in the expression of (x + 1)^n, the coefficient of the x^3 term is two times the coefficient of the x^2 term. Find the value of n.
Answers
According to Khayyam-Pascal triangle ,
the value of n is 8.
\( look \: \\ \\ {(x + 1)}^{8} = {x}^{8} + 8 {x}^{7} + 28 {x}^{6} + 56 {x}^{5} + 70 {x}^{4} + 56 {x}^{3} + 28 {x}^{2} + 8x + 1 \\ \frac{56}{28} = 2 \)
Recall the binomial theorem:
\((a+b)^n=\displaystyle\sum_{k=0}^n\binom nk a^k b^{n-k}\)
where
\(\dbinom nk=\dfrac{n!}{k!(n-k)!}\)
is the binomial coefficient. So
\((x+1)^n=\displaystyle\sum_{k=0}^n\binom nk x^k\)
The x² term occurs for k = 2, which has coefficient
\(\dbinom n2=\dfrac{n!}{2!(n-2)!}=\dfrac{n(n-1)}2\)
The x³ occurs for k = 3, with coefficient
\(\dbinom n3=\dfrac{n!}{3!(n-3)!}=\dfrac{n(n-1)(n-2)}6\)
The latter coefficient is twice the other one, so that
\(\dfrac{n(n-1)(n-2)}6=2\cdot\dfrac{n(n-1)}2\)
Solve for n :
\(\dfrac{n(n-1)(n-2)}6=n(n-1)\)
\(n(n-1)(n-2)=6n(n-1)\)
\(n-2=6\)
\(\boxed{n=8}\)
Related Questions
what is 3 1/6 divided by 1 3/8
Answers
The result of the division of dividend 3 1/6 by divisor 1 3/8 is computed as 76/33.
Define the method of division?A division method is the process used to divide something. According to the degree of difficulty, there are three different sorts of division procedures. These include the bus stop method, extended division method, and the chunking approach, often known as division by repeated subtraction.For the stated question;
The dividend is 3 1/6, which is given in mixed fraction.
In proper fraction; 3 1/6 = 19/6.
The divisor is 1 3/8 = 11/8.
Thus,
= 19/6 / 11/8
= 19*8 / 6*11
= 76/33
Thus, the result of the division of dividend 3 1/6 by divisor 1 3/8 is computed as 76/33.
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Moe is 8 years older than Zach. If twenty years ago Moe was 3 times as old as Zach, what are their present ages.Moe: 32 Zach: 24Please include the steps taken to get the answer.
Answers
Moe's present age = 32
Zach's present age = 24
Explanations:Let Moe's present age be represented by m
Let Zach's present age be represented by z
Moe is 8 years older than Zach now
m = z + 8..........(1)
Twenty years ago, Moe's age would be m - 20
Twenty years ago, Zach's age would be z - 20
Twenty years ago, Moe's age was three times that of Zach
m - 20 = 3(z - 20)
m - 20 = 3z - 60
m = 3z - 60 + 20
m = 3z - 40..........(2)
Let equations (1) and (2) be equal to each other:
z + 8 = 3z - 40
3z - z = 8 + 40
2z = 48
z = 48 / 2
z = 24
Substitute the value of z into equation (1)
m = z + 8
m = 24 + 8
m = 32
Moe's present age = 32
Zach's present age = 24
A (2,5 and B(-3,9)
1.The Midpoint of the straight
Answers
Answer:
Step-by-step explanation:
To figure out the midpoint, you need to plug the coordinates into the midpoint formula.
M (x1 + x2 / 2, y1 + y2/2)
M [ 2 + (-3)/2, 5 + 9/2]
M ( -1/2, 7)
The continuous random variable V has a probability density function given by: 6 f(v) = for 3 ≤ ≤7,0 otherwise. 24 What is the expected value of V? Number
Answers
The expected value of the continuous random variable V is 5. The expected value of V is 5, indicating that, on average, we expect the value of V to be around 5.
To calculate the expected value of a continuous random variable V with a given probability density function (PDF), we integrate the product of V and the PDF over its entire range.
