An A.P has common difference d.If the sum of of the first twenty terms is twenty five times the first term, find in terms of d, the sum of thirty terms.

p(f | e) = p(e | f) * p(f) / p(e) = (1/5) / (1/3) = 3/5

Therefore, p(f | e) = 3/5.

We can use Bayes' theorem to find p(f | e):

p(f | e) = p(e | f) * p(f) / p(e)

We know that p(e) = 1/3 and p(f) = 1/2. To find p(e | f), we can use the conditional probability formula:

p(e | f) = p(e ∩ f) / p(f)

We are given that p(e | f) = 2/5, so we can rearrange the formula to get:

p(e ∩ f) = p(e | f) * p(f) = (2/5) * (1/2) = 1/5

Now we have all the information we need to apply Bayes' theorem:

p(f | e) = p(e | f) * p(f) / p(e) = (1/5) / (1/3) = 3/5

Therefore, p(f | e) = 3/5.

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There is only 1 way to draw a line segment with endpoint p that is perpendicular to line n.

If line m is parallel to line n and point p is between lines m and n, there is only one line segment with endpoint p that is perpendicular to line n.

To visualize this, consider the lines m and n as two horizontal parallel lines, and point p is located between these lines. There can be only one vertical line segment with an endpoint at point p that is perpendicular to both lines m and n, since a perpendicular line to line n will also be perpendicular to line m due to their parallel nature.

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Therefore, there are 81,920 different pizzas that this parlor offers.

Since there are five different sizes of pizza, a customer can choose any one of the five sizes. For each size, the customer can choose to have any combination of the 14 toppings, or no toppings at all. This means that for each size of pizza, there are $2^{14}$ different possible topping combinations, including the option of having no toppings. So the total number of different pizzas that the parlor offers is:

=5*2¹⁴

=5*16,384

=81,920

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The surface area of the sphere of radius of 7cm is 616 square centimeters.

We know that the surface area of a sphere of radius r is given by the formula:

S = 4*(22/7)*r²

Here we want to find the surface area of a sphere whose radius is r = 7 cm.

Replacing it in the formula above, we will get:

S = 4*(22/7)*7²

S = 4*22*7

S = 616

And the units are square centimeters, so the correct option is C.

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The requried relative frequency of rolling an odd number is 3/5.

An odd number was rolled 15 times.

The total number of rolls was 25.

The relative frequency of rolling an odd number is found by dividing the number of times an odd number was rolled by the total number of rolls:

Relative frequency = number of odd rolls / total number of rolls

Substituting the values, we get:

Relative frequency = 15/25

Relative frequency = 3/5

Therefore, the relative frequency of rolling an odd number is 3/5.

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For `C = 0, d = 1 therefore C + d = 0 +1 = 1

For C = 27, d =  0.2161 therefore C+d = 27 + 0.2161 = 27.2161

We can use the substitution method to find the values of x and y.

We can rearrange the first equation to solve for x in terms of y:

Cx + dy = 12

Cx = 12 - dy

[tex]x = \frac{ (12 - dy)}{C}[/tex]

This expression for x can then be substituted into the second equation:

2x + 7y = 4

2([tex]\frac{(12 - dy)}{C}[/tex]) + 7y = 4

To eliminate the denominator, multiply both sides by C:

2(12 - dy) + 7Cy = 4C

Increasing the size of the brackets:

24 - 2dy + 7Cy = 4C

Rearranging and calculating y:

-2dy + 7Cy = 4C - 24

y(7C - 2d) = 4C - 24

y = [tex]\frac{(4C - 24)}{(7C - 2d)}[/tex]

We can then plug this y expression back into the first equation to find x:

Cx + dy = 12

C([tex]\frac{(4C - 24)}{(7C - 2d)}[/tex]) + d([tex]\frac{(4C - 24)}{(7C - 2d)}[/tex]) = 12

Multiplying to eliminate the denominator, multiply both sides by (7C - 2d):

12(7C - 2d) = C(4C - 24) + d(4C - 24).

