Using the binomial theorem, we get the value of n to be 1.
The given expression can be simplified by factoring out (1 + x)² from each term, which gives
(1 + x)²[1 + (1 + x) + (1 + x)² + ... + (1 + x)⁴⁷ + (1 + x)⁴⁸]
We can then use the formula for the sum of a geometric series to simplify the expression inside the brackets
1 + (1 + x) + (1 + x)² + ... + (1 + x)⁴⁷ + (1 + x)⁴⁸ = (1 - (1 + x)⁴⁹)/(1 - (1 + x))
= ((1 - (1 + x)⁴⁹)/x)(1/x - 1)
Now, we can use the binomial theorem to expand (1 + x)⁴⁹ and simplify the resulting expression
(1 + x)⁴⁹ = ⁴⁹C₀ + ⁴⁹C₁x + ⁴⁹C₂x² + ... + ⁴⁹C₄₈x⁴⁸ + ⁴⁹C₄₉x⁴⁹
The coefficient of x² is ⁴⁹C₂ + ⁴⁹C₃, which simplifies to (3n + 1)⁵¹C₃. Therefore, we have
⁴⁹C₂ + ⁴⁹C₃ = (3n + 1)⁵¹C₃
We can solve for n by using the formula for binomial coefficients
ⁿCᵣ = n!/(r!(n-r)!)
Substituting this into the above equation and simplifying, we get
(49 × 48)/(2 × 1) + (49 × 48 × 47)/(3 × 2 × 1) = (3n + 1) × (3n) × (3n - 1) × ... × (3n - 50)
Simplifying the left-hand side and dividing both sides by (3n + 1), we get
(49 × 16) + (49 × 24 × 47)/(3 × 2) = (3n) × (3n - 1) × ... × (3n - 50)
784 + 2,352 = (3n) × (3n - 1) × ... × (3n - 50)
3,136 = (3n) × (3n - 1) × ... × (3n - 50)
Since 3n is a factor of the right-hand side, we can write
3n × k = 3,136
where k is a product of the remaining factors on the right-hand side. Since n is the smallest positive integer that satisfies this equation, we can start by letting n = 1 and finding the smallest factor k that is divisible by 3. We find that k = 16 × 31 × 32 × ... × 99. Therefore, the value of n is 1
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--The given question is incomplete, the complete question is given
" Let m be the smallest positive integer such that the coefficient of x²
in the expansion of (1 + x)² + (1 + x)³ + (1 + x)⁴ + ........... + (1 + x)⁴⁹ + (1 + x)⁵⁰
is (3n+1)⁵¹C₃
for some positive integer n. Then the value of n is-"--
ASAP NEED HELP!!!!!!!!!!
Answer:
37.5 liters of water
Step-by-step explanation:
We Know
Mrs.Lim bought 25 bottles of water.
Each bottle contained 1[tex]\frac{1}{2}[/tex] liter of water.
How many liters of water did she buy altogether?
We Take
1.5 · 25 = 37.5 liters of water
So, she bought 37.5 liters of water.
you have a bag of orange, green and yellow skittles. there are 9 of each color in the bag. what is the probability of pulling two skittles of the same color in a row?
The probability of pulling two skittles of the same colour in a row is 4/13.
Finding the likelihood that the first skittle will be any colour and then the likelihood that the second skittle will be the same colour as the first will allow you to calculate the likelihood of pulling two Skittles of the same colour in a row.
There are a total of 27 Skittles in the bag because there are 9 of each colour there. As a result, there is a one-in-one chance that the first skittle will be of any hue.
There will be a total of 26 skittles left in the bag when the first skittle is pulled, along with 8 skittles of the same colour. The likelihood that the second skittle will be the same colour as the first is thus
Probability = 8/26 or 4/13.
By adding these probabilities, we obtain:
1 x 4/13 = 4/13
Therefore, the probability of pulling two Skittles of the same colour in a row is 4/13.
