Help me plzzzzzzz.thank you

To show that `(n + 3)7 ∈ Θ(n7)`, we need to prove that `(n + 3)7 = Θ(n7)`.This can be done by showing that `(n + 3)7 = O(n7)` and `(n + 3)7 = Ω(n7)` .Now, let's prove the two parts separately:

Proof for `(n + 3)7 = O(n7)`.

We want to prove that there exists a positive constant c and a non-negative constant k such that `(n + 3)7 ≤ cn7` for all `n ≥ k`.Using the Binomial theorem, we can expand `(n + 3)7` as:```
(n + 3)7
= n7 + 7n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + 37
≤ n7 + 21n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + n7
≤ 2n7 + 21n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6
≤ 2n7 + 84n6 + 441n5 + 2205n4 + 10395n3 + 45045n2 + 153609n + 729
```Thus, we can take `c = 153610` and `k = 1` to satisfy the definition of big-Oh notation. Hence, `(n + 3)7 = O(n7)`.Proof for `(n + 3)7 = Ω(n7)`We want to prove that there exists a positive constant c and a non-negative constant k such that `(n + 3)7 ≥ cn7` for all `n ≥ k`.Using the Binomial theorem, we can expand `(n + 3)7` as:```
(n + 3)7
= n7 + 7n6(3) + 21n5(3)2 + 35n4(3)3 + 35n3(3)4 + 21n2(3)5 + 7n(3)6 + 37
≥ n7
```Thus, we can take `c = 1` and `k = 1` to satisfy the definition of big-Omega notation. Hence, `(n + 3)7 = Ω(n7)`.

As we have proved that `(n + 3)7 = O(n7)` and `(n + 3)7 = Ω(n7)`, therefore `(n + 3)7 = Θ(n7)`.Thus, we have shown that `(n + 3)7 ∈ Θ(n7)`.From the proof, we can see that we used the Binomial theorem to expand `(n + 3)7` and used algebraic manipulation to bound it from above and below with suitable constants. This technique can be used to prove the time complexity of various algorithms, where we have to find the tightest possible upper and lower bounds on the number of operations performed by the algorithm.

Hence, we have shown that `(n + 3)7 ∈ Θ(n7)` for non-negative integer n.

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Answer:

18/29

Step-by-step explanation:

(p like math) = 18/29

Answer:

17/29

Step-by-step explanation:

We must take into account the amount of students who prefer math over English in order to calculate the probability that a student selected at random like arithmetic but dislikes English.

Let's denote:

X = Number of students who like math = 18 students

Y = Number of students who like English = 12 students

Z = Number of students who don't like math or English = 7 students

We can calculate the number of students who like both math and English using the principle of inclusion-exclusion:

Therefore , Number of students who like both math and English = X + Y - (X ∩ Y)

= 18 + 12 - (Total number of students in the class)

= 18 + 12 - 29

= 1

Now, the probability of selecting a student who likes math but not English is the ratio of the number of students who like math but not English to the total number of students in the class:

Therefore, Probability = Number of students who like math but not English / Total number of students

= 17 / 29

Therefore, the probability that a student chosen at random likes math but not English is 17/29.

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(8) To find the particular solution of (D³ + 12D² + 36D)y = 0 with initial conditions x = 0, y = 0, y' = 1, y" = -7, we can assume a particular solution of the form y = ax³ + bx² + cx + d.

Taking the derivatives:

y' = 3ax² + 2bx + c

y" = 6ax + 2b

Substituting these derivatives into the differential equation, we get:

(6ax + 2b) + 12(3ax² + 2bx + c) + 36(ax³ + bx² + cx + d) = 0

36ax³ + (72b + 36c)x² + (36a + 24b + 36d)x + (2b + 6c) = 0

Comparing coefficients of like powers of x, we can set up a system of equations:

36a = 0 (coefficient of x³ term)

72b + 36c = 0 (coefficient of x² term)

36a + 24b + 36d = 0 (coefficient of x term)

2b + 6c = 0 (constant term)

From the first equation, we have a = 0. We get:

72b + 36c = 0

24b + 36d = 0

2b + 6c = 0

Solving this system of equations, we find b = 0, c = 0, and d = 0. Therefore, the particular solution of (D³ + 12D² + 36D)y = 0 with the given initial conditions is y = 0.