The PDF of V is defined as:
f(v) = 6/24 = 0.25 for 3 ≤ v ≤ 7, and 0 otherwise.
The expected value of V, denoted as E(V), can be calculated as:
E(V) = ∫v * f(v) dv
To find the expected value, we integrate v * f(v) over the range where the PDF is non-zero, which is 3 to 7.
E(V) = ∫v * (0.25) dv, with the limits of integration from 3 to 7.
E(V) = (0.25) * ∫v dv, with the limits of integration from 3 to 7.
E(V) = (0.25) * [(v^2) / 2] evaluated from 3 to 7.
E(V) = (0.25) * [(7^2 / 2) - (3^2 / 2)].
E(V) = (0.25) * [(49 / 2) - (9 / 2)].
E(V) = (0.25) * (40 / 2).
E(V) = (0.25) * 20.
E(V) = 5.
Therefore, the expected value of the continuous random variable V is 5.
The expected value represents the average value or mean of the random variable V. It is the weighted average of all possible values of V, with each value weighted by its corresponding probability. In this case, the expected value of V is 5, indicating that, on average, we expect the value of V to be around 5.
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e ohio lottery has a game called pick 4 where a player pays $1 and picks a four-digit number. if the four numbers come up in the order you picked, then you win $3900. a) write the probability distribution for a player's winnings. fill in the table below. for the computer to grade this one correctly make sure that your x values are from smallest to largest.
Answers
The probability of winning $3,899 is 0.0001, which is a very small probability, but still possible.
To write the probability distribution for a player's winnings in the Pick 4 game, we need to consider all the possible outcomes and their probabilities.
There are a total of 10,000 possible four-digit numbers that can be drawn in the game. Since the player has to match the numbers in the exact order, there is only one winning combination for each four-digit number. Therefore, the probability of winning is 1/10,000.
To calculate the player's winnings, we need to subtract the $1 cost of playing from the $3,900 prize. Thus, the player's net winnings can be calculated as follows:
Net Winnings = $3,900 - $1 = $3,899
The probability distribution for the player's winnings can be summarized in the following table:
| Winnings (x) | Probability (P) |
|--------------|-----------------|
| $0 | 0.9999 |
| $3,899 | 0.0001 |
Note that the table shows the possible winnings (x) in ascending order, as requested. The probability of winning $0 is 0.9999, which means that the player is most likely to lose their $1 bet.
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what is the domain of a graph below?
Answers
Answer:
-4,0
Step-by-step explanation:
Which symbol correctly compares the fractions below?
1/8 ? 1/2
Answers
Answer:
< "less than sign"
Step-by-step explanation:
1/8 is 0.125 in decimal form.
1/2 is 0.5 in decimal form.
Therefore, 1/2 is greater than 1/8 so you would use the less than symbol (<) for a final answer of 1/8<1/3.
Which number line represents the solution set for the inequality 3(8 - 4x) < 6(x - 5)?
+
4
-5
-4
-3
-2
-1 0
1 2 3
5
+
+
-5
+ + +
-2 -1 0
+
2 3 4 5
-4
-3
1
út
+
+
+
-2 -1 0
-4
+
3
-5
-3
+
4
1
2
5
-5
-4
-3
-2 -1 0
1
2 3
4 5
Zoom in to see number line clearly
Answers
so I guess it should be the THIRD ONE and sorry if my answer is wrong but i tried my best so enjoy!
What expression is equivalent to 1/3 x -1/5
Answers
a travel agency is giving away a free trip as part of a grand opening. the trip will be to one of the ten locations listed. london paris rome sydney athens nassau hong kongmadridtokyorio de janeiro what is the probability that a trip will be to either nassau or paris?
Answers
Answer: 1/5
Step-by-step explanation:
The probability that a trip will be to either Nassau or Paris is 0.2 or 20%.
Given, a travel agency is giving away a free trip as part of a grand opening.
The trip will be to one of the ten locations listed. The 10 locations are London, Paris, Rome, Sydney, Athens, Nassau, Hong Kong, Madrid, Tokyo, and Rio De Janeiro.