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7. Why might someone say that you only need to know how to multiply whole numbers to be able to multiply decimals?
What is the area of this figure 9mm 7mm 2mm 3mm 10mm
The area of this figure based on the information will be 10mm²
How to calculate the areaThe area of a shape simply means the total space that is taken by the shape. It simply expresses the extent of the region on a particular plane as well as a curved surface.
From the information, the shape gas sides 5mm and 2mm. The area of this figure will be:
= Length × Width
= 5mm × 2mm
= 10mm²
In conclusion, the area of this figure based on the information will be 10mm²
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A shape gas sides 5mm and 2mm. What is the area of this figure 9mm 7mm 2mm 3mm 10mm
draw the a concave hexagon with two pairs of congruent sides
Concave hexagon with two pairs of congruent sides is:
/\
/ \
/ \
/ \
/ \
/_____\
\ /
\ /
\ /
\ /
\/
How to draw a concave hexagon with two pairs of congruent sides?A hexagon is a six-sided polygon. A concave hexagon is a hexagon with at least one interior angle greater than 180 degrees.
We can start with a regular hexagon, which has six congruent sides and six interior angles of 120 degrees each.
We can then extend two opposite sides of the regular hexagon outwards to create two pairs of congruent sides. The extended sides should not be parallel to each other, but rather at an angle, so that the resulting hexagon is concave.
Here is an example of a concave hexagon with two pairs of congruent sides:
/\
/ \
/ \
/ \
/ \
/_____\
\ /
\ /
\ /
\ /
\/
In this hexagon, sides AB and CD are congruent, and sides EF and GH are also congruent. However, the angles at vertices A and H are greater than 180 degrees, making this a concave hexagon.
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What is the best estimate of V18?
O 4.2
0 4.4
O 3.6
O 4.9
HELP MEEE HURRY PLEASE
Answer:
(a) 4.2
Step-by-step explanation:
You want an estimate of √18 to one decimal place.
Square rootThe squares on either side of 18 are 4² = 16 and 5² = 25. The value 18 is about (18 -16)/(25 -16) = 2/9 of the way between these numbers. So, an estimate of the square root is ...
4 + 2/9 ≈ 4.2
__
Additional comment
A fast way to improve the estimate is to take the average of this value and the quotient of 18 and this value:
(4.2 +18/4.2)/2 ≈ 4.2429
Compare this to the actual root of 4.2426.
Using that same method again gives a value that matches the root when rounded to 8 decimal places.
There are 20 counters in a bag.
8 of the counters are yellow.
12 of the counters are green.
Asif takes at random two of the counters.
Work out the probability that the two counters are different colours.
The probability of selecting two different counters that are of two different colors is 0.193
What is the probability that the two containers counters are different colors?In the given question, we have a total of 20 counters, 8 are yellow and 12 are green.
The probability of choosing a 2 different colors will be;
P = 7/20 * 11/20
P = 77 / 400
P = 0.193
This is because the first probability of choosing a green is 11/20 and the second probability of choosing a yellow is 7/20. This is an example of probability without replacement
When we multiply both together to determine the probability of choosing different colors, we would have 0.193
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Solve for x.
Set up the proportion.
The value of x is 12
What are similar triangles?Similar triangles are triangles that have the same shape, but their sizes may vary. For two triangles to be similar, the corresponding angles of the triangles must be equal.
Also the ratio of the corresponding sides of the similar triangles are equal.
Therefore ;
x/4 = 36/x
x² = 36×4
x² = 144
x = √144
x = 12
therefore the value of x using similar triangles is 12.
