by-step explanation:
since it is an equal pair,
x+1 = 3
or, x = 3 - 1
x = 2
y + 2 = 4
or, y = 4 - 2
y = 2
4.
Which point is NOT part of the solution of the inequality y < (3x + 1?
(-4, 12)
(0,0)
(2, -9)
(3, 15)
Answer: The non solutions of the inequality are A and D.
Step-by-step explanation:
One dozen cookies cost $5.50. At that rate, how much does one cookie cost?
Answer:
0,458$
Step-by-step explanation:
there are unspecified number of cows and hens in a den. if there are total of 30 heads and 80 legs in a den, find the number of cows and hens in a den.
Let x be the number of cow.
Number of hens=180-x.
Since cows have 4 legs and hens 2,
So, 4x+2(180-x)=420.
or, 4x+360–2x=420
or, 2x=420–360
or, 2x=60
or,x=30.
So, number of cows is 30.
Step-by-step explanation:
Hope this helps you^_^ Please mark as the brainliest.
What is the volume of the cone? Round to the nearest cubic cm.
Answer:
the answer will be
184.31
Answer if you want tho...
Answer:
3 loaves 2 bananas left.
Step-by-step explanation:
The answer is asking how many loaves of three bananas Seth can make. 11 can be divided by 3 a total of 3 times, leaving a remainder of 2 bananas that would not be enough to make another loaf.
What is the 100th term of the arithmetic sequence below?
4, 11, 18, 25
Answer:
a₁₀₀ = 697
Step-by-step explanation:
The nth term of an arithmetic sequence is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
Here a₁ = 4 and d = a₂ - a₁ = 11 - 4 = 7 , then
a₁₀₀ = 4 + (99 × 7) = 4 + 693 = 697
QUESTION 2 Write the slope-intercept form of the equation of the line through the gran point micia the grea slope through: (5,-2), siope--1 Oay=-2=6 Oby=-2-3 O cy=-x=3 Ody=-z-6
Answer:
y = -x+3
Step-by-step explanation:
Slope intercept form of a line is
y = mx+b where m is the slope and b is the y intercept
y = -1x +b
Using the point for x and y we can calculate b
-2 = -1(5)+b
-2 = -5+b
Add 5 to each side
-2+5 = b
3 = b
y = -x+3
Una empresa que trabaja en un mercado de competencia perfecta tiene una función de costos totales CT=Q^3-45Q^2+600Q+1.200 Las funciones de oferta y demanda de este mercado son: Qo=5P-300 y Qd=2.860-8P. Recuerde que la condición para maximizar beneficios dice que la empresa habrá de producir una cantidad tal que haga igualar sus costos marginales con el precio de equilibrio de mercado.
Answer:
La cantidad producida para maximizar las ganancias es de 915 unidades
Step-by-step explanation:
Pregunta: Calcule cuánto producirá para maximizar las ganancias.
La función de costo dada se presenta de la siguiente manera;
CT = Q³ - 45 · Q² + 600 · Q + 1200
La función de suministro es Qo = 5 · P - 300
La función de demanda es Qd = 2860 - 8 · P
A precio de equilibrio, tenemos;
Qo = Qd
Por lo tanto;
5 · P - 300 = 2860 - 8 · P
13 · P = 2.860 + 300 = 3.160
P = 3160/13
Por lo tanto;
Q = 2860 - 8 × (3160/13) = 11900/13 ≈ 915,38
La cantidad producida que maximiza las ganancias es de aproximadamente 915 unidades.
El costo total CT = 915³ - 45 × 915² + 600 × 915 + 1200 = 728,935,950
Find the average rate of change of a function that contains the points (-2,3) and (2,5). -2, -1/2, 1/2, 2
Answer:
The answer is "[tex]\frac{1}{2}[/tex]".
Step-by-step explanation:
Given points:
[tex](-2,3)\\\\(2,5)\\\\[/tex]
average rate=?
[tex]x_1=-2\\\\y_1=3\\\\x_2=2\\\\y_2=5[/tex]
[tex]average\ rate=\frac{y_2-y_1}{x_2-x_1}[/tex]
[tex]=\frac{5-3}{2-(-2)}\\\\=\frac{5-3}{2+2}\\\\=\frac{2}{4}\\\\=\frac{1}{2}[/tex]
Solve the inequality.
q + 12 - 2(q - 22) > 0
a.)9 <-32
b.)9 < 56
c.)9> -32
d.)9 > 56
[tex]\\ \sf\longmapsto q+12-2(q-22)>0[/tex]
[tex]\\ \sf\longmapsto q+12-2q+44>0[/tex]
[tex]\\ \sf\longmapsto q-2q+12+44>0[/tex]
[tex]\\ \sf\longmapsto -q+56>0[/tex]
[tex]\\ \sf\longmapsto -q>-56[/tex]
[tex]\\ \sf\longmapsto q<56[/tex]
