Use the binomial theorem to find: (a) the coefficient of x4 in the expansion of (x + 3)20 n n n (b) the sum 1 + 3n +9 -) + - +3") +...--(.) + 3*-'n + 3". ( (3 : 3k 3n-2 2 3n" 2 k n
Answers
(a) The coefficient of x^4 in the expansion of (x + 3)^20 can be found using the binomial theorem.
(b) To find the sum of the series 1 + 3n + 9n^2 + 27n^3 + ... + 3^n, we can use the formula for the sum of a geometric series.
(a) According to the binomial theorem, the expansion of (x + y)^n can be written as the sum of the terms:
C(n, 0) * x^n * y^0 + C(n, 1) * x^(n-1) * y^1 + C(n, 2) * x^(n-2) * y^2 + ... + C(n, n) * x^0 * y^n,
where C(n, r) denotes the binomial coefficient, which is calculated as C(n, r) = n! / (r! * (n-r)!).
In this case, we are looking for the coefficient of x^4 in the expansion of (x + 3)^20. Using the binomial theorem, we can determine that the term with x^4 is obtained when we choose x^4 from the first binomial factor (x) and 3^16 from the second binomial factor (3), since 4 + 16 = 20. The coefficient of this term is given by the binomial coefficient C(20, 4). Therefore, the coefficient of x^4 in the expansion is C(20, 4) = 4845.
(b) The given series is a geometric series with the first term (a) equal to 1 and the common ratio (r) equal to 3n. The formula for the sum of a geometric series is S = a * (1 - r^n) / (1 - r), where S represents the sum of the series.
In this case, plugging in the values, we have a = 1 and r = 3n. Substituting these values into the formula, we get the sum of the series as S = 1 * (1 - (3n)^(n+1)) / (1 - 3n).
Note that the given series has infinitely many terms, so the sum depends on the value of n. By using this formula, you can find the sum of the series for any specific value of n.
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Related Questions
Kelsey bought an oil painting online. It cost $709 plus 25% shipping and handing. What was the total cost?
Answers
question 1/ Find the slope of the line that passes through the points (2, 5) and (-7, -4). If you get stuck, try plotting the points on graph paper and drawing the line through them with a ruler.
Answers
Answer:
I believe the slope is 1
Step-by-step explanation:
Answer:
The slope is 1
Step-by-step explanation:
m=y²-y¹/x²-x¹
m=(-4)-5/(-7)-2
m=-9/-9
m= 1
1. Use the following graph to determine the coordinated of the y-intercept.
A. (5,0)
B. (4,0)
C. (0,4)
D. (0,5)
HURRY!!! Please.
Answers
Answer:
I believe it's C. (0.4)
Step-by-step explanation:
A scuba diver descends (goes down) to a location that is −10 m from sea level. He then descends (goes down) another -12 m. What is the scuba diver’s final location relative to sea level
Answers
Answer:
-22 m
Step-by-step explanation:
I need help this is my homework I will brainless you
Answers
Answer:
4-10.5
5-no because the nearest thousandth is not 5 or more
6- Tu is correct because the tenth is 6 which is more than 5 so it can be rounded up to $79
A scatter plot is shown on the coordinate plane.
scatter plot with points plotted at 1 comma 5, 1 comma 8, 2 comma 4, 3 comma 5, 3 comma 6, 5 comma 6, 6 comma 4, 7 comma 2, 9 comma 1, and 10 comma 1
Which two points would a line of fit go through to best fit the data?
(6, 4) and (9, 1)
(3, 5) and (10, 1)
(1, 8) and (5, 6)
(1, 5) and (7, 3)
Answers
Answer:
I believe (3, 5) and (10, 1) is the answer
Step-by-step explanation:
please help meee brainliest
Answers
Answer: 19.
Step-by-step explanation:
9 -3 / 1/3 + 1
6 / 1/3 + 1
(6 * 3/1) + 1 also written as (6 * 3) + 1
18 + 1
= 19.
form a system of equations 1 and 2. without graphing explain how you can tell how many solutions the system has?
Equation#1: y=2x+3
Equation#2 y=2x-3
Answers
The manager of a city recreation center wants to estimate the percent of city residents who favor a proposal to build a new dog park. To gather data, the manager will select a random sample of city residents. Which of the following is the most appropriate interval for the manager to use for such an estimate?
a. A one-sample zz-interval for a sample proportion.
b. A one-sample zz-interval for a population proportion.
c. A one-sample zz-interval for a difference between population proportions.
d. A two-sample zz-interval for a difference between sample proportions.