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Kevin bought a new plan the plan girls 2. 8 cm in the first week and 1. 97 CM the second week. Select all of the statements that are true about the plant
The statements that are true about the plant are:
The plant has grown a total of 4.77 cm in two weeksThe plant grew at a faster rate in the first week than the second weekWhat is true of the plant ?The plant did grow 4. 77 cm in two weeks as shown below :
= 2. 8 cm in first week + 1. 97 cm in second week
= 2. 8 + 1. 97
= 4. 77 cm
The plant grew faster in the first week than the second as the plant grew 2.8 cm in the first week and 1.97 cm in the second week, indicating a slower growth rate in the second week compared to the first week.
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Options for this question include:
The plant has grown a total of 4.77 cm in two weeks
The plant grew at a faster rate in the first week than the second week
The plant will be 100 cm tall after 50 weeks
The plant will require at least 5 cm of water per week to continue growing
The plant will die if it doesn't receive enough sunlight each day
Based on the information above what persent of students type less than 30 words per minute
The percentage of students that type less than 30 words per minute are given as follows:
30.61%.
What is an histogram?An histogram is a graph that shows the number of times each element of x was observed, or the number of observations in each range of a data-set.
For the histogram in this problem, we have that:
5 students type between 0 and 14 words per minute.10 students type between 15 and 29 words per minute.14 students type between 30 and 44 words per minute.20 students type between 45 and 59 words per minute.The total number of students is given as follows:
5 + 10 + 14 + 20 = 49.
15 students type less than 30 words per minute, hence the percentage is given as follows:
P = 15/49 x 100%
P = 30.61%.
Missing InformationThe histogram is given by the image presented at the end of the answer.
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actoring Quadratic Expressions. Factor each completely. 1) x. 2 − 7x − 18. 2) p. 2 − 5p − 14. 3) m. 2 − 9m + 8.
Completely factored expressions are:
(x - 9)(x + 2)(p - 7)(p + 2)(m - 1)(m - 8)How to evaluate each part of the question?1. x² - 7x - 18 can be factored as:
(x - 9)(x + 2)
Expand the expression using FOIL:
(x - 9)(x + 2) = x² + 2x - 9x - 18 = x² - 7x - 18
2. p² - 5p - 14 can be factored as:
(p - 7)(p + 2)
Expand the expression using FOIL:
(p - 7)(p + 2) = p² + 2p - 7p - 14 = p² - 5p - 14
3. m² - 9m + 8 can be factored as:
(m - 1)(m - 8)
Expand the expression using FOIL:
(m - 1)(m - 8) = m² - 8m - m + 8 = m² - 9m + 8
Therefore, the completely factored expressions are:
(x - 9)(x + 2)(p - 7)(p + 2)(m - 1)(m - 8)Learn more about factored expressions.
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35 Two solids are described in the list below. • The other solid is a cylinder with a radius of 6 inches and a height of 6 inches. What is the difference between the volumes, in cubic inches, of the solids in terms of it? A 72π B C D • One solid is a sphere and has a radius of 6 inches. 144π 216π 288T
The difference between the volumes of the solids is given as follows:
A. 72π.
How to obtain the volumes?The volume of a sphere of radius r is given by the equation presented as follows:
V = 4πr³/3.
The radius is of 6 inches, hence the volume is given as follows:
V = 4π x 6³/3
V = 288π cubic inches.
The volume of a cylinder of radius r and height h is given by the equation presented as follows:
V = πr²h.
Both the radius and the height are of 6 inches, hence the volume is given as follows:
V = π x 6³
V = 216π cubic inches.
Then the difference of the volumes is given as follows:
π(288 - 216) = 72π.
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The largest numbers that can be divided by (without a remainder) both the numbers being compared (can be 1, can't be bigger than either number) is called the ______.
Answer:
GCF, Greatest Common Factor
Step-by-step explanation:
Hope this helped
What is the value of x? Responses 6 6 62√ 6 square root 2 12 12 122√
The value of x as given in the above triangle is x = 12
Why is this so?A hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle, in geometry.
The Pythagorean theorem, states that the square of the length of the hypotenuse is the same as the sum of the squares of the lengths of the other two sides, may be utilized in the calculation the length of the hypotenuse.