Increasing the size of the brackets:

84C - 24d = 4C2 - 24C + 4Cd - 24C

Simplifying:

[tex]4C^2 - 108C = 0[/tex]

Taking 4C into account:

4C(C - 27) = 0

As a result, either C = 0 or C = 27.

If C is equal to zero, the first equation becomes:

dy = 12

The second equation is as follows:

2x + 7y = 4

Adding dy = 12 to the first equation:

d(12) = 12

d = 1

Adding d = 1 to the second equation:

2x + 7(12) = 4

2x = -80

x = -40

As a result, if C = 0, x = -40, and y = 1.

If C = 27, the first equation is as follows:

27x + dy = 12

The second equation is as follows:

2x + 7y = 4

Adding dy = 12 - 27x to the first equation:

27x + d(12 - 27x) = 12

-27dx + 27x = 12 - 27x

d = (12 - 27x)/-27x + 1

In the second equation, substitute d = (12 - 27x)/-27x + 1:

2x + 7((12 - 27x)/-27x + 1) = 4

To eliminate the denominator, multiply both sides by -27x:

-54x + 84 - 7x(-27x + 27x + 1) = -108x

Simplifying:

-54x + 84 + 7x = -108x

-47x = -84

x = 84/47

Adding x = 84/47 to the formula for d:

d = (12 - 27(84/47))/-27(84/47) +1

d = (12 - 1.7872)/ -27(1.7872) +1

d = 10.2128/-47.2544

d = 0.2161

For `C = 0, d = 1 therefore C + d = 0 +1 = 1

For C = 27, d =  0.2161 therefore C+d = 27 + 0.2161 = 27.2161

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By answering the presented question, we may conclude that As a result, fraction 3/8 cup of syrup remained in the measuring cup.

A whole can be represented by any number of equal pieces, or fractions. In standard English, fractions denote the number of units of a specific size. 8, 3/4. Fractions are included in a whole. In mathematics, numbers are stated as a ratio of the numerator to the denominator. In simple fractions, each of these is an integer. A fraction can be found in the numerator or denominator of a complex fraction. True fractions have denominators that are greater than their numerators. A fraction is a sum that represents a percentage of a total. You may assess it by splitting it down into smaller chunks. For example, 12 represents half of a whole number or object.

Troy started with half a cup of syrup in the measuring cup. Then he poured 1/8 cup back into the jug.

To find out how much syrup is still in the measuring cup, subtract the quantity put back into the jug from the amount that was originally in the measuring cup.

1/2 - 1/8

We need to discover a common denominator to remove these two fractions. 8 is the lowest common multiple of 2 and 8.

As a result, we may rewrite 1/2 as 4/8:

4/8 - 1/8 = 3/8

As a result, 3/8 cup of syrup remained in the measuring cup.

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Answer:

28 Units

Step-by-step explanation:

The perimeter is the distance around the object.

Info: The area is 48 units squared and you're given a 6 for side BE.
This means that side CD is also 6.
6+6 = 12.
(This is the width)

However, you need 2 more sides.
You need to do 48 divided by 6 since the area requires the width multiplied by the length.
48/6 = 8
(This is the length)

Sides BC and ED is 8 meaning 8 + 8 = 16.
Now given our calculations: 12 + 16 is 28 units.

You can verify by doing 6+6+8+8 for perimeter
To check if the 8 matches with the area: Do 8 x 6 which equals 48 Units Squared.

The given series is convergent, and its sum is approximately 0.1524.

To determine whether the series ∑n=1∞ 1/(9+[tex]e^{(-n)}[/tex]) is convergent or divergent, we can use the comparison test with the series 1/n.

Since for all n, [tex]e^{(-n)}[/tex] > 0, we have [tex]9 + e^{(-n)}[/tex] > 9, and so [tex]1/(9+e^{(-n)})[/tex] < 1/9.