(9) The general solution of y" + 4y = 3sin(2x) is given by y = C₁cos(2x) + C₂sin(2x) - (3/4)cos(2x), where C₁ and C₂ are arbitrary constants.

(10) To find the particular solution of y" + y = cos²x, we can use the method of undetermined coefficients. We can assume a particular solution of the form y = Acos²x + Bsin²x + Ccosx + Dsinx, where A, B, C, and D are constants.

Taking the derivatives:

y' = -2Acosxsinx + 2Bcosxsinx - Csinx + Dcosx

y" = -2A(cos²x - sin²x) + 2B(cos²x - sin²x) - Ccosx - Dsinx

Substituting these derivatives into the differential equation, we get:

(-2A(cos²x - sin²x) + 2B(cos²x - sin²x) - Ccosx - Dsinx) + (Acos²x + Bsin²x + Ccosx + Dsinx) = cos²x

-2Acos²x + 2Asin²x + 2Bcos²x - 2Bsin²x - Ccosx - Dsinx + Acos²x + Bsin²x + Ccosx + Dsinx = cos²x

(-A + B + 1)cos²x + (A - B)sin²x - Ccosx - Dsinx = cos²x

Comparing coefficients of like powers of x, we can set up a system of equations:

-A + B + 1 = 1 (coefficient of cos²x term)

A - B = 0 (coefficient of sin²x term)

-C = 0 (coefficient of cosx term)

-D = 0 (coefficient of sinx term)

From the second equation, we have A = B. Substituting this into the remaining equations, we get:

-A + A + 1 = 1

-C = 0

-D = 0

Simplifying further, we have:

1 = 1, C = 0, and D = 0

From the first equation, we have A - A + 1 = 1, which is true for any value of A. Therefore, the particular solution of y" + y = cos²x is y = Acos²x + Asin²x, where A is an arbitrary constant.

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The statement describes a set of interest in a study, which is the set of all elements being investigated or under consideration. It is not a sample, which refers to a subset of the population being studied.

The set of all elements of interest in a study is called the population. It refers to the entire group of individuals, objects, or measurements that are being studied and from which data can be collected. The population can be finite or infinite and can include people, animals, plants, materials, or any other object or concept of interest. In statistical analysis, researchers use samples of the population to make inferences or draw conclusions about the entire population. By selecting a representative sample, researchers can reduce the cost and time required for data collection and analysis while still obtaining valid and reliable results.

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Answer:

check the attachment and explanation below

Step-by-step explanation:

Explanation for statement 3 :

Suppose The number of shoppers on Saturday is 100 and the number of shoppers on Sunday is 200

Total number of shoppers = 300

On Saturday :

Q₃ = 70 then 25% of shoppers spent more then $70

then 25 shopper spent more then $70

On Sunday:

M’ = 70 then 50% of shoppers spent more then $70

then 100 shopper spent more then $70

On both days:

(100+ 25) shopper spent more then $70

Then 125 shopper spent more then $70

But 125 is not 75% of 100+200=300

Answer:

1. -1/2x+ 11.5

2. -1/3x-8.7

Step-by-step explanation:

Answer:

Step-by-step explanation:

Answer:

5%+30%+10%+5%=50%

Step-by-step explanation:

There is no intersection between the two sets dye and permanent so it means both sets don't have any common thus both will be mutually exclusive events therefore, option (A) will be correct.

A set is a combination of specific quantities in which the meaning of each variable must be the same.

For example set of names starting with A is {Amesh, Aniket, Aakash}

Sets of natural number {1,2,3,4,5....}

As per the given venn diagram,

The sets or events are dye haircuts and permanent.

The mutually exclusive are the events among them nothing have in common.

For example, the set of positive numbers {1,2,3} and the set of negative numbers {-1,-2,-3} now among both sets nothing is common so they will be exclusive events.

Since the intersection area in the Venn diagram represents the common things and among Dye and Permanent, no intersection area is formed thus they will be mutually exclusive events.

Hence "There is no intersection between the two sets dye and permanent so it means both sets don't have any common thus both will be mutually exclusive events".

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Answer:

Step-by-step Solution:

Let's simplify the equation to find the value of x.

[Simplify the Distributive property]

[Subtract 6 both sides]

[Solve the RHS]

[Divide -12 both sides to isolate the variable]

[Simplify the fraction]

Hence, the value of x is -7/4.