Since we have a 45-45-90 triangle, so the sides opposite of the 45-degree angle are the same, while the hypotenuse is √2
6√2(√2) = 6(2) = 12
x = 12
Recall that two of the same roots will become one whole number, for example
√2(√2) = 2
√3(√3) = 3
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the ages of the members of a gym have a mean of 48 years and a standard deviation of 10 years. what can you conclude from chebyshev's theorem about the percentage of gym members aged between 26 and 70?
Using Chebyshev's theorem, we can conclude that at least 56% of the gym members' ages fall between 26 and 70.
Chebyshev's theorem states that for any given dataset, regardless of its distribution, at least 1 - 1/k² of the data will fall within k standard deviations from the mean.
We can use Chebyshev's theorem to determine the percentage of gym members aged between 26 and 70. We need to determine how many standard deviations away from the mean these ages are.
To do this, we calculate the z-scores for both ages using the formula z = [tex](x - μ) / σ[/tex], where x is the value (in this case, either 26 or 70), [tex]μ[/tex] is the mean (48), and σ is the standard deviation (10). For x = 26: z = (26 - 48) / 10 = -2.2 For x = 70: z = (70 - 48) / 10 = 2.2
Both ages are 2.2 standard deviations away from the mean. Using Chebyshev's theorem, we know that at least 1 - 1/2.2² = 56% of the gym members' ages fall within this range.This is a lower bound - the actual percentage of gym members aged between 26 and 70 could be higher than 56%, as the distribution of ages may not be perfectly symmetrical or may have outliers.
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Consider the graph of the function f(z) = ² + 1.
Y
-6
-2
tum. All rights reserved.
8
6
2:
-2-
0
2
4
-X
The inverse of function f is a logarithmic function. The inverse of function f has a domain of
and a range of
The domain of f^(-1)(x) is [1, infinity), and the range of f^(-1)(x) is all real numbers.
We are given that;
Function f(z) = Y² + 1.
Now,
Step 1: Replace f(z) with y
y = z^2 + 1
Step 2: Swap x and y
z = y^2 + 1
Step 3: Solve for y
y^2 = z - 1
y = sqrt(z - 1)
Step 4: Replace y with f^(-1)(x)
f^(-1)(x) = sqrt(x - 1)
This is the inverse function of f(z).
Therefore, by domain and range answer will be all real numbers, and the range of f(z) is [1, infinity).
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James
Juan buys 3 packs of jelly beans.
If each package costs $3.97, how much will Juan pay before tax?
The cost of producing pens is $200 at the break-even point. What is the amount of income from selling pens at the break-even point? Explain your reasoning.
Answer: No profit
Step-by-step explanation: because break even point are when produce costs and selling costs are the same therefore no profit
If this helped brainliest would help! :)
Help wanted! A triangular pyramid has a base shaped like an equilateral triangle. The legs of the equilateral triangle are all 11. 7 millimeters long, and the height of the equilateral triangle is 10. 1 millimeters. The pyramid's slant height is 12. 3 millimeters. What is its surface area?
The surface area of the triangular pyramid is approximately 435.66 square millimeters.
The surface area of the triangular pyramid can be calculated by finding the area of the equilateral triangle base and the area of the three triangular faces, and then adding them together.
To find the area of the equilateral triangle base, we use the formula A = (√(3)/4) * a², where a is the length of one of the legs.
A = (√(3)/4) * 11.7² = 72.9444 mm²
To find the area of each triangular face, we use the formula
A = (1/2) * base * height
where the base is the side length of the equilateral triangle and the height is the slant height of the pyramid.
A = (1/2) * 11.7 * 12.3 = 71.9425 mm²
So, the surface area of the triangular pyramid is:
SA = 3A(Base) + 3A(face)
SA = 3(72.9444) + 3(71.9425)
SA = 435.6621 mm²
Therefore, the surface area of the triangular pyramid is approximately 435.66 square millimeters.