Now, we can compare the given series with the series ∑n=1∞ 1/9, which is a convergent p-series with p=1.

By the comparison test, since the terms of the given series are smaller than those of the convergent series 1/9, the given series must also converge.

To find the sum of the series, we can use the formula for the sum of an infinite geometric series:

S = a/(1-r)

where a is the first term and r is the common ratio. In this case, the first term is 1/10 (since [tex]e^{(-1)}[/tex] is very small compared to 9.

We can approximate [tex]9+e^{(-n)}[/tex] as 9 for large n), and the common ratio is [tex]e^{(-1)} < 1[/tex]. Therefore, the sum of the series is:

S = (1/10)/(1 - [tex]e^{(-1)}[/tex]) = (1/10)/(1 - 0.3679) ≈ 0.1524

Therefore, the given series is convergent, and its sum is approximately 0.1524.

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Answer:

2 5/8

Step-by-step explanation:

1.5*1.75=2.625=2 5/8

The probability of at least one male getting an earring is approximately 1 - 0.0173 ≈ 0.9827 or 98.27%.

To find the probability of at least one male getting an earring, we'll use the complementary probability.

There are 12 crackers with 4 containing earrings, so the probability of a cracker not having earrings is 2/3.

With 6 males at the party, the probability of all males not getting earrings is (2/3)^6 ≈ 0.0173.

Therefore, the probability of at least one male getting an earring is approximately 1 - 0.0173 ≈ 0.9827 or 98.27%.

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The perimeter and the area of the triangle are given as follows:

First we obtain the area, as we have the two parameters, as follows:

The area is half the multiplication of the base and the height, hence it is given as follows:

[tex]A = 0.5 \times 16 \times 8\sqrt{3}[/tex]

[tex]A = 64\sqrt{3}[/tex] cm².

For the perimeter, we must obtain the lateral segments, considering the bisection and the Pythagorean Theorem, as follows:

[tex]l^2 = 8^2 + (8\sqrt{3})^2[/tex]

l² = 64 + 192

l² = 256

l = 16.

Hence the perimeter is given as follows:

P = 3 x 16

P = 48 cm.

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Answer:

3.2 miles

Step-by-step explanation:

[tex]\frac{5684.106yds}{1}[/tex] · [tex]\frac{3ft}{1yd}[/tex] · [tex]\frac{1mile}{5280ft}[/tex] You can cross cancel words just like numbers.  Cross cancel the words: yards and feet.  That will leave you with just miles

[tex]\frac{5684.106}{1 }[/tex] ·[tex]\frac{3}{1}[/tex] · [tex]\frac{1mile}{5280}[/tex]

[tex]\frac{17052.318}{5280}[/tex]

3.22960568182

This rounded to the nearest tenth would be: 3.2

Helping in the name of Jesus.

Which is 1/λ = 1/4.25 ≈ 0.2353 hours (or approximately 14.12 minutes).

We can model the time until the first delivery is made as an exponential distribution with parameter λ = 4.25 orders per hour.

(a) Let Y be the time until the first delivery is made. Then we need to find P(Y > 0.8). Using the cumulative distribution function of the exponential distribution, we have:

P(Y > 0.8) = 1 - P(Y ≤ 0.8) = 1 - F(0.8) = 1 - (1 - e^(-λt))|{t=0.8} = e^(-λt)|{t=0.8} = e^(-4.25*0.8) ≈ 0.332

So the probability that the time until the first delivery exceeds 0.8 hours is approximately 0.332.

(b) The average time it takes to deliver the first order is given by the expected value of Y, which is 1/λ = 1/4.25 ≈ 0.2353 hours (or approximately 14.12 minutes).

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A particular solution of the given differential equation y'' + 3y' + 4y = 8x + 2 is: y_p = 2x - 1 (option c).