Answers
Considering the hypothesis tested, the most appropriate interval for the manager to use for such an estimate is given by:
b. A one-sample z-interval for a population proportion.
What is the most appropriate test?The manager will sample some residents, and if the proportion who is in favor of the new park is high enough, he builds the new park.
For the most appropriate test, we have that:
There is only one sample, hence it is a one-sample test.He will use the sample proportion to estimate the population proportion.Hence, option B is correct.
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help me solve this please i will give someone who responds a crown
Answers
Answer:
4.
Step-by-step explanation:
You are so smart. 5 - (-7) = 5+7 which = 12. 12/3 is 4.
Answer:
\( \boxed{x = -4} \)
Step-by-step explanation:
This drawing illustrates the
equation: -3x – 7 = 5.
Since:
(-x + -x + -x) + (-1 + -1 + -1 + -1 + -1 + -1 + -1) =
(1 + 1 + 1 + 1 + 1) → (-3x) + (-7) = (5) →
-3x – 7 = 5.
To reduce the equation and find x:
[add 7 to both sides to isolate the constant term on the right side]
-3x – 7 + 7 = 5 + 7 →
-3x = 12 →
[divide both sides by -3 to cancel out the coefficient of -3 in -3x to get x]
-3x / -3 = 12 / -3 →
x = -4
Jill has 4 one dollar bills, 3 quarters, 4 dimes, no 3 pennies. Mark has 3 one dollar bills, 4 dimes, and 2 pennies. What is the difference between the amount of money Jill has and the amount of money mark has?
Answers
Answer: $1.73
Step-by-step explanation:
Jill has $5.15 and Mark has $3.42. Subtract. Voilà.
Is Mabel correct?
A. Mabel is correct.
B. No, she made a mistake in step 2
C. No, she made a mistake in step 3
D. No, she made a mistake in step 4
Answers
(
4
x
−
3
)
(
(
x
−
2
)
2
)
=
4
x
3
−
19
x
2
+
4
x
−
12
4
x
3
−
19
x
2
+
28
x
−
12
=
4
x
3
−
19
x
2
+
4
x
−
12
Step 1: Subtract 4x^3 from both sides.
4
x
3
−
19
x
2
+
28
x
−
12
−
4
x
3
=
4
x
3
−
19
x
2
+
4
x
−
12
−
4
x
3
−
19
x
2
+
28
x
−
12
=
−
19
x
2
+
4
x
12
Step 2: Add 19x^2 to both sides.
−
19
x
2
+
28
x
−
12
+
19
x
2
=
−
19
x
2
+
4
x
−
12
+
19
x
2
28
x
−
12
=
4
x
−
12
Step 3: Subtract 4x from both sides.
28
x
−
12
−
4
x
=
4
x
−
12
−
4
x
24
x
−
12
=
−
12
Step 4: Add 12 to both sides.
24
x
−
12
+
12
=
−
12
+
12
24
x
=
0
Step 5: Divide both sides by 24.
24
x
24
=
0
24
x
=
0
Answer:
x
=
0
Answer:
D. No, she made a mistake in step 4
Step-by-step explanation:
5. a) Verify that the altitude from vertex J
bisects side KL in the triangle with
vertices J(-5, 4), K(1, 8), and L(−1, −2).
b) Classify AJKL. Explain your reasoning.
Answers
Answer:
a) Yes, the altitude from vertex J bisects side KL in the triangle with vertices J(-5, 4), K(1, 8), and L(−1, −2).
b) AJKL is an isosceles triangle. This is because side KL has the same length, which is 6 units.
Step-by-step explanation:
An isosceles triangle has two sides of equal length and two equal angles opposite to those sides. The angles between the two equal sides are called the base angles of the triangle. In the case of AJKL, the two equal sides are KL and JK, which have a length of 6 units. The two equal angles opposite to these sides are angle AJK and angle ALK.
Domain?
Range?
Function?
Answers
Answer:
domain is all real numberange
R = 3 < y < R
F = f(x) = 1x
I hope this is right :D
2. question 2 a data analyst is creating a chart for a presentation. the data they will display shows a correlation between variables. why should they be careful when presenting their chart to an audience? 1 point correlation can only be represented in bar charts. correlation can be misunderstood as causation. correlation should be avoided in charts. correlation causes accessibility issues.