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please help urgently
The size of the angle TAC is calculated to be equal to 70.52° to the nearest hundredth degree using trigonometric ratios of cosine and tangent.
What is trigonometric ratios?The trigonometric ratios is concerned with the relationship of an angle of a right-angled triangle to ratios of two side lengths.
The basic trigonometric ratios includes;
sine, cosine and tangent.
Considering the right triangle ABM;
cos A = AM/AB {adjacent/hypotenuse}
cos 45° = AM/6
AM = 6 × cos 45 {cross multiplication}
AM = 6 × √2/2 {recall cos 45° = √2/2}
AM = 3√2
For the right triangle ATM;
tan A = TM/AM {opposite/adjacent}
tan A = 12/3√2
tan A = 2√2
A = tan⁻¹(2√2)
A = 70.5288
Therefore, the size of the angle TAC is calculated to be equal to 70.53° to the nearest hundredth degree using trigonometric ratios of cosine and tangent.
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driving 268 miles in 4 hours is a total rate of _ miles per hour
Answer:
67 miles per hour
Step-by-step explanation:
We can set up a proportion to find how many miles per hour this person travels:
268/4
=67
So, the rate is 67 miles per hour.
Hope this helps :)
— driving 268 miles in 4 hours is a total rate of 67 miles per hour
why???
268/4 = 67
The temperature in an oven changes from 350 to 362. what is the percent increase in temperature, to the nearest tenth of a percent
The percent increase in temperature when an oven changes from 350 to 362 is approximately 3.4%.
To calculate the percent increase in temperature, we need to find the difference between the initial temperature and the final temperature, divide that difference by the initial temperature, and then multiply the result by 100 to convert it to a percentage.
The difference between the initial temperature of 350 and the final temperature of 362 is
362 - 350 = 12
To find the percent increase, we divide the difference by the initial temperature
12 / 350 = 0.0342857...
Multiplying by 100, we get
0.0342857... * 100 = 3.42857...
Rounding to the nearest tenth of a percent, we get
3.4%
Therefore, the percent increase in temperature is approximately 3.4%.
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Pls help I’m so confused
Answer:
∠ E = 70° , ∠ F = 40° , ∠ G = 70°
Step-by-step explanation:
given EF = FG then Δ EFG is isosceles with base angles congruent, that is
∠ G = ∠ E = 5n
the sum of the 3 angles in a triangle = 180°
sum the 3 angles and equate to 180
4n - 16 + 5n + 5n
14n - 16 = 180 ( add 16 to both sides )
14n = 196 ( divide both sides by 14 )
n = 14
Then
∠ E = 5n = 5(14) = 70°
∠ F = 4n - 16 = 4(14) - 16 = 56 - 16 = 40°
∠ G = 5n = 5(14) = 70°
You can use a system of equations to graph and solve the polynomial equation 3x³ + x-2x+1. Which statement is
true?
O The system has one solution, and the equation has three zeroes.
The y-coordinates of the solutions to the system and the zeroes of the equation are not equal.
The x-coordinates of the solutions to the system and the zeroes of the equation are not equal.
O The equation has one zero, and the system has three solutions.
The statement which is true include the following: B. The y-coordinates of the solutions to the system and the zeroes of the equation are not equal.
What is a polynomial graph?In Mathematics and Geometry, a polynomial graph simply refers to a type of graph that touches the x-axis at zeros, roots, solutions, x-intercepts, and factors with even multiplicities.
Generally speaking, the zero of a polynomial function simply refers to a point where it crosses or cuts the x-axis of a graph.
