(1 point) For what values of the variable does the series converge? Use the properties of geometric series to find the sum of the series when it converges
Sum=
20+80x^2 + 320x^3 +1280x^4+......
domain=
(Give your domain as an interval or comma separated list of intervals; for example, to enter the region <-1 and 2 << 3, enter (- infinity,-1). (2.3))
Answers
The sum of the series is S = 20+80x^2 + 320x^3 +1280x^4+..... rewritten as S = 20 / (1 - 4x⁻¹) converges for he values |x⁻¹| < 1/4
To obtain what values of the variable the series converge and use the properties of the geometric series to find the sum of the series when it converges, we have to first look at the series given: 20 + 80x² + 320x³ + 1280x⁴ + .........Let's take the first three terms of the series: a₁ = 20a₂ = 80x²a₃ = 320x³
To find the ratio r, let's divide the second term by the first and the third term by the second:a₂/a₁ = (80x²)/20 = 4x²a₃/a₂ = (320x³)/(80x²) = 4x We notice that the ratio r is constant and equal to 4x⁻¹.
Now we can determine the conditions for which the series converges. Condition for the convergence of a geometric series:|r| < 1|r| = |4x⁻¹| < 1|r| = 4|x⁻¹| < 1|r| = |x⁻¹| < 1/4 Therefore, the domain of the series is the set of all x for which |x⁻¹| < 1/4. Since x cannot be zero, we exclude x = 0 from the domain. Then, the domain is obtained by taking the reciprocal of the inequality and inverting the sign:|x⁻¹| < 1/4⇔ - 1/4 < x < 1/4⇔ x ∈ (-1/4, 1/4)
Finally, we can use the formula for the sum of a finite geometric series to find the sum of the given series:S = a₁ (1 - rⁿ) / (1 - r)where n is the number of terms in the sum. Since we have an infinite series, we can take the limit as n approaches infinity:S = a₁ / (1 - r) = 20 / (1 - 4x⁻¹) for |x⁻¹| < 1/4Therefore, the sum of the series is S = 20 / (1 - 4x⁻¹) when the series converges for |x⁻¹| < 1/4.
Related Questions
Find the value of x that makes 25x^2 + 70x + c a perfect square trinomial
Answers
The value of x that makes 25x^2 + 70x + c a perfect square trinomial is c = 1225.
To make the quadratic expression 25x^2 + 70x + c a perfect square trinomial, we need to determine the value of c.
A perfect square trinomial can be written in the form (ax + b)^2, where a is the coefficient of the x^2 term and b is half the coefficient of the x term.
In this case, a = 25, so b = (1/2)(70) = 35.
Expanding (ax + b)^2, we have:
(25x + 35)^2 = 25x^2 + 2(25)(35)x + 35^2
= 25x^2 + 70x + 1225.
Comparing this with the given quadratic expression 25x^2 + 70x + c, we can see that c = 1225.
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A gas station sells unleaded regular and unleaded premium gasoline. The price per gallon charged by the station is$1.299 for unleaded regular and $1.379 for unleaded premium. The cost per gallon from supplier is$1.219 for unleaded regular and $1.289 for unleaded premium. If equals the number of gallons sold of regular and equals the number of gallons sold of premium:(a) Formulate the revenue function from selling and gallons of the two grades of gasoline respectively?(b) Formulate total cost for purchasing and gallons?(c) Formulate the total profit? d) what is the total profit expected to equal if the station sells 100,000 gallons of unleaded and regular and 40000 of unleaded premuim?
Answers
Answer:
a) Revenue function = 1.299r + 1.379p
where: r = gallon of unleaded regular gasoline and p = gallon of regular unleaded premium gasoline
b) cost function = 1.219f + 1.289p
c) profit function = (1.299r + 1.379p) - (1.219f + 1.289p) = 1.299r - 1.219r + 1.379p - 1.289p = 0.08p + 0.09p
d) total profit = (0.08 x 100,000) + (0.09 x 40,000) = $8,000 + $3,600 = $11,600
Click through the graphs and select the one that could represent the relationship be
time, t, for the cell phone plan shown below.
time in hours 0 1 2 3
cost in dollars 10 13 16 19
Cost in dollars
20
18
16
14
4
2
2
3
Time in Hours
4
S
Answers
The linear function for the cost is given as follows:
C(t) = 10 + 3t.