The particular solution of the given differential equation can be found by using the method of undetermined coefficients. We assume that the particular solution has the same form as the right-hand side of the equation, i.e., y_p = Ax + B, where A and B are constants. We then substitute this into the differential equation and solve for A and B.

y" + 3y' + 4y = 8x + 2

y_p = Ax + B
y'_p = A
y"_p = 0

Substituting these into the equation, we get:

0 + 3A + 4Ax + 4B = 8x + 2

Comparing the coefficients of x and the constant term, we get:

4A = 8  =>  A = 2
4B = 2  =>  B = 1/2

Therefore, the particular solution is y_p = 2x + 1, which is option a.

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a. The finite population correction factor is not necessary. The expected value of the sample proportion is 0.34 and the standard deviation of the sample proportion is 0.0508.

b. The probability that the sample proportion is greater than 0.37 is approximately 0.2776.

a. To determine if the finite population correction factor is necessary, we need to check if the sample size is large enough in relation to the population size. If the sample size is less than 5% of the population size, then the correction factor is not necessary. In this case, n = 100 is less than 5% of N = 3,000, so we don't need to apply the finite population correction factor.

The expected value of the sample proportion is equal to the population proportion, so E(p) = p = 0.34.

The formula for the standard deviation of the sample proportion is

σ(p) = sqrt[p(1-p)/n]

Substituting in the values, we get:

σ(p) = sqrt[(0.34)(1-0.34)/100] = 0.0508

Therefore, the expected value of the sample proportion is 0.34 and the standard deviation of the sample proportion is 0.0508.

b. We want to find the probability that the sample proportion is greater than 0.37. We can use the z-score formula and standard normal distribution to find this probability.

The z-score formula is:

z = (P - p) / σ(P)

Substituting in the values, we getp

z = (0.37 - 0.34) / 0.0508 = 0.591

Using a standard normal distribution table or calculator, we can find that the probability of z being greater than 0.591 is approximately 0.2776.

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Since the limit is a finite value, the integral is convergent. Furthermore, the value of the convergent integral is 21e^(-1), which is approximately 7.713.

whether the integral is convergent or divergent.

First, let's rewrite the integral using proper notation:

∫(1 to ∞) 21e^(-x) dx

Now, to determine if the integral is convergent or divergent, we'll perform the following steps:

1. Apply the limit as the upper bound approaches infinity:

lim(b→∞) ∫(1 to b) 21e^(-x) dx

2. Evaluate the improper integral using the antiderivative:

F(x) = -21e^(-x)

Now, we need to find the limit as b approaches infinity:

lim(b→∞) (F(b) - F(1))

3. Calculate the limit:

lim(b→∞) (-21e^(-b) - (-21e^(-1)))

As b approaches infinity, e^(-b) approaches 0. Therefore, the limit is:

-(-21e^(-1)) = 21e^(-1)

Since the limit is a finite value, the integral is convergent. Furthermore, the value of the convergent integral is 21e^(-1), which is approximately 7.713.

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We are given the differential equation [tex]y" + 2y = -18x^2e^2x[/tex]n:. To find the complementary function, we first solve the homogeneous equation:[tex]y" + 2y = 0[/tex]. The answer is the particular solution is:[tex]y_p(x) = -3/2*x^2e^2x[/tex].

The characteristic equation is:[tex]r^2 + 2 = 0[/tex]

Which has the roots:[tex]r = ±√(-2)[/tex]

Since the roots are complex, we can write them as:[tex]r1 = i√2[/tex]

and [tex]r2 = -i\sqrt{2}[/tex]

Thus, the complementary function is: y_c(x) = [tex]c1cos(\sqrt{2x} )[/tex] + [tex]c2sin(\sqrt{2}x )[/tex]

To find the particular solution, we assume a solution of the form:[tex]y_p(x) = Ax^2e^2x[/tex]

Taking the first and second derivatives of y_p(x), we get:

[tex]y'_p(x) = 2Axe^2x + 2Ax^2e^2x[/tex]