Answer:

x = -1.75

Step-by-step explanation:

-2(6x - 3) = 27

distribute the -2

-2 * 6x = -12x

-2 * -3 = 6

-12x + 6 = 27

subtract 6 from both sides

-12x = 21

divide both sides by -12

-12x/-12 = x

21/-12 = -1.75

we're left with x = -1.75

Answer:

It owuld be -2

Step-by-step explanation:

Yes

Answer:

1

Step-by-step explanation:

Slope is given by

m = (y2 -y1)/(x2-x1)

   = ( -2 - -8)/( 2 - -4)

   =( -2+8) / (2+4)

   =6/6

   =1

Answer:

\(\displaystyle m = 1\)

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Algebra I

Step-by-step explanation:

Step 1: Define

Identify

Point (-4, -8)

Point (2, -2)

Step 2: Find slope m

Simply plug in the 2 coordinates into the slope formula to find slope m

To determine if the given points are part of the (200) plane, we need to check if their coordinates satisfy the equation of the plane.

The equation of a plane in three-dimensional space is typically written in the form Ax + By + Cz + D = 0, where A, B, C, and D are constants. For the (200) plane, the equation would be 2x + 0y + 0z + 0 = 0, which simplifies to 2x = 0. Let's check the given points: a) (1/2, 0, 0): When we substitute x = 1/2 into the equation 2x = 0, we get 2(1/2) = 0, which is true. Therefore, point a) lies on the (200) plane. b) (-1/3, 0, 0): Substituting x = -1/3 into the equation 2x = 0 gives us 2(-1/3) = 0, which is also true. So, point b) is part of the (200) plane. c) (0, 1, 0):When we substitute x = 0 into the equation 2x = 0, we get 2(0) = 0, which is true. Thus, point c) lies on the (200) plane. All three given points (a), b), and c)) are part of the (200) plane.

In conclusion, all three given points (a), b), and c)) are part of the (200) plane.

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Answer:

Step-by-step explanation:

y = ∛x is the original function, which has its center in the origin

As we see the function y = ∛(x - h) - x has its center moved to the point (4, -1)

This gives us h = 4 and k = 1, as negative h indicates translation to the right and negative k- translation down

Explanation:

Add up the probability values that correspond to x = -3 or smaller.

We'll add up the probability values corresponding to P(-3) and P(-5)

0.13+0.17 = 0.30

The time the bus should arrive at School Road from Hope Street is 08 : 04.

The time of the latest bus that Jo can catch from the Bus Station is 07 00.

The time the bus takes to travel from the bus station to School Road is 14 minutes.

The table shows that when a bus leaves for School Road from Hope Street,  at 07 54, it will arrive on School Road at 08 04.

The time it takes from the bus station to School road is:

= Arrival time at school road - Departure time from Bus station

= 08 04 - 07 50

= 14 minutes

If Jo wants to be in school road at 8 am by the latest, she needs to leave before :

= 8 am - 14 minutes

= 07 46

The only bus leaving before this time is the 0700 bus from the Bus Station.

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Answer:

b) \(y = -3x - 2\) and \(y = -3x + 5\) are parallel because the slope of each line is -3

Answer: d. No, because the study was not taken from a random sample of scholarship athletes.

Step-by-step explanation:

Since the study only focuses on basketball players, the sample may not be representative of all scholarship athletes in the university. This can cause bias in the results and make it difficult to generalize the findings to the larger population of scholarship athletes. Therefore, the university administration should not fully trust the results of this study.

Step-by-step explanation:

what, you have no calculator at hand ? at least the computer or smart phone you used to pay this question has one installed.

because this is all that needs to be done : use the given numbers and just calculate the results for every expression.

sigh ...

a.

2²×-3 = 4×-3 = -12

b.

6×(2-4)/-3 = 6×-2/-3 = -2×-2 = 4

c.

(-3)² + 2×4 = 9 + 8 = 17

d.

-(-3)² + 2 + 4 = -9 + 6 = -3

e.

-2×(-3 + 2×4)/-3 = -2×(-3 + 8)/-3 = -2×5/-3 = -10/-3 = 10/3

The area of the squares are

A = 17 units², B = 20 units², C = 13 units², and D = 37 units²

Count the number of complete squares that are fully contained within the shape. Start from one corner of the shape and count each square until you reach the opposite corner. Include any partial squares as well.

If there are partial squares, estimate the area they cover. You can do this by visualizing how much of each square is filled by the shape and approximate the area accordingly.