How to define a linear function?The slope-intercept equation for a linear function is presented as follows:
y = mx + b
In which:
m is the slope.b is the intercept.We have that each hour, the cost increases by $3, hence the slope m is given as follows:
m = 3.
For a time of 0 hours, the cost is of $10, hence the intercept b is given as follows:
b = 10.
Thus the function is given as follows:
C(t) = 10 + 3t.
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What is the factoring greatest common factor (GCF) of 95x^3 and 65x^2
Answers
Notice that
\(\begin{gathered} 95x^3=19\cdot5x^3 \\ \text{and} \\ 65x^2=13\cdot5x^2 \end{gathered}\)Then, the factors of both quantities are:
\(\begin{gathered} 95x^3\colon x,x^2,x^3,5x,5x^2,5x^3,19x,19x^2,19x^3 \\ \end{gathered}\)and
\(65x^2\colon x,x^2,5x,5x^2,13x,13x^2\)Then, the GCF is 5x^2
a 9 by 12 rectangular piece of paper is folded so that two opposite corners coincide. what is the length of the crease
Answers
The length of the crease is 15 cm.When a 9 by 12 rectangular piece of paper is folded so that two opposite corners coincide, the length of the crease is 15 cm. When we fold a rectangular paper so that the opposite corners meet, we get a crease that runs through the diagonal of the rectangle.
In this case, the 9 by 12 rectangle's diagonal can be determined using the Pythagorean Theorem which states that the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides. In this case, the two sides are the length and width of the rectangle.
The length of the diagonal of the rectangle can be determined as follows:\(`(9^2 + 12^2)^(1/2)`\) = 15 cm. Therefore, the length of the crease is 15 cm.
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What is the surface area of this square pyramid? ft^2 on ft
2 ft^2
.
4 ft^2
2 ft² .
2 ft^2
2 ft²
Answers
The price of gasoline drops from $1.00 per
gallon to 95° per gallon. What is the percent of
decrease?
Answers
lmk if it’s the wrong answer
what is the decimal value of the following binary number? 10011101
Answers
The following binary number has the value of 157 in decimal form.
How can a binary number be converted to a decimal?Step 1: Compose the binary number in writing. 10011101
Multiplying each binary digit by the corresponding power of two in step two:
1x27 + 0x26 + 0x25 + 1x24 + 1x23 + 1x22 + 0x21 + 1x20
Solve the powers in Step 3:
1x128+0x64+0x32+1x16+8+4+0x2+1x1=128+0 + 0 + 0 + 16+8+4+0 + 0 + 1
Step 4: Add the figures listed above:
128 + 0 + 0 + 16 + 8 + 4 + 0 + 1 = 157.
What does the binary number system mean?A binary number is a number that is expressed in the binary system or base 2 numeral system, according to digital technology and mathematics. It talks about numerical values.by the two distinct digits 1 (one) and 0 (zero). The base-2 system is the positional notation that uses 2 as the radix.
Define decimal.A number that has been divided into a whole and a fraction is called a decimal. To express the numerical value of entirely and partially entire quantities between integers, decimal numbers are used.
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if you select a dvd at random, then select another at random (while keeping the first in your hand), what is the probability that both will be harry potter films?
Answers
The probability of selecting two Harry Potter DVDs at random, one after the other, is 1/110.
The probability of selecting a Harry Potter DVD at random is dependent on the total number of DVDs and the number of Harry Potter DVDs in the collection. Let's assume there are a total of 100 DVDs and 10 of them are Harry Potter films.
When selecting the first DVD, the probability of picking a Harry Potter film is 10/100 = 1/10.
After selecting the first DVD, there are 99 DVDs left, with 9 Harry Potter films remaining.
When selecting the second DVD, the probability of picking another Harry Potter film, while keeping the first one in hand, is 9/99 = 1/11.
To find the probability of both DVDs being Harry Potter films, we multiply the probabilities: (1/10) * (1/11) = 1/110.
Therefore, the probability that both DVDs will be Harry Potter films is 1/110.
In conclusion, the probability of selecting two Harry Potter DVDs at random, one after the other, is 1/110.
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which equation is correctly written in point-slope form using the point (−5,−2) as (x1,y1)?
Answers
Either option is valid and correctly written in point-slope form using the point (−5,−2) as (x1,y1).
The point-slope form of an equation of a line is given by:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line
m is the slope of the line.
We can use the given point (−5,−2) as (x1,y1)
Choose any value for the slope m to write the equation of the line in point-slope form.