To find the probability, we need to find how many outcomes are favorable to the event "the trip is to Nassau or Paris" and divide by the total number of possible outcomes.
So, favorable outcomes = 2 (Nassau, Paris)
Total possible outcomes = 10
Probability = (Favorable outcomes)/(Total possible outcomes)= 2/10= 0.2 or 20%.
Therefore, the probability that a trip will be to either Nassau or Paris is 0.2 or 20%.
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Identify the expression equivalent to 2x 8 by substituting x = 7 and x = 5. a. 2x 10 10 b. 2(x 2) c. (x 4) (x 4) d. (x 4) e. 2(2x 4)
Answers
the expression equivalent to 2x + 8 is (x+4) + (x+4)
using the commutative property we can say that
2x +8 = x + x + 8
also 2x + 8 = x + 4 + x + 4
hence 2x + 8 = (x+4) + (x+4)
What is an expression?Mathematical statements are called expressions if they have at least two terms that are related by an operator and contain either numbers, variables, or both.Addition, subtraction, multiplication, and division are all possible mathematical operations.There are two sorts of expressions in mathematics: numerical expressions, which only contain numbers, and algebraic expressions, which also include variables.In a mathematical expression, the following terms are used:An absolute numerical number is referred to as a constant.Variable: A symbol without a set value is referred to as a variable.Term: A term can be made up of a single constant, a single variable, or a mix of variables and constants multiplied or divided.a number that is multiplied by a variable is referred to as a coefficient.To know more about variable with the given
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solve: -11 = -2 + x/4
Answers
Answer:
x = -36
Step-by-step explanation:
-11 + 2 = x/4
-9 = x/4
-9 * 4 = x
-36
Finding Missing Sides
These two figures are the image and pre-image of a
dilation.
Find the value of x
4 m
6 m
O 4 m
O 6 m
* 8 m
09 m
6 m
Answers
Answer:
the answer is 9
Step-by-step explanation:
I did this quiz the other day and I'm pretty sure that is the answer I hope this helps
Which of these lists four factors of 12?
O 2,3,4,6
0 4,5,7,8
O 4, 6, 9, 12
O 24, 36, 48, 60
Answers
Answer:
2346 is the factor of 12 hope it helps
(9).Suppose(r,s) satisfy the equation r+5s=7and-2r-7s=-5 .Find the value of s.
a)-8 b)3 c) 0 d) -1/4 e) none of these (10). Which of the following matrices are orthogonal 20 117 iii) 13-5 iv) 0 02 -1
Answers
A rectangular array of characters, numbers, or phrases arranged in rows and columns is known as a matrix. It is a fundamental mathematical idea that is applied in many disciplines, such as physics, mathematics, statistics, and linear algebra.
To solve the system of equations:
r + 5s = 7 ...(1)
-2r - 7s = -5 ...(2)
We can use the method of elimination or substitution. Let's use the method of elimination:
Multiply equation (1) by 2:
2r + 10s = 14 ...(3)
Now, add equation (2) and equation (3) together:
(-2r - 7s) + (2r + 10s) = -5 + 14
3s = 9
s = 9/3
s = 3
Therefore, the value of s is 3.
Answer: b) 3
Regarding the matrices:
i) 20 11
7 -5
ii) 13 -5
-1 2
iii) 0 0
2 -1
iv) 0 0
-1 0
To determine if a matrix is orthogonal, we need to check if its transpose is equal to its inverse.
i) The transpose of the first matrix is:
20 7
11 -5
The inverse of the first matrix does not exist, so it is not orthogonal.
ii) The transpose of the second matrix is:
13 -1
-5 2
The inverse of the second matrix does not exist, so it is not orthogonal.
iii) The transpose of the third matrix is:
0 2
0 -1
The inverse of the third matrix is also:
0 2
0 -1
Since the transpose is equal to its inverse, the third matrix is orthogonal.
iv) The transpose of the fourth matrix is:
0 -1
0 0
The inverse of the fourth matrix does not exist, so it is not orthogonal.
