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The explicit formula for the sequence [tex]\(a_n\)[/tex] is:

[tex]\(a_n = \begin{cases} 4n + 3 & \text{if } n \text{ is even} \\ 4n - 2 & \text{if } n \text{ is odd} \end{cases}\)[/tex]

To find an explicit formula for the sequence [tex]\(a_n\)[/tex] that satisfies the recurrence relation [tex]\(a_n = 2n-1 + 2a_{n-2}\)[/tex] with initial conditions [tex]\(a_1 = 2\)[/tex] and [tex]\(a_2 = 7\)[/tex], we can proceed as follows:

First, let's examine the first few terms of the sequence:

[tex]\(a_1 = 2\)\\\(a_2 = 7\)\\\(a_3 = 2(3) - 1 + 2a_1 = 5 + 2(2) = 9\)\\\(a_4 = 2(4) - 1 + 2a_2 = 8 + 2(7) = 22\)\\\(a_5 = 2(5) - 1 + 2a_3 = 9 + 2(9) = 27\)\\[/tex]

We can observe that the even-indexed terms [tex]\(a_2, a_4, a_6, \ldots\)[/tex] are increasing by a factor of 2, while the odd-indexed terms [tex]\(a_1, a_3, a_5, \ldots\)[/tex] are increasing by a factor of 3. This pattern suggests that we can split the sequence into two separate sequences:

For even-indexed terms:

[tex]\(b_n = a_{2n}\)[/tex]

For odd-indexed terms:

[tex]\(c_n = a_{2n-1}\)[/tex]

Let's find explicit formulas for both [tex](\(b_n\))[/tex] and [tex](\(c_n\))[/tex]:

1. Even-indexed terms [tex](\(b_n\))[/tex]:

The recurrence relation becomes:

[tex]\(b_n = 2(2n) - 1 + 2b_{n-1}\)[/tex]

To simplify the formula, let's rewrite [tex]\(b_n\)[/tex] as [tex]\(b_{n+1}\)[/tex] (i.e., shifting the index by 1):

[tex]\(b_{n+1} = 2(2n + 2) - 1 + 2b_{n}\)[/tex]

Subtracting the two equations, we get:

[tex]\(b_{n+1} - b_n = 4\)[/tex]

This is a simple arithmetic progression with a common difference of 4. To find an explicit formula for [tex]\(b_n\)[/tex], we can use the formula for the nth term of an arithmetic progression:

[tex]\(b_n = b_1 + (n - 1) \cdot \text{{common difference}}\)[/tex]

Substituting [tex]\(b_1 = a_2 = 7\)[/tex] and the common difference of 4, we have:

[tex]\(b_n = 7 + (n - 1) \cdot 4 = 4n + 3\)[/tex]

2. Odd-indexed terms [tex](\(c_n\))[/tex]:

The recurrence relation becomes:

[tex]\(c_n = 2(2n-1) - 1 + 2c_{n-1}\)[/tex]

Similar to before, let's rewrite [tex]\(c_n\)[/tex] as [tex]\(c_{n+1}\)[/tex]:

[tex]\(c_{n+1} = 2(2n + 1) - 1 + 2c_{n}\)[/tex]

Subtracting the two equations, we get:

[tex]\(c_{n+1} - c_n = 4\)[/tex]

Again, this is an arithmetic progression with a common difference of 4. Applying the formula for the nth term of an arithmetic progression:

[tex]\(c_n = c_1 + (n - 1) \cdot \text{{common difference}}\)[/tex]

Substituting [tex]\(c_1 = a_1 = 2\)[/tex] and the common difference of 4, we have:

[tex]\(c_n = 2 + (n - 1) \cdot 4 = 4n-2[/tex]

1) [tex]\cdot 4 = 4n - 2\)[/tex]

Now that we have explicit formulas for both [tex]\(b_n\)[/tex] and [tex]\(c_n\)[/tex], we can combine them to obtain the explicit formula for the original sequence [tex]\(a_n\)[/tex]:

For even-indexed terms, [tex]\(a_{2n} = b_n = 4n + 3\)[/tex]

For odd-indexed terms, [tex]\(a_{2n-1} = c_n = 4n - 2\)[/tex]

Therefore, the explicit formula for the sequence [tex]\(a_n\)[/tex] is:

[tex]\(a_n = \begin{cases} 4n + 3 & \text{if } n \text{ is even} \\ 4n - 2 & \text{if } n \text{ is odd} \end{cases}\)[/tex]

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Answer:

The second one

Step-by-step explanation:

She started with x dollars and then used 8 dollars to buy a football game ticket, so x-8. Then, she is left with 56 dollars, so x-8=56. Therefore, the second story represents the equation.

Answer:

They have air-filled pockets in their leaves

Step-by-step explanation:

The given statement is true  (i) implies (ii) and (ii) implies (i).

The statement in the question that needs to be proven is :C & F, then X-cef = G is G an open set in C and CnH+ 0 s G for each Hef

We will prove that (i) implies (ii) and (ii) implies (i).

Proof: (i) C & F, then X-cef = G is G an open set in C and CnH+ 0 s G for each Hef

Let X \ {C & F} = U, then U is open, since C & F is closed.

Let H be any point of U.

By hypothesis, there exists an open set G such that CnH+ 0 s G.

Let x in G. If x ∈ C & F, then x ∉ H, so x ∉ U.

Thus, G ⊆ C, and so G ∩ U = ∅.

Hence, U is open(ii) G is G an open set in C and CnH+ 0 s G for each Hef

Let x ∈ X-C & F.

Then x ∉ C & F, so x ∉ C.

Since C is closed, there exists a neighborhood G of x that is disjoint from C.

Let H be any point of X-C & F.

Then H ∈ G and so CnH+ 0 s G.

Thus, C & F is closed.

Therefore, X-C & F is open, since C & F is closed.

Thus, X-C & F = G.

Hence, (ii) implies (i).

Therefore, the statement in the question is proven.

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Answer: One of the ways you can do all this is by zearn or by listening by your teacher.

Answer:

1 e

2 a

3 b

4 c

5 d

Step-by-step explanation:

i did the answers to the first question i dont know the rest sorry

The theoretical mean for a draw from a Poisson distribution with parameter λ is equal to λ itself. In this case, λ = 37, so the theoretical mean is also 37.

Explanation:

The Poisson distribution is commonly used to model the number of events occurring within a fixed interval of time or space, when these events occur with a known average rate λ. The probability mass function of the Poisson distribution is given by P(X=k) = (e^(-λ) * λ^k) / k!, where X represents the random variable representing the number of events and k is the observed value.

The Central Limit Theorem states that when independent random variables are added, their sum tends toward a normal distribution, regardless of the shape of the original distribution. For a Poisson distribution, as the parameter λ increases, the distribution becomes more symmetric and bell-shaped, resembling a normal distribution.

Since the mean of a Poisson distribution is equal to its parameter λ, the theoretical mean for a draw from a Poisson distribution with parameter 1 = 37 is 37. This means that, on average, 37 events are expected to occur within the given interval.

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Answer:

y=1

Step-by-step explanation:

6=2(y+2)

6=2y+4

2=2y

y=1

Answer:

y=1

Step-by-step explanation:

Answer:

C

Step-by-step explanation:

Ur welcome

Step-by-step explanation:

The answer will be zero

here are the steps:

cos(90-10)xcos(90-80)-sin(90-10)xsin(90-10)=

(cos10xcos10)-(sin80xsin80)=

0.965111-0.965111=0

have a good day :)

I hope it will benefit you.