[tex]y''_p(x) = 4Axe^2x + 4Ax^2e^2x + 4Ae^2x[/tex]

Substituting y_p(x), y'_p(x), and y''_p(x) back into the original differential equation, we get:

[tex](4Axe^2x + 4Ax^2e^2x + 4Ae^2x) + 2(Ax^2e^2x) = -18x^2e^2x[/tex]

Simplifying and collecting like terms, we get:[tex](6A + 4Ax)xe^2x + (4A + 2A)x^2e^2x = -18x^2e^2x[/tex]

Equating coefficients of like terms, we get:[tex]6A + 4Ax = 0, 4A + 2A = -18[/tex]

Solving for A, we get:

A =[tex]\frac{-3}{2}[/tex]

Therefore, the particular solution is:[tex]y_p(x) = -3/2*x^2e^2x[/tex]

The general solution is the sum of the complementary function and the particular solution:

[tex]y(x) = y_c(x) + y_p(x)[/tex]

[tex]y(x) = c1cos(√2x) + c2sin(√2x) - 3/2*x^2e^2x[/tex]

Where c1 and c2 are constants determined by initial conditions.

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The distance between the points (4, 3), (7, 5) using the distance formula is sqrt(13) and using integration is also sqrt(13).

(a) Using the distance formula:

d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

= sqrt((7 - 4)^2 + (5 - 3)^2)

= sqrt(9 + 4)

= sqrt(13)

Therefore, the distance between the points (4, 3) and (7, 5) is sqrt(13).

(b) Using integration:

The distance between two points can also be found by integrating the magnitude of the velocity function that connects the two points.

Let P1 = (4, 3) and P2 = (7, 5), and let f(t) be the position function of an object moving from P1 to P2 along some path. Then the velocity function is given by:

v(t) = f'(t)

The magnitude of the velocity is given by:

|v(t)| = sqrt((dx/dt)^2 + (dy/dt)^2)

We can find the position function by integrating the velocity function:

f(t) = ∫ v(t) dt

For the points P1 and P2, we have:

P1 = (4, 3) and P2 = (7, 5)

Therefore,

dx/dt = 3, dy/dt = 2

Thus,

|v(t)| = sqrt(3^2 + 2^2) = sqrt(13)

Integrating this over the interval [0,1], we get:

d = ∫0^1 |v(t)| dt

= ∫0^1 sqrt(13) dt

= sqrt(13) * t |0^1

= sqrt(13)

Therefore, the distance between the points (4, 3) and (7, 5) is sqrt(13), using integration as well.

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Basis containing the given set of vectors is as follows:

(a) { (1, 0, 1, 0), (0, 1, 1, 0) }; (b) { (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 0, 1) }

(c) { [1 0 0 0], [0 2 0 0], [0 0 0 1] }

To construct a basis of V, we can use Gaussian elimination. We can start by creating an augmented matrix with the given vectors as columns:

(a)

| 1 0 1 0 |

| 0 1 1 0 |

| 1 0 -1 0 |

| 0 0 0 0 |

Perform elementary row operations to get matrix in row echelon form:

| 1 0 1 0 |

| 0 1 1 0 |

| 0 0 -2 0 |

| 0 0 0 0 |

Therefore, a basis for V is:

{ (1, 0, 1, 0), (0, 1, 1, 0) }

(b)

| 1 0 0 0 |

| 1 0 0 0 |

| 0 1 0 0 |

| 0 1 0 0 |

| 0 0 0 1 |

| 0 0 0 1 |

Perform elementary row operations to get matrix in row echelon form:

| 1 0 0 0 |

| 0 1 0 0 |

| 0 0 0 1 |

| 0 0 0 0 |

| 0 0 0 0 |

| 0 0 0 0 |

Therefore, a basis for V is:

{ (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 0, 1) }

(c) We can see that given set of vectors is already a basis for M22, since they are linearly independent. Therefore, a basis for V is:

{ [1 0 0 0], [0 2 0 0], [0 0 0 1] }

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The value of x at the 70th percentile of a standard normal distribution is approximately 0.52

In a standard normal distribution, the mean (μ) is 0 and the standard deviation (σ) is 1. To find the value of x that corresponds to the 70th percentile, we need to find the z-score that corresponds to the 70th percentile and then use that z-score to find the corresponding value of x.