Therefore, the only matrix among the options that is orthogonal is:
iii) 0 2
0 -1
Answer: iii) 0 2
0 -1
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Find the total surface area.
Answer Options:
A. 422 in²
B. 517 in²
C. 554 in²
D. 638 in²
Answers
Answer:
D. 638 in^2
Step-by-step explanation:
total surface area is just the sum of areas of each side of the cube
opposite sides have the same area
top and bottom = 9*11
sides = 11*9
front and back = 11*11
SA = 2(9*11 + 11*9 + 11*11)
SA = 2(99 + 99 + 121)
SA = 2(319)
SA = 638 in^2
what is the indentation diagonal length when a load of 0.700 kg produces a vickers hv of 650
Answers
the indentation diagonal length is approximately 0.0686 units.
What is Intention Diagonal Length?
The indentation diagonal d is determined by the mean value of the two diagonals d 1 and d 2 at right angles to each other: To avoid the risk of bulging of the material on the opposite side of the sample, the thickness should not fall below a certain minimum value. value. The minimum thickness depends on the expected hardness of the material and the test load.
To calculate the indentation diagonal length using the Vickers hardness value, you need to know the applied load and the hardness number. The Vickers hardness test measures the resistance of a material to indentation using a diamond indenter.
In this case, you have the following information:
Load: 0.700 kg
Vickers HV: 650
The Vickers hardness number (HV) is defined as the applied load divided by the surface area of the indentation.
The formula to calculate the indentation diagonal length (d) is:
d = 1.854 * sqrt(L / HV)
Where:
d = indentation diagonal length
L = applied load in kg
HV = Vickers hardness number
Plugging in the values:
d = 1.854 * sqrt(0.700 / 650)
Calculating the square root and performing the division:
d ≈ 1.854 * 0.0370262
d ≈ 0.0686
Therefore, the indentation diagonal length is approximately 0.0686 units. Please note that the specific unit (e.g., millimeters) was not provided in the question, so the answer is given in relative units.
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You have a mortagege of $275,000
after down payment with an interest
rate of 3% for 30 years.
What does P, r,n, and t are equal to?
Answers
Answer:
Our calculator limits your interest deduction to the interest payment that would be paid on a $1,000,000 mortgage. Interest rate: Annual interest rate for this
Step-by-step explanation:
A costs $25000 and each year its value drops by 8%. How much will the car be worth in 3 years? PLEASE I NEED HELP I AM ON A MATH TEST
Answers
Answer:
19467.2
Step-by-step explanation:
Step 1:
25000 * (1 - .08) = 23000 end of year 1 value
Step 2:
23000 * (1 - .08) = 21,160 end of year 2 value
Step 3:
21,160 * (1-.08) = 19467.2 end of year 3 value
What is the speed of a sailboat traveling 120m in 60s?
Answers
A sailboat traveling 120m in 60s is 20.
what is 2/5 close to?
Answers
What is the derivative of sin x with respect to x?
What is the derivative of cos x with respect to x?
What is the antiderivative of sin x with respect to x?
What is the antiderivative of cos x with respect to x?
Answers
Cosx, -sinx, -cosx + C, sinx + C are the derivative answers to the question.
The derivative of sin x with respect to x is cos x.
The derivative of cos x with respect to x is -sin x.
The antiderivative of sin x with respect to x is -cos x + C, where C is the constant of integration.
The antiderivative of cos x with respect to x is sin x + C, where C is the constant of integration.
1. The derivative of sin(x) with respect to x is cos(x).
2. The derivative of cos(x) with respect to x is -sin(x).
3. The antiderivative of sin(x) with respect to x is -cos(x) + C, where C is the constant of integration.
4. The antiderivative of cos(x) with respect to x is sin(x) + C, where C is the constant of integration.
A derivative in mathematics is a measurement of how much a function alters as its input alters. It refers to how quickly a function alters in relation to its input variable. The slope of the tangent line to the function at a certain point is what it is, in other words.