Answers
The statistical concept of correlation describes how closely two variables move in tandem with one another.
What is correlation ?Any statistical association between two random variables or bivariate data, whether causal or not, is referred to in statistics as correlation or dependency.
Although "correlation" can mean any kind of association in the broadest sense, in statistics it typically refers to the strength of a pair of variables' linear relationships.
Examples of common dependent phenomena include the relationship between parent and child height and the relationship between a good's price and the number of units buyers are prepared to buy, as shown in the so-called demand curve.
Because they can reveal a link that can be used in practice, correlations are helpful.
Hence, The statistical concept of correlation describes how closely two variables move in tandem with one another.
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How can I make an equation that equals 74 with using the digits/numbers 1 2 3 4 and 0 once and only once in the equation. Every number has to be used, with the exception on 0.
Answers
Answer:
A mathematical equation is an equation with zero or one variables (e.g., 9 = 6 + 3; 9 = x + 3), whereas an algebraic equation
Step-by-step explanation:
Answer:
9=6+3, 9=x+3
Hope it helped you
You flip two coins and roll a number cube. What is the probability of flipping two tails and rolling as even number?
The probability of flipping two tails and rolling an even number is
.
Answers
The probability of flipping two tails and rolling an even number is 1/8.
What is Probability?Probability is the possibility of getting an event. We are predicting the chance of getting an event.
The value of probability will be always in the range from 0 to 1.
Suppose you flip a coin.
Total number of outcomes = 2
Number of outcomes getting a tail = 1
The probability of getting a tail = 1/2
Now, suppose you flip two coins.
Probability of getting two tails = 1/2 × 1/2 = 1/4
Suppose you roll a die.
Total number of outcomes = 6
Number of outcomes which are even number = 3
Probability of getting an even number = 3/6 = 1/2
The probability of flipping two tails and rolling an even number is,
P = 1/4 × 1/2 = 1/8
Hence the required probability is 1/8.
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Which of the following algebraic represents shows a dilation that is an enlargement ?
Answers
The algebraic representation that shows a dilation that is an enlargement is (5/2 x,5/2 y). (Option D)
A dilation is a type of transformation that changes the size of the shape or object. It refers to a process of changing an object’s size by decreasing or increasing its dimensions by a scaling factor. A dilation produces an image that has the same shape as the original image but is a different size.
A dilation that results in a larger image is called an enlargement while a dilation that generates a smaller image is called a reduction. A dilation is described using the scale factor and the center of the dilation (which is a fixed point in the plane).
For a scale factor > 1, the image is an enlargement; for a scale factor < 1 and > 0, the image is a reduction; and for a scale factor = 1, the figure and the image are congruent. Hence, for a point (x,y), algebraic representation that shows a dilation that is an enlargement is (5/2 x,5/2 y) as the scale factor is greater than 1. For the remaining options, the scale factor is between 0 and 1, hence they are reduction.
Note: The question is incomplete. The complete question probably is: Which of the following algebraic representation shows a dilation that is an enlargement? A) (1/3 x,1/3 y) B) (0.1x, 0.1y) C) (5/6 x,5/6 y) D) (5/2 x,5/2 y)
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A 60-gallon tank is full of water when it begins to leak at a
rate of 2 gallons per minute. Which equation provides t, the
number of minutes it takes the tank to become empty?
Answers
Answer:
60/2=t
Step-by-step explanation:
Simple algebra. You need to divide the total by the per minute to find the length in time that it will take. t=30.
The base edge of the regular triangular pyramid is b=10 cm and altitude of the base hb ≈ 8.66 cm. The slant height of the pyramid is k=8 cm. Find: a Lateral area of the pyramid
Answers
Answer:
Lateral Area: 120
Surface Area: 163.3
which number line graph matches the inequality solution to –6x 3 < 3 6x?
Answers
The number line graph that matches the inequality solution to -6x + 3 < 3 + 6x is the one where all values to the left of the point representing -1 are shaded. This indicates that any value less than -1 satisfies the inequality.