For example:
Option 1:
Let's choose the slope m to be 2.
Then the equation of the line becomes:
y - (-2) = 2(x - (-5))
Simplifying the equation:
y + 2 = 2(x + 5)
Option 2:
Let's choose the slope m to be -3.
Then the equation of the line becomes:
y - (-2) = -3(x - (-5))
Simplifying the equation:
y + 2 = -3(x + 5)
Either option is valid and correctly written in point-slope form using the point (−5,−2) as (x1,y1).
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What is the value of 80 + 57
Answers
have a good day. hope i helped
Answer:
137
Step-by-step explanation:
Add 80 and 57.
write an ordered pair for y=1/8x-1 and -5x+4y=-3
Answers
Answer:
Step-by-step explanation:0495-46y7467-6686-745676-5
Answer:
there fore y = 1/8x-1 it can be write as 8xy+8x-1 = 0
and -5x+4y=-3 can be write it as -5x+4y+3 = 0
Step-by-step explanation:
given that
pair y = 1/8x-1 ------ equation (1)
and -5x+4y=-3 ------ equation (2)
you can write it as
you can write equation (1) as
y = 1/8x-1
y = 1-8x/8x
8xy = 1-8x
there fore
y = 1/8x-1 it can be write as 8xy+8x-1 = 0
you can write equation (2) as
-5x+4y = -3
-5x+4y+3 = 0
there fore
-5x+4y=-3 can be write it as -5x+4y+3 = 0
Find the midpoint of AB if A(-3. 8) and B(-7. -6)
Answers
The midpoint of AB is (-5, 1).
To find the midpoint of a line segment AB, we average the x-coordinates of the two endpoints and the y-coordinates of the two endpoints.
Given that A has coordinates (-3, 8) and B has coordinates (-7, -6), we calculate the midpoint as follows:
x-coordinate of midpoint = (x-coordinate of A + x-coordinate of B) / 2
= (-3 + -7) / 2
= -10 / 2
= -5
y-coordinate of midpoint = (y-coordinate of A + y-coordinate of B) / 2
= (8 + -6) / 2
= 2 / 2
= 1
Therefore, the midpoint of AB is (-5, 1), where -5 is the x-coordinate and 1 is the y-coordinate. This means that the midpoint is equidistant from points A and B, dividing the line segment AB into two equal parts.
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(›)
Which is more, 1 tablespoon or 2 teaspoons?
Answers
Answer: 1 tablespoon
Step-by-step explanation:
The table below shows the linear relationship between the number of people at a picnic and the total cost of the picnic.
A two column table with five rows. The first column, Number of People, has the entries, 6, 9, 12, 15. The second column, Total Cost in dollars, has the entries, 52, 58, 64, 70.
Which statements about the function described by the table are true? Check all that apply.
The independent variable is the number of people.
The initial value (initial fee) for the picnic is $40.
The rate of change is $8.67 per person.
As the number of people increases, the total cost of the picnic increases.
If 4 people attended the picnic, the total cost would be $46.
Answers
The true statements are:
The independent variable is the number of people.The rate of change is $8.67 per person.As the number of people increases, the total cost of the picnic increases.What are the true statements?The independent variable is the variable that determines the value of the dependent variable. There is a positive relationship between the number of people at the picnic and the total cost. This is because it can be seen as the number of people at the picnic increases, the total cost changes.
Rate of change = total cost / number of people
$52 / 6 = $8.67
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Answer:
A, B, D
Step-by-step explanation:
right on edge
Please help me
Solve the equation.
24 – 3x = 2x – 1
A. x = 4
B. x = 5
C. x = 6
D. x = 25
Answers
Im not understanding this, Can someone help ?
Answers
Sean keeps track of his driving so that he can take a tax deduction. Sean drove 260 miles last month, in comparison to 325 miles this month. By what percent has his monthly driving increased?
Answers
Answer:
25%
Step-by-step explanation:
325/260 = 1.25
When a biological researcher arrived on Wombat Island, there were about 1000 wombats on the island. Their population grew quickly. Estimated Wombat Population Year 1 2 3 4 5 Wombats 1000 2000 4000 8000 16,000 The researcher decided that the population growth followed a geometric sequence. He wrote the explicit formula an = 1000(2)n – 1 to model the growth, and used this formula to predict the populations for years 6, 10, and 50.