The limit of the ratio of the change in the output to the change in the input, as the change in the input approaches zero, is known as the derivative of a function, indicated by the symbols f'(x) or \(dy/dx\).
The opposite of differentiation is an antiderivative, usually referred to as an indeterminate integral. It is a function that yields the original function when differentiated. In other words, f(x) follows if \(f'(x) = g(x)\).
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The cost, c, of supplies for school depends
on other factors. list at least two
independent variables that could affect
the cost of school supplies.
Answers
Social researchers treat gender identity, ethnicity, race, income, and education as independent variables because they are all significant subject factors.
What is an example of an independent variable?
It is an independent variable that doesn't alter as a result of the other variables you're attempting to assess. An independent variable could be something like a person's age.
Other elements (such as what they consume, how much they attend school, and how much television they watch) won't alter a person's age.
What distinguishes a variable from a dependent one?
The cause is the independent variable. The other factors in your study have no bearing on its value. Effect is the dependent variable.
Changes in the independent variable affect how much it is worth. Characteristics called variables can have various values.
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1.Round 3 601 111 to the nearest million
2. Round 17.81 to the nearest whole number
3. Round 59.52 to the nearest 1 d.p.
4. Round 7.176 to the nearest 2d.p.
(Pls help me…)
Answers
Answer:
See explanation
Step-by-step explanation:
Rules to follow when rounding
1. If the number you are rounding is followed by 5, 6, 7, 8, or 9, round the number up. Eg: 56 rounded to the nearest ten is 60
2. If the number you are rounding is followed by 0, 1, 2, 3, or 4, round the number down. Example: 23 rounded to the nearest ten is 20
3. All the numbers to the right of the place you are rounding to become zero
1.Round 3 601 111 to the nearest million
= 4,000,000
2. Round 17.81 to the nearest whole number
= 18
An whole number is a number without a decimal point
3. Round 59.52 to the nearest 1 d.p.
= 59.5
1 d.p means one digits after the decimal point
4. Round 7.176 to the nearest 2d.p.
= 7.18
2 d.p means two digits after the decimal point
Examine rectangle JKLM, shown below.
If JN=x+3 and JL 3x + 1, determine which of the following values is correct
Answers
Answer:NM=8
Step-by-step explanation:
JN is half of JL so x + 3 times 2 is equal to 3x + 1
then u just do the work
2x+6=3x+1
5=x
then you just plug in the answers and JN is equal to 8 therefore NM is also equal to 8
To solve the problem we must know about the properties of the diagonals of a rectangle.
What are the properties of the diagonals of a rectangle?The diagonals of a rectangle are of equal length.The diagonals of the rectangle bisect each other at equal distances.The correct statement from all the given options is D.
Given to us
JN = x+3
JL = 3x+1
What is the value of x?We know that the diagonals of the rectangle bisect each other at two equal parts, therefore, JN=NL, also
\(JL = JN + NL\\\\\text{Substituting the values,}\\\\3x+1 = x+3 + NL\\\\3x+1 = x+3 + NL\\\\3x-x=3-1+NL \ \ \ \ \ [JN=NL]\\\\2x = 2 + JN\\\\2x = 2 + x+3\\\\2x-x=5\\\\x = 5\)
Value of diagonalsAs we know the values of the diagonals are equal, therefore,
JL = KM = 3x + 1
JL = KM = 3(5) + 1
JL = KM = 16
Value of half DiagonalsWe know that the diagonals of the rectangle bisect each other at two equal parts, also the diagonal are equal therefore all the four parts of the two diagonals will be equal.
JN = NL = KN = NM = x+3
JN = NL = KN = NM = 5+3
JN = NL = KN = NM = 8
Hence, the correct statement from all the given options is D.
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Please please plea help me please
Answers
Answer:
it think its -3,-4
Step-by-step explanation:
i hope its right
Dina can run 4 miles in 30 minutes. If she maintains this pace, how many miles can she run in 90 minutes?
Answer choices
A. 12
B. 16
C. 8
D. 15
Answers
Answer:
15minutes / 2miles = 7.5 minutes to run a single mile.