Step-by-step explanation:

Mohammed age = x

Holly age = x + 3

Karen age = 2x

given,

[tex]x + (x + 3) + 2x = 51 \\ x + x + 3 + 2x = 51 \\ 4x + 3 = 51 \\ 4x = 51 - 3 \\ 4x = 48 \\ x = 48 \div 4 \\ = 12[/tex]

Answer:

2/3 and 3/5 is same, then 5/6

Step-by-step explanation:

you can convert the fractions to decimals to find their value and then arrange them from least to the greatest.

Answer:

3/5, 2/3, 5/6 [From Least to Greatest]

Step-by-step explanation:

First you're going to want to know which one is "the bigger piece of pie".

I made a few drawing and look at the pictures (Just in case you have a different opinion from my answer)

Answer:

The angle at B is the same as the angle at E so equate them to each other to find x

2x+4=40°

2x=40-4

2x=36

x=36/2=18

Step-by-step explanation:

Hope this is helpful! stay safe and God Bless:)))

The strophoid is a curve represented by the polar equation r = sec(θ) − 2cos(θ), where -2 < θ < 2. In Cartesian coordinates, the strophoid equation can be written as (x^2 + y^2)^2 = 4y^2(x + 2).

The strophoid has a unique shape characterized by its looped structure.

The strophoid is symmetric with respect to the y-axis, as changing θ to -θ gives the same value of r. It has two branches that intersect at the origin (0, 0). As θ increases from -2 to 2, the curve starts from the rightmost point of the loop, extends to the left, and then returns back to the rightmost point.

The loop of the strophoid is created by the interplay of the secant function, which stretches the curve away from the origin, and the cosine function, which pulls it towards the origin. The strophoid exhibits interesting geometric properties and is often used in mathematical modeling and visualization.

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Answer:

jamel brother has 48.00 dollors

Step-by-step explanation:

Answer: He has 32$ 24 divided by 6 is 4 multiply 4 by 8 and you get 32

Answer:

3 CDs

Step-by-step explanation:

If we have $65 and buy a $23 DVD, we will have $42 left.

So how many $14 CDs can we buy with $42?

All we have to do is divide 42 into 14, so we know how many groups of $14 we can make with $42.

42 ÷ 14 = 3

Therefore, Michella can purchase 3 CDs.

Answer:

(6,8)

Step-by-step explanation:

midpoint=(x1+x2)÷2,(y1+y2)÷2

a(4,15) b(8,1)

x=4+8=12÷2=6

y=15+1=16÷2=8

Answer=(6,8)

Answer:

DB = 10

m∡C = 106°

Step-by-step explanation:

DE = EB

20x - 8 = 16x + 12

4x = 20

x = 5

DB = 5 doubled, or 10

m∡A + m∡D = 180

3y + 7 + 2y + 8 = 180

5y + 15 = 180

5y = 165

y = 33

m∡A = m∡C

m∡A = 3(33)+7 = 106°

m∡C = 106° also

Answer:

Option B

Step-by-step explanation:

We have to complete the table given in the question,

                         One headed               Two headed                    Total

Green                      100                       230 - 125 = 105        105 + 100 = 205

Purple            270 - 100 = 170                    125                      170 + 125 = 295

Total                         270                      500 - 270 = 230                500

By analyzing the given table,

Number of green Jolos in Balan's colony = Total of one headed green Jolos and Two headed green Jolos

= 205

Therefore, number of green Jolos in Balan's colony are 205.

Option B will be answer.

Answer:

When you put together the whole chart you will see the total is 205.

Step-by-step explanation:

The rule that best represents the dilation applied to Circle | to create Circle || is the scale factor. The scale factor determines the ratio of corresponding lengths between the original figure (Circle |) and the dilated figure (Circle ||).

In a dilation, all lengths in the original figure are multiplied by the scale factor to obtain the corresponding lengths in the dilated figure. This includes the radii of the circles.

For example, if the scale factor is 2, it means that every length in the original figure is doubled in the dilated figure. If the scale factor is 1/2, it means that every length is halved. The scale factor can be greater than 1, less than 1 (but greater than 0), or even negative, indicating a reflection.