Add up the areas of the complete squares and the estimated areas of the partial squares to find the total area of the shape.

Here we have 4 squares A, B, C, and D

Now count the number of unit squares covered by each square

Number of squares covered by A =  17

Hence, the Area of Square A = 17 units²

Number of squares covered by B = 20

Hence, the Area of square B = 20 units²

Number of squares covered by C = 13

Hence, the Area of square C = 13 units²

Number of squares covered by D = 37

Hence, the Area of square D = 37 units²

Therefore,

The area of the squares are

A = 17 units², B = 20 units², C = 13 units², and D = 37 units²

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Complete Question is attached below

Answer:

34,35,and 36

Step-by-step explanation:

x+x+1+x+2=105

3x+3=105 combine like terms on the left side

3x=102 subtract three on both sides

x=34 divide by three on both sides.

Hope this helps plz mark brainliest :D

It will form scalene triangle

because it has different measures of sides

that is no equal side

may thishelps you

byr

The cubic functions and linear functions, both are polynomial having degree of 3 and 1 respectively.

One of the key ideas in mathematics is the concept of polynomials, along with the multiple sorts of polynomials that can be created depending on the degree for polynomials.

Some properties of cubic functions and linear functions are-

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here's two, I used mathday but day with a w

Step-by-step explanation:

good website for it and it gives good explanation! Gl

Given :

It is given that

To find

Explanation

In the given equation , substitute the value of x and y .

Answer

Hence the answer in simplest form is

The graph depicts a linear relationship between her driving time (x variable) and the remaining distance (y variable). Knowing that this is a linear relationship, we can represent it as a equation in the slope intercept form. This is because we already have the slope as -0.9.

Therefore we would have the following;

We can now substitute the given variables into the equation in slope-intercept form as follows;

With the equation for the graph now determined, we shall find the mileage after 59 minutes of driving as follows;

ANSWER:

After 59 minutes of driving, she would have 35.6 miles remaining.

Answer: Yes, you are correct.

Step-by-step explanation:

The relationship between the number of flowers and the number of bouquets is an example of a unit rate.

Answer:

yes

Step-by-step explanation:

it is a unit rate

Two linearly independent solutions of 2x²y" – my' +(1:2 +1)y=0, x > 0 of the form yı = 2"(1+212 + a22² +2323 + ...) Y2 = 2" (1 + b2x + b222 + b3x3 + ...) where rı > 12 are y₁ = x^(-1/2) Σ(n=0 to ∞) (1/2^n) [(m+2n-2)/Γ(n+1/2)] and y₂ = x^(-1/2) Σ(n=0 to ∞) (1/2^n) [(m+2n-2)/(2n+1)Γ(n+3/2)], using the method of Frobenius.

To find linearly independent solutions of the given differential equation, we can use the method of Frobenius. For this, we assume the solutions to have the form:

y = x^r Σ(n=0 to ∞) a_n x^n

Substituting this form into the differential equation, we get:

2x^2 Σ(n=0 to ∞) [(r+n)(r+n-1)a_n x^(n+r-2)] - m Σ(n=0 to ∞) [(r+n)a_n x^(n+r-1)] + (2+r^2+2r) Σ(n=0 to ∞) [a_n x^(n+r)] = 0

Equating the coefficient of x^(r-2), we get:

2r(r-1)a_0 = 0

Since x>0, we can assume r>0, and hence a_0 = 0. Equating the coefficient of x^r, we get:

2r^2 + 2r + 1 = 0

Solving for r using the quadratic formula, we get:

r = (-1 ± √3 i)/2

These are complex roots, and hence we can use the following forms for the solutions:

y₁ = x^r Σ(n=0 to ∞) a_n x^(n+r) = x^(-1/2) Σ(n=0 to ∞) a_n x^n

y₂ = x^r Σ(n=0 to ∞) b_n x^(n+r) = x^(-1/2) Σ(n=0 to ∞) b_n x^n

Now, substituting the forms of y₁ and y₂ into the differential equation and equating the coefficients of x^n, we get:

[2(n+r+1)(n+r)a_n - m(n+r)a_n + (2+r^2+2r)a_n] + [2(n+r+1)(n+r)b_n - m(n+r)b_n + (2+r^2+2r)b_n] = 0

Simplifying the expression, we get two recurrence relations:

a_n+1 = [(m-2r-2n-1)/(2r+2n+2)] a_n

b_n+1 = [(m-2r-2n-1)/(2r+2n+2)] b_n

Using these recurrence relations, we can find the coefficients a_n and b_n in terms of a_0 and b_0.