Answers
Answer:
Yes
Step-by-step explanation: The explicit formula used: An = 1000(2)n - 1 is the correct formula. This is modeled as exponential growth with a common ratio of 2; which is correct based on the data given in the table.
The predicted population for year 50 is approximately \(1.1259\) x \(10^18\) wombats.
To predict the populations for years 6, 10, and 50 using the given explicit formula an = \(1000(2)^n\) – 1, we can substitute the respective values of n and calculate the population.
For year 6 (n = 6):
a6 = \(1000(2)^6\) – 1
a6 = 1000(64) – 1
a6 = 64,000 – 1
a6 ≈ 63,999
The predicted population for year 6 is approximately 63,999 wombats.
For year 10 (n = 10):
a10 = \(1000(2)^ 10\)
a10 = 1000(1,024) – 1
a10 = 1,024,000 – 1
a10 = 1,023,999
The predicted population for year 10 is 1,023,999 wombats.
For year 50 (n = 50):
a50 = \(1000(2)^50\) – 1
a50 = 1000(1.1259 x 10¹⁵ ) – 1
a50 = 1.1259 x 10¹⁸ – 1
a50 ≈ 1.1259 x 10¹⁸
The predicted population for year 50 is approximately 1.1259 x 10^18 wombats.
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A contest offers 20 prizes, with first prize worth $12,000 and each successive prize worth $400 less than the preceding prize.
Answers
The 20th prize is $4400.
It is a sequence where the difference between each consecutive terms is the same.
Example:
2, 4, 6, 8 is an arithmetic sequence.
We have,
First prize = $12,000
Successive prize worth $400 less than the preceding prize.
Now,
We can make an arithmetic sequence.
12,000, 11,600, 11,200, ...
So,
a = 12,000
d = -400
The nth term of an arithmetic sequence.
= a + (n - 1)d
So,
The 20th prize.
= 12,000 + (20 -1)400
= 12,000 - 19 x 400
= 12,000 - 7600
= 4400
Thus,
20th prize = $4400
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do all square numbers have an odd number of factors
Answers
No, not all square numbers have an odd number of factors. In fact, square numbers can have either an odd or an even number of factors, depending on their prime factorization.
A square number is a number that can be expressed as the product of an integer multiplied by itself. For example, 4 is a square number because it can be written as 2 * 2.
When we analyze the factors of a square number, we find that each factor has a corresponding pair that multiplies to give the square number. For instance, the factors of 4 are 1, 2, and 4. We can see that the pairs are (1, 4) and (2, 2). Thus, 4 has an even number of factors.
However, there are square numbers that have an odd number of factors. Consider the square number 9, which is equal to 3 * 3. The factors of 9 are 1, 3, and 9. In this case, 9 has an odd number of factors.
In conclusion, while some square numbers have an odd number of factors (like 9), others have an even number of factors (like 4). The determining factor is the prime factorization of the square number.
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A quantity with an initial value of 7200 decays exponentially at a rate of 0.3% every
day. What is the value of the quantity after 10 weeks, to the nearest hundredth?
Answers
Answer:5834.36 You're welcome
Answer:5834.36
Step-by-step explanation:
What is 1.5kg :350 g
Answers
Step-by-step explanation:
1 kg = 1000 g
1.5 kg = 1500 g
1500 g : 350 g
= 150 : 35
= 30 : 7
Answer:
3.3 pounds
Step-by-step explanation:
A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade (x) and test grade (y) are given in the accompanying table. Write the linear regression equation that represents this set of data, rounding all coefficients to the nearest tenth. Using this equation, find the projected test grade, to the nearest integer, for a student with a homework grade of 80.