In the context of the given scenario, since the origin is the center of dilation, the scale factor determines how the distances from the origin to any point on Circle | are scaled to obtain the corresponding distances on Circle ||.

The comparison of systolic blood pressure in the sitting and standing positions is an example of testing the difference between matched pairs from two dependent populations.

The scenario described involves measuring the systolic blood pressure of the same set of patients in two different positions (sitting and standing). This creates a dependency between the measurements because each patient serves as their own control. In this case, the appropriate statistical test would be a paired t-test or a related test for dependent samples.

Two means from independent populations: This option would be suitable if the measurements were taken from two different groups of patients who were independent of each other, but in this case, the same individuals were measured in both positions. Two population proportions: This option would be applicable if the data involved proportions or categorical variables, rather than continuous measurements like blood pressure.

Matched pairs from two dependent populations: This option accurately represents the scenario described, as the measurements were taken from the same individuals in both positions, making them dependent on each other.

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Answer:

The answer is 25 degree because there is ( 8x-1)

Answer:

119

Step-by-step explanation:

Answer:

x= 61

Step-by-step explanation:

i think

Answer:

(D) 3/10

Step-by-step explanation:

So its split into 10 different rates each and its on the third split So 3/10.

An eigenvector corresponding to λ₂ = 2 - i is v₂ = [-1, 1].

To find the eigenvalues of matrix A, we need to solve the characteristic equation det(A - λI) = 0, where I is the identity matrix.

Let's compute the determinant:

det(A - λI) = |[2 - λ -1]|

|[ 1 2 - λ]|

Expanding along the first row, we have:

(2 - λ)(2 - λ) - (-1)(1) = (2 - λ)² + 1 = λ² - 4λ + 5 = 0

To solve this quadratic equation, we can use the quadratic formula:

λ = (-(-4) ± √((-4)² - 4(1)(5))) / (2(1))

= (4 ± √(16 - 20)) / 2

= (4 ± √(-4)) / 2

Since we are working over the complex numbers, the square root of -4 is √(-4) = 2i.

λ₁ = (4 + 2i) / 2 = 2 + i

λ₂ = (4 - 2i) / 2 = 2 - i

Now, let's find the eigenvectors corresponding to each eigenvalue.

For λ₁ = 2 + i, we solve the equation (A - (2 + i)I)v = 0:

[2 - (2 + i) -1] [x] [0]

[ 1 2 - (2 + i)] [y] = [0]

Simplifying, we have:

[0 -1 -1] [x] [0]

[ 1 0 - i] [y] = [0]

From the first equation, we have -x - y = 0, which implies x = -y.

Choosing y = 1, we have x = -1.

Therefore, an eigenvector corresponding to λ₁ = 2 + i is v₁ = [-1, 1].

For λ₂ = 2 - i, we solve the equation (A - (2 - i)I)v = 0:

[2 - (2 - i) -1] [x] [0]

[ 1 2 - (2 - i)] [y] = [0]

Simplifying, we have:

[0 -1 -1] [x] [0]

[ 1 0 i] [y] = [0]

From the first equation, we have -x - y = 0, which implies x = -y.

Choosing y = 1, we have x = -1.

In summary:

λ₁ = 2 + i has eigenspace span {[-1, 1]}

λ₂ = 2 - i has eigenspace span {[-1, 1]}

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Answer:

x=3.5

Step-by-step explanation:

To make DEF similar to XYZ, the sides have to be in the same ratio. EF corresponds to YZ. EF=3, and YZ=4.5. The ratio 3:4.5 can be simplified to 2:3. Side DF corresponds to XZ.  DF=7 and XZ=3x. So, the ratio is 7:3x.

To  find x, we first find out what 3x is.  In this case 3x is 3(7/2)=10.5. So, x=10.5/3=3.5.

Answer:

-5

Step-by-step explanation:

Rewrite x - 2 as -3 - 2, which comes out to -5.