The z-score corresponding to the 70th percentile can be found using a standard normal distribution table or calculator. The table or calculator will give the value of the cumulative distribution function (CDF) for a given z-score. We want to find the z-score such that the CDF is 0.70. From the standard normal distribution table, we can find that the z-score is approximately 0.52.

Once we have the z-score, we can use the formula

x = μ + zσ

Substituting the values of μ = 0, σ = 1, and z = 0.52, we get

x = 0 + 0.52(1) = 0.52

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It believed that self interest was the greatest motivator of the people.

What about self interest?

Self-interest refers to the motivation or desire of an individual to pursue their own benefit or well-being. It is a fundamental human behavior that drives individuals to make decisions and take actions that are likely to result in personal gain or advantage.

Self-interest can manifest in various forms, such as seeking financial gain, pursuing personal happiness, or striving for success and recognition. While self-interest can be seen as a positive force that drives individuals to work hard and achieve their goals, it can also lead to negative consequences if pursued at the expense of others or the common good.

In economic theory, self-interest is often viewed as a key driver of market behavior, as individuals and businesses seek to maximize their profits or utility. However, many argue that a purely self-interested approach can lead to negative externalities and social problems, and that considerations of the greater good and moral principles should also be taken into account.

According to the given information:

Alexander Hamilton believed that self interest was the greatest motivator of people.

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It believed that self interest was the greatest motivator of the people.

What about self interest?

Self-interest refers to the motivation or desire of an individual to pursue their own benefit or well-being. It is a fundamental human behavior that drives individuals to make decisions and take actions that are likely to result in personal gain or advantage.

Self-interest can manifest in various forms, such as seeking financial gain, pursuing personal happiness, or striving for success and recognition. While self-interest can be seen as a positive force that drives individuals to work hard and achieve their goals, it can also lead to negative consequences if pursued at the expense of others or the common good.

In economic theory, self-interest is often viewed as a key driver of market behavior, as individuals and businesses seek to maximize their profits or utility. However, many argue that a purely self-interested approach can lead to negative externalities and social problems, and that considerations of the greater good and moral principles should also be taken into account.

According to the given information:

Alexander Hamilton believed that self interest was the greatest motivator of people.

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The magnitude of the electric field is 1,454,000 N/C and the SI unit for this answer is Newtons per Coulomb (N/C).

a) To find the net electric flux through the Gaussian surface, we can use Gauss's law which states that the electric flux through any closed surface is proportional to the charge enclosed by the surface.

So, [tex]Φ = q/ε0[/tex], where Φ is the electric flux, q is the charge enclosed by the Gaussian surface, and ε0 is the permittivity of free space.

Since 6.78 pC of charge is enclosed by the Gaussian surface, the electric flux through the surface is [tex]Φ = (6.78 × 10^-12 C) / ε0.[/tex]

The area of the Gaussian surface is

[tex](2.16 × 10^-6 m) × (2.16 × 10^-6 m) = 4.6656 × 10^-12 m^2.[/tex]

So, the electric flux through the Gaussian surface is

[tex]Φ = (6.78 × 10^-12 C) / ε0 = (ε0 × E × 4.6656 × 10^-12 m^2) / ε0[/tex],

where E is the electric field.

Solving for E, we get

[tex]E = (6.78 × 10^-12 C) / (4.6656 × 10^-12 m^2) = 1.454 × 10^6 N/C.[/tex]

Therefore, the magnitude of the electric field is [tex]1.454 × 10^6 N/C[/tex] and the SI unit for this answer is Newtons per Coulomb (N/C).