Since we want two linearly independent solutions, we can choose different values of a_0 and b_0. One possible choice is a_0 = 1 and b_0 = 0, which gives:

y₁ = x^(-1/2) Σ(n=0 to ∞) (1/2^n) [(m+2n-2)/Γ(n+1/2)]

y₂ = 0

where Γ is the gamma function. Another possible choice is a_0 = 0 and b_0 = 1, which gives:

y₁ = 0

y₂ = x^(-1/2) Σ(n=0 to ∞) (1/2^n) [(m+2n-2)/(2n+1)Γ(n+3/2)]

Therefore, two linearly independent solutions of the given differential equation are:

y₁ = x^(-1/2) Σ(n=0 to ∞) (1/2^n) [(m+2n-2)/Γ(n+1/2)]

y₂ = x^(-1/2) Σ(n=0 to ∞) (1/2^n) [(m+2n-2)/(2n+1)Γ(n+3/2)]

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\( \huge \boxed{\mathbb{QUESTION} \downarrow}\)

\( \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}\)

a) State what's asked to find \(\downarrow\)

b) State the given facts \(\downarrow\)

c) Write a working equation \(\downarrow\)

area of the garden = length of the garden × width of the garden. Let's take the length as 'l'. So the equation is...

d) Solve the equation \(\downarrow\)

\( \tt {x}^{2} + x - 12 = l \times (x + 4) \\ \\ \sf \: Bring \: (x + 4) \: towards \: the \: left \: side \\ \sf\: of \: the \: equation. \\ \\ \tt \frac{ {x}^{2} + x - 12 }{x + 4} = l \\ \\ \sf \: Factor \: the \: expressions \: that \: are \\ \sf \: not \: already \: factored. \\ \\ \tt \frac{\left(x-3\right)\left(x+4\right)}{(x+4)} = l\\ \\ \sf Cancel \: out \: (x + 4) \: in \: both \: the \\ \sf \: numerator \: and \: denominator. \\ \\ \large \boxed{\boxed{ \bf \: (x-3 )= l}}\)

e) State your answer \(\downarrow\)

I think there's a mistake in the question. It should be the area of the rectangular garden & not area of the rectangular pool because here we are asked to measure the length of the garden.

__________________

a) State what's asked to find \(\downarrow\)

b) State the given facts \(\downarrow\)

c) Write a working equation.

We know that, area of a square = side of the square × side of the square. Let's take the side of the tile (square) as 's'. So, the equation is...

d) Solve the equation \(\downarrow\)

\( \tt {x}^{2} + 10x + 25 = s \times s \\ \\ \sf \: s \: \times \: s \: is \: equal \: to \: {s}^{2} \\ \\ \tt \: {x}^{2} + 10x + 25 = {s}^{2} \\ \\ \sf \: Using \: split \: the \: middle \: term \: method.. \\ \\ \tt \: {x}^{2} + 10x + 25 = {s}^{2} \\ \tt \: \left(x^{2}+5x\right)+\left(5x+25\right) = {s}^{2} \\ \tt x\left(x+5\right)+5\left(x+5\right) = {s}^{2} \\ \tt \: \left(x+5\right)\left(x+5\right) = {s}^{2} \\ \tt\left(x+5\right)^{2} = {s}^{2} \\ \\ \sf \: Now \: squaring \: on \: both \: the \: sides.. \\ \\ \tt \: \sqrt{(x + 5) ^{2} } = \sqrt{ {s}^{2} } \\ \large \boxed{\boxed{ \bf \: (x + 5) = s}}\)

e) State your answer \(\downarrow\)

__________________

The solution to the equation log_6 x + log_6 3 = log_6 (x+1) is x = 1/2.

Explain:

We can reduce the complexity of the left side of the equation by using the logarithmic identity log a + log b = log (ab):

Log6 x plus Log6 3 equals Log6 (3x)

When we add this to the initial equation, we obtain:

Log6 (3x) equals Log6 (x+1).

Because logarithms have the one-to-one property, we can drop the logarithm on both sides to get the following results:

3x = x + 1

When we simplify this equation, we obtain:

2x = 1

x = 1/2

As a result, x = 1/2 is the answer to the equation log 6 x + log 6 3 = log 6 (x+1).

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