7.5minutes * 5miles = 37.5minutes
37.5minutes to run 5 miles
Step-by-step explanation:
Answer:
A. 12
Step-by-step explanation:
Set up a proportion: \(\frac{4}{30} = \frac{x}{90}\) Cross multiply, then divide: 4 × 90 = 360, 360 ÷ 30 = 12So, Dina can run 12 miles in 90 minutesI hope this helps!
1. A piece of wire is to be bent to form an arc of a circle. The central angle is 105°and the radius is 14.8 cm. Find the length of the wire. Round to tenths. o 1050 14.8 cm
Answers
To find the length of the wire, we need to use the formula: Length of arc = (central angle / 360) x 2πr Plugging in the given values, we get:
length of arc = (105/360) x 2π(14.8)
length of arc = (0.2917) x (2 x 3.14 x 14.8)
length of arc = 25.9 cm (rounded to tenths)
Therefore, the length of the wire needed to form the arc of the circle is 25.9 cm.
To find the length of the wire, we will use the formula for the arc length of a circle:
Arc length = (Central angle / 360°) × 2π × Radius
1. First, plug in the given values for the central angle (105°) and the radius (14.8 cm):
Arc length = (105° / 360°) × 2π × 14.8 cm
2. Divide 105 by 360:
0.2917 = 105° / 360°
3. Multiply the result by 2π:
0.2917 × 2π = 1.8326π
4. Finally, multiply the result by the radius (14.8 cm):
Arc length = 1.8326π × 14.8 cm ≈ 85.5 cm (rounded to tenths)
So, the length of the wire that forms the arc is approximately 85.5 cm.
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Determine the open intervals on which the graph of the function is concave upward or concave downward. f(x) = x^4 - 3x^3
Answers
The open intervals on which the graph of \(f(x) = x^4 - 3x^3\) is concave upward are (-∞, 0) and (3/2, ∞), and the open interval on which the graph of f(x) is concave downward is (0, 3/2).
What is graph?A graph is a visual representation of data, information, or functions. In mathematics, a graph typically refers to a set of points and lines or curves connecting them, which can be used to represent mathematical relationships and functions.
According to given information:To determine the intervals where the function \(f(x) = x^4 - 3x^3\) is concave upward or concave downward, we need to find the second derivative of the function, which gives us the concavity of the function.
\(f(x) = x^4 - 3x^3\\\\f'(x) = 4x^3 - 9x^2\\\\f''(x) = 12x^2 - 18x\)
For the function f(x) to be concave upward, f''(x) > 0, and for f(x) to be concave downward, f''(x) < 0.
So, we need to solve the inequality f''(x) > 0 and f''(x) < 0:
\(f''(x) > 0:\\\\12x^2 - 18x > 0\\\\6x(2x - 3) > 0\)
The critical points are x = 0 and x = 3/2. We can test each interval:
Interval (-∞, 0):
\(f''(-1) = 12(-1)^2 - 18(-1) = 30 > 0\), so f(x) is concave upward on this interval.
Interval (0, 3/2):
\(f''(1) = 12(1)^2 - 18(1) = -6 < 0,\) so f(x) is concave downward on this interval.
Interval (3/2, ∞):
\(f''(2) = 12(2)^2 - 18(2) = 12 > 0\), so f(x) is concave upward on this interval.
Therefore, the open intervals on which the graph of \(f(x) = x^4 - 3x^3\) is concave upward are (-∞, 0) and (3/2, ∞), and the open interval on which the graph of f(x) is concave downward is (0, 3/2).
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HELPP Write the equation of the given line in slope-intercept form:
Answers
Answer:
y = -3x - 1
Step-by-step explanation:
The slope-intercept form is y = mx + b
m = the slope
b = y-intercept
Slope = rise/run or (y2 - y1) / (x2 - x1)
Point (-1, 2) (1, -4)
We see the y decrease by 6 and the x increase by 2, so the slope is
m = -6 / 2 = -3
Y-intercept is located at (0, - 1)
So, the equation is y = -3x - 1