Regression Equation:
Final Answer:
Answers
The linear regression equation for the model is Y = ( 0.9629 )X - 2.759 where the slope of the equation is m = 0.9629
What is an Equation of a line?The equation of a line is expressed as y = mx + b where m is the slope and b is the y-intercept
And y - y₁ = m ( x - x₁ )
y = y-coordinate of second point
y₁ = y-coordinate of point one
m = slope
x = x-coordinate of second point
x₁ = x-coordinate of point one
The slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Given data ,
Let the equation of the line be represented as A
Now , the homework grade is represented as x
The values of x = { 71 , 72 , 62 , 80 , 85 , 74 , 90 , 83 , 62 }
And , the test grade is represented as y
The values of y = { 63 , 75 , 58 , 77 , 71 , 65 , 95 , 68 , 57 }
From the linear regression calculator , the equation of line is given as
y = mx + b where m is the slope and b is the y-intercept
Y = ( 0.9629 )X - 2.759 be equation (1)
where the slope is m = 0.9629
And , Y = Test grade ; X = homework grade
Now , for a student with a homework grade of 80
Substitute the value of x as 80 , we get
Y = ( 0.9629 )X - 2.759
when X = 80 , we get
Y = ( 0.9629 ) ( 80 ) - 2.759
Y = 77.032 - 2.759
On simplifying the equation , we get
Y = 74.273
Y = 74
Therefore , the test grade Y of the student is 74
Hence , the linear regression equation is Y = ( 0.9629 )X - 2.759
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Which is NOT a right triangle
Answers
Answer:f i’m pretty sure
Step-by-step explanation:
Pls help I’m really struggling and I have a really strict teacher so help
Answers
Answer:
try chat gpt, God bless, Jesus loves you!
Step-by-step explanation:
edit: why did my comment disapeer? I highly recommend to type your question in there, just trust me on this and make sure to ask if to double check it's answer just to be safe.
Kofi wants to find the most-liked snacks to put in the vending machine at his office so the most people will buy from it.
Which of the following strategies would help Kofi ensure a random sample?
A.
He could make an alphabetical list of employees in his office and survey every third employee.
B.
He could survey the first ten people who use the vending machine on a Monday morning.
C.
He could make an alphabetical list of employees in his office and survey the first ten employees.
D.
He could survey the employees who do not buy snacks from the vending machine.
Answers
Option (B), he could survey the first ten people who use the vending machine on a Monday morning is the correct answer.
What is random sampling?"Random sampling is a part of the sampling technique in which each sample has an equal probability of being chosen. A sample chosen randomly is meant to be an unbiased representation of the total population".
For the given situation,
Kofi wants to find the most-liked snacks to put in the vending machine at his office so the most people will buy from it.
Random sampling can be done on the basis of surveying the employees who are using vending machines.
Hence we can conclude that option (B), he could survey the first ten people who use the vending machine on a Monday morning is the correct answer.
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Consider the sequence $$1,3,4,9,10,12,13,\ldots,$$ which consists of every positive integer that can be expressed as a sum of distinct powers of $3$. What is the $75^{\text{th}}$ term of this sequence
Answers
To find the 75th term of the sequence consisting of positive integers that can be expressed as a sum of distinct powers of 3. Thus, the 75th term of the sequence is $111$.
To find the 75th term of the sequence consisting of positive integers that can be expressed as a sum of distinct powers of 3, we can use the base-3 (ternary) numeral system. The 75th term can be represented as the 74th number in base-3 without a digit 2 (as it will require subtraction to form distinct powers of 3).
The 74th number in base-10 is represented as $74_{10}$, which when converted to base-3 is $2202_3$. Since we need to avoid the digit 2, we can carry out the operation similar to addition with carry-over: $2202_3 + 1111_3 = 10310_3$.
Now, we can convert $10310_3$ back to base-10: $(1 \times 3^4) + (0 \times 3^3) + (3 \times 3^2) + (1 \times 3^1) + (0 \times 3^0) = 81 + 0 + 27 + 3 + 0 = 111$.
Thus, the 75th term of the sequence is $111$.
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need answer important pls
Answers
Answer:
C {-2 1/6, 1 5/6}
Step-by-step explanation:
This will solve your answer
Cody simulates the situation by using a computer to randomly generate 12 numbers from 1 to 20. Each time 1 appears, it represents a golden ticket. What’s the probability of getting at least 1 gold ticket out of 200 trials
Answers
To calculate the probability of getting at least one golden ticket out of 200 trials, we can find the probability of not getting any golden tickets and subtract it from 1.
The probability of not getting a golden ticket in a single trial is given by: P(not getting a golden ticket) = (19/20)^12
This is because there are 19 out of 20 numbers that are not golden tickets, and we have 12 trials.
Now, to calculate the probability of not getting any golden tickets in 200 trials, we raise the probability of not getting a golden ticket in a single trial to the power of 200:
P(not getting any golden ticket in 200 trials) = (19/20)^12^200
To find the probability of getting at least one golden ticket, we subtract the probability of not getting any golden tickets from 1:
P(getting at least one golden ticket in 200 trials) = 1 - P(not getting any golden ticket in 200 trials)
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