In this inequality, we have -6x + 3 on the left side and 3 + 6x on the right side. To solve the inequality, we can start by subtracting 6x from both sides:
-6x + 3 - 6x < 3 + 6x - 6x
This simplifies to:
-12x + 3 < 3
Next, we can subtract 3 from both sides:
-12x + 3 - 3 < 3 - 3
This further simplifies to: -12x < 0
To isolate x, we divide both sides of the inequality by -12, remembering to reverse the inequality sign because we are dividing by a negative number:
(-12x)/-12 > 0/-12
This yields: x > 0
Thus, any value of x greater than 0 satisfies the inequality. On the number line graph, we shade all values to the right of 0, excluding 0 itself, indicating that these values are solutions to the inequality. The values to the left of 0 are not part of the solution.
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Find two consecutive odd integers such that the sum of the
smaller and twice the larger is equal to 55.
Answers
Answer:
they're 17 and 19
Step-by-step explanation:
x+2(x+2)=55
x+2x+4=55
51=3x
x=17
x+2=19
2 prime numbers are represented by f and g what is the greatest common factor of f and g please answer this fast its for a timed test
Answers
Answer:
1
Step-by-step explanation:
The greatest common factor of f and g is 1.
What is the greatest common factor?The greatest number can divide the numbers into equal parts.
It is called the greatest common factor.
Given:
2 prime numbers are represented by f and g.
That means the numbers are only factored by 1 and the number itself.
Let f = 2 and g = 3
Here, 2 and 3 are the prime numbers.
The GCF of 2 and 3 is 1.
That means the numbers are only factored by 1 and the number itself.
So, the greatest common factor of f and g is 1.
Therefore, 1 is GCF.
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Your teacher wants to use a point system to select the winning pet. She wants each pet to get a certain number of points for each 1st choice vote and a certain number of points for each 2nd choice vote.
Your teacher decides to use these rules for her point system:
Points need to be positive whole numbers
Points for a 1st choice vote have to be greater than or equal to the points for a 2nd choice vote.
Determine point values for the 1st and 2nd choice that would result in the turtle winning. Use words and numbers to explain how this point system results in the turtle winning
Answers
In the above case, the possible point system will be:
1st choice vote: 4 points
2nd choice vote: 2 points
What is the point system?In arranging for the turtle to win in this point framework, the point values relegated for 1st choice and 2nd choice votes have to be meet the taking after criteria:
The point got to be positive entire numbersThe point for a 1st choice vote need to be more noteworthy than or rise to to the focuses for a 2nd choice voteLets say:
1st choice vote: 4 points
2nd choice vote: 2 points
Thus by the use of these point values, the turtle would have the next chance of winning since 1st choice votes would be worth more point (4 point ) compared to 2nd choice votes (3 point ). This would push individuals to select the turtle as their 1st choice, because it would allow the turtle a better add up to point gains and increase its chances of winning.
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Consider the function g defined by g(x)=2x ^2
−9 a. Evaluate g(6)= b. Evaluate g(−7)= c. Solve g(x)=0. The answer is x= d. Solve g(x)=−9. The answer is x= e. Solve g(x)=−37. The answer is x=
Answers
The evaluation of the quadratic function g(x) = 2·x² - 9, indicates that we get;
a. g(6) = 63
b. g(-7) = 89
c. x = ±3·(√2)/2
d. x = 0
e. x = ±√(14)·i
What are quadratic functions?Quadratic functions are functions that can be expressed in the form f(x) = a·x² + b·x + c, where a ≠ 0, and a, b, and c, are numbers.
The quadratic function is; g(x) = 2·x² - 9
a. The value of g(6), obtained from the function g(x) = 2·x² - 9, can be found from plugging in x = 6, in the function as follows;
g(6) = 2 × 6² - 9 = 63
b. The value of g(-7), obtained from the function g(x) = 2·x² - 9, can be found from plugging in x = -7, in the function as follows;
g(-7) = 2 × (-7)² - 9 = 89
c. The function, g(x) = 0, when we get;
g(x) = 2·x² - 9 = 0
2·x² = 9
x² = (9/2)
x = ±√(9/2) = ±3/√2 = ±3·(√2)/2
d. The function, g(x) = -9, when we get;
g(x) = 2·x² - 9 = -9
2·x² = -9 + 9 = 0
x² = 0/2 = 0
x = √0 = 0
e. The function, g(x) = -37, when we get;
g(x) = 2·x² - 9 = -37
2·x² = -37 + 9 = -28
x² = -28/2 = -14
x = ±√(-14) = ±(√14)·i
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What geometric figure has two circular bases lying in parallel planes? A. cone B. cylinder C. prism D. pyramid?