By critically observing the graph of the polynomial function shown in the image attached below, we can logically deduce that the y-coordinates of the solutions to the system of equations and the zeros of the equation are not equal;
3x³ + x = 2x + 1
P(x) = 3x³ + x - 2x - 1
3x³ + x - 2x - 1 = 0
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Una empresa maderera necesita estimar el volumen de madera que producirá una carga de madera. El supervisor sabe que el volumen de madera en un árbol varía conjuntamente con la altura h y el cuadrado de la circunferencia del árbol g. El supervisor observa que un árbol de 40 metros de altura con una circunferencia de 1,5 metros produce 288 metros cúbicos de madera. Escribe una ecuación que represente esta situación
In the above case, the law of joint proportionality is the one I am going to use to write the equation that stands for the above situation:
V = k x h x g²
What is the above equation about?Note that from the equation written:
V = the volume of wood produced by the tree
h = the height of the tree
g = the circumference of the tree
k = a constant of proportionality.
For one to know the value of k, we have to make use the data given in the problem.
Since h = 40 m
g = 1.5 m
V = 288 m³
Hence the volume of wood produced is:
288 = k x 40 x (1.5)²
Simplifying this equation, we have:
288 = k x 90
When we Divide both sides by 90, we have :
k = 3.2
So one can say the equation that stands the situation is:
V = 3.2 x h x g²
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Solve the system of equations using elimination -3y - 4x = 15 and 9y+7x = 25
Answer:
y=41/3 and x = -14
Step-by-step explanation:
system of the linear equation :
-3y-4x=15 eq[i]
9y+7x=25 eq[ii]
multiplying eq[i] by -7 and eq[ii] by 4,we get
21y+28x= -105 eq[iii]
36y+28x= 100 eq[iv]
subtracting eq[iii] from eq[iv],we get
15y=205
y=205/15=41/3
and x= -14
what is the surface area of this solid?
A: 37.68
B: 40.82
C: 28.26
D: 31.4
The surface area of the given shape above would be = 47.2
How to calculate the surface area of the given solid shape?To determine the surface area of the solid shape above,the surface area of a cone and a cylinder is first determined and then added together.
The formula for the surface area (SA) of cone = πrs + πr²
where;
r = 1
s = 6
S.A = 3.14×1×6 + 3.14×1×1²
= 18.84+3.14
= 21.98
Formula for Surface area of cylinder = 2πr(r+h)
where;
r = 1
h = 3
SA = 2×3.14×1 (1+3)
= 25.12
Therefore, the surface area of the solid shape = 21.98+25.22 = 47.2
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The list price of a new printer is $8,000. The chain discount is 20/15/10. What's the net price?
A. $3,204
B. $3,304
C. $4,103
D. $4,896
Please help! Easy question, answer pls, 15 pts.
Answer:
360 in
Step-by-step explanation:
To figure out how many inches the dressmaker has in 10, 3 ft rolls, we can multiply by the conversion ratio:
[tex]\dfrac{12 \text{ in}}{1\text{ ft}} \\ \\ \text{} \ \ \implies (10 \cdot 3 \text{ ft}) \cdot \dfrac{12 \text{ in}}{1\text{ ft}} \\ \\ \text{} \ \ \implies 30 \text{ ft} \cdot \dfrac{12 \text{ in}}{1\text{ ft}} \\ \\ \text{} \ \ \implies 30\cdot 12 \text{ in} \\ \\ \text{} \ \ \implies \boxed{360 \text{ in}}[/tex]
So, the dressmaker has 360 in of ribbon.
help me answer these equations PLEASEEE
The values of the missing sides are;
1. 6. 7 Option C
2. 6. 9. Option D
3. 7. 6 Option C
4.6.3 Option C
How to determine the valuesFrom the information given, we have that;
1. Using the sine identity;
sin 72 = x/7
cross multiply the values
x = 6. 66
2. Using the sine identity;
sin 60 = x/8
cross multiply the values
x = sin 60(8)
find the value
x = 6. 93
3. Using the cosine identity;
cos 23 = 7/x
cross multiply the values
x = 7/0. 9205
x = 7. 60
4. Using the cosine identity;
cos 25 = x/7
cross multiply the values
x = 6. 34
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