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Answer:

710 670 630 580 550

Step-by-step explanation:

670-630 = 40

630 - 590 = 40

We are subtracting 40 from each term.

590 - 40 = 550

The last term is 550

x - 40 = 670

x = 670+40

x = 710

The first term is 710.

We have proved that if x is a nontrivial square root of 1, modulo n, then gcd(r-1, n) and gcd(s+1, n) are both nontrivial divisors of n.

Let us assume that x is a nontrivial square root of 1, modulo n, then we have x^2 ≡ 1 (mod n).

This implies that (x+1)(x-1) ≡ 0 (mod n).

So, either (x+1) ≡ 0 (mod n) or (x-1) ≡ 0 (mod n), since n is a composite number and not a prime.

If (x+1) ≡ 0 (mod n), then n|(x+1), which implies that x+1 = kn for some integer k. So, we have x = kn-1.

Now, let r = gcd(k-1, n). Since r|n and r|k-1, we have r|(k-1) + 1 = k. So, we have r|k and r|n.

Therefore, we have gcd(r-1, n) is a nontrivial divisor of n.

On the other hand, if (x-1) ≡ 0 (mod n), then n|(x-1), which implies that x = kn+1.

Now, let s = gcd(k+1, n). Since s|n and s|k+1, we have s|(k+1) - 1 = k. So, we have s|k and s|n.

Therefore, we have gcd(s+1, n) is a nontrivial divisor of n.

Hence, we have proved that if x is a nontrivial square root of 1, modulo n, then gcd(r-1, n) and gcd(s+1, n) are both nontrivial divisors of n.

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The solution to the initial value problem y' = 2yx, y(1) = 1/4 is  [tex]y = (1/(4e)) * e^(^x^2^)[/tex]

To find the solution, follow these steps:

Step 1: Identify the given differential equation and initial condition.
The differential equation is y' = 2yx, and the initial condition is y(1) = 1/4.

Step 2: Separate variables.
Divide both sides of the equation by y to isolate dy/dx:

(dy/dx) / y = 2x

Now, multiply both sides by dx to separate the variables:

(dy/y) = 2x dx

Step 3: Integrate both sides.
Integrate the left side with respect to y, and the right side with respect to x:

[tex]∫(1/y) dy = ∫(2x) dx[/tex]

ln|y| = x^2 + C₁ (Remember to add the constant of integration, C₁)

Step 4: Solve for y.
To remove the natural logarithm, take the exponent of both sides:

[tex]y = e^(x^2 + C₁)[/tex]

We can rewrite this as:

[tex]y = e^(^x^2^) * e^(^C^_1)[/tex]
Since e^(C₁) is also a constant, let C = e^(C₁):

[tex]y = C * e^(^x^2^)[/tex]

Step 5: Apply the initial condition to find the constant C.
Use the initial condition y(1) = 1/4 and substitute x = 1:

1/4 = C * e^(1^2)

1/4 = C * e

Now, solve for C:

C = 1/(4e)

Step 6: Write the solution.
Substitute the value of C back into the equation for y:

[tex]y = (1/(4e)) * e^(^x^2^)[/tex]

This is the solution to the initial value problem y' = 2yx, y(1) = 1/4.

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we have shown that if f(n) = f(n2), then n = n2, which means that f is one-to-one.

The question asks us to define a function f from the set of integers to itself, where f(n) = 2 - 3n for each integer n. We then need to prove that this function is one-to-one.

To prove that f is one-to-one, we need to show that for any two integers n and n2, if f(n) = f(n2), then n = n2. Here's how we can do that:

Proof:

1. Suppose n and n2 are any integers such that f(n) = f(n2).

2. Substituting from the definition of f gives us:

2 - 3n = 2 - 3n2

3. Simplifying this equation, we get:

-3n = -3n2

4. Dividing both sides by -3, we get:

n = n2

5. Therefore, we have shown that if f(n) = f(n2), then n = n2, which means that f is one-to-one.

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