Answers
A cylinder which is option B is a geometric figure which has two circular bases lying in parallel planes.
One of the fundamental three-dimensional forms in geometry is the cylinder, which has two distant, parallel circular bases. At a certain distance from the centre, a curved surface connects the two circular bases. The axis of the cylinder is the line segment connecting the centres of two circular bases. The height of the cylinder is defined as the distance between the two circular bases.
The cylinder has two key characteristics, namely surface area and volume, because it is a three-dimensional form. The combined curved surface area of the cylinder and the areas of the two circular bases make up its whole surface area. A cylinder's volume is the term used to describe the three-dimensional space it takes up.
A cylinder is a three-dimensional solid in mathematics that maintains, at a given distance, two parallel bases connected by a curving surface. These bases often have a circular form (like a circle), and a line segment known as the axis connects the centres of the two bases.
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What is the slope of the graph of y = 12x - 197
Answers
Answer:
Slope = 12Step-by-step explanation:
Since the equation above is a linear graph
Equation of a line is y = mx + c
where
m is the slope
c is the y intercept
From the question the equation is
y = 12x - 197
Comparing the equation with the general equation above
Slope / m = 12Hope this helps you
Ten different families were tested for the average number of gallons of water they used per day before and after viewing a conservation video. A 90% confidence interval for the difference of the means after and before the training, was determined to be (−10.8,−4.2)
a. Based on this sample, we are 90% confident that the average decrease in daily water consumption after viewing the conservation video is between 4.2 and 10.8 gallons.
b. We are 90% confident that a randomly selected family who has viewed the video will use between 4.2 and 10.8 fewer gallons of water per day compared to a randomly selected family who has not viewed the video.
c. We know that 90% of families will use between 4.2 and 10.8 fewer gallons of water each day after viewing the conservation video.
d. We are 90% confident that a randomly selected family who has viewed the video will use between 4.2 and 10.8 more gallons of water per day compared to a randomly selected family who has not viewed the video.
e. Based on this sample, we are 90% confident that the average increase in daily water consumption after viewing the conservation video is between 4.2 and 10.8 gallons.
Answers
Answer:
a. Based on this sample, we are 90% confident that the average decrease in daily water consumption after viewing the conservation video is between 4.2 and 10.8 gallons.
Step-by-step explanation:
From the given information:
we learnt that :Ten different families were tested for the average number of gallons of water they used per day before and after viewing a conservation video.
A 90% confidence interval for the difference of the means after and before the training, was determined to be (−10.8,−4.2)
Confidence interval shows the range of values with the likelihood to contain a true population value with a certain degree of confidence. In confidence interval, a true population mean lies within the interval of a lower limit and upper limit.
From the given information; the lower limit is -10.8 and the upper limit is -4.82; based on this negative sign, it means they are both decreasing.
Therefore; we can conclude from the given option that :
Based on this sample, we are 90% confident that the average decrease in daily water consumption after viewing the conservation video is between 4.2 and 10.8 gallons.
help i don’t get this
Answers
Answer:
0.42
Step-by-step explanation:
4.2 * 1/10 = 4.2/10
= 0.42
Given that f(x)=4x−8 and g(x)=4−x^2
, calculate (a) f(g(0))= (b) g(f(0))=
Answers
The values of f(g(0)) and g(f(0)) are 8 and -60, respectively.
Given that f(x)=4x−8 and g(x)=4−x²
Calculate:(a) f(g(0))(b) g(f(0))
Solution:(a)
To find f(g(0)), we first need to calculate g(0) and then use the result in the f(x) function.
The calculation is shown below:
g(x) = 4 - x²g(0)
= 4 - 0²g(0)
= 4f(g(0))
= f(4)f(x)
= 4x - 8f(4)
= 4(4) - 8f(4)
= 16 - 8f(g(0))
= f(g(0))
= 16 - 8
= 8(b)
To find g(f(0)), we first need to calculate f(0) and then use the result in the g(x) function.
The calculation is shown below:
f(x) = 4x - 8f(0)
= 4(0) - 8f(0)
= -8g(f(0)) = g(-8)g(x)
= 4 - x²g(-8)
= 4 - (-8)²g(-8)
= -60g(f(0))
= g(-8)
= -60
Therefore, the values of f(g(0)) and g(f(0)) are 8 and -60, respectively.
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