Determine if the series converges or diverges. If the series converges, find its sum.
80
Σ
5
n(n+3)
OA. The series converges to
B. The series converges to
55
18
25
6

The exponential function is y=5.[tex]2^x[/tex].

What is exponential function?

A mathematical function called an exponential function is employed frequently in everyday life. It is mostly used to compute investments, model populations, determine exponential decline or exponential growth, and so forth.

Here the exponential function is [tex]y=ab^x[/tex]

Since (-3,5/8) is on the graph, -[tex]\frac{5}{8}[/tex]=[tex]ab^{-3}[/tex] -----> 1

Since (3, 40) is on the graph, 40=[tex]ab^3[/tex] ------> 2

So, [tex]\frac{ab^3}{ab^{-3}}=\frac{40}{\frac{-5}{8}}[/tex]

=> [tex]b^{3+3}=8\times8[/tex]

=> [tex]b^6=2^6[/tex]

=> b = 2

put b=2 into 2 then,

=> 40= [tex]a\times2^3[/tex]

=> 8a=40

=> a =5

Then the exponential function is y=5.[tex]2^x[/tex].

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The distance between city A and city B is approximately 442.3 km.

What is the law of cosine?

The Law of Cosines, also known as the Cosine Rule, is a mathematical formula that relates the lengths of the sides of a triangle to the cosine of one of its angles. Specifically, it states that:

c² = a² + b² - 2ab cos(C).

We can use the Law of Cosines to find the distance between City A and City B. Let's call this distance d.

From the information given, we know that:

The distance between the satellite and city A is 450 km.

The distance between the satellite and city B is 340 km.

The angle between city A, the satellite, and city B is 1.5 degrees.

Using the Law of Cosines, we have:

d² = 450² + 340² - 2(450)(340)cos(1.5)

d² = 202500 + 115600 - 2(450)(340)cos(1.5)

d² = 318100 - 122328.8

d² = 195771.2

d = √195771.2

d ≈ 442.3

Therefore, the distance between city A and city B is approximately 442.3 km (rounded to the nearest tenth).

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Answer:

x = 58.03°

Step-by-step explanation:

Given:

Solve for x:

1. [tex]x=cos^{-1}(\frac{9}{17})=58.03[/tex]

Answer:

So, the measure of angle x is equal to 58.03.

Using factorization and simplifying the equations, the points of intersections are (-2, 0), ( [ -1 - 3√(7) ] / 2, 4[ -1 - 3√(7) ] / 2 - 11 ) and ( [ -1 + 3√(7) ] / 2, 4[ -1 + 3√(7) ] / 2 - 11 )

We are given two equations:

y = 4x² - 3x + 3

y = x³ + 7x² - 3x + d

and we know that they intersect at x = -4, so we can substitute -4 for x in both equations:

y = 4(-4)² - 3(-4) + 3 = 49

y = (-4)³ + 7(-4)² - 3(-4) + d = -64 + 112 + 12 + d = 60 + d

So, at x = -4, we have y = 49 and y = 60 + d. Since the graphs intersect, these two equations must be equal:

49 = 60 + d

Solving for d, we get:

d = -11

Therefore, the two equations become:

y = 4x² - 3x + 3

y = x³ + 7x² - 3x - 11

We can now set them equal to each other:

4x² - 3x + 3 = x³ + 7x² - 3x - 11

Simplifying and rearranging, we get:

x³ + 3x² - 8x - 14 = 0

We can try to factor this expression by testing possible roots. One possible root is x = 2, because if we substitute 2 for x, we get:

2³ + 3(2)² - 8(2) - 14 = 8 + 12 - 16 - 14 = -10

Since this expression evaluates to a non-zero value, x = 2 is not a root. Similarly, we can test x = -1:

(-1)³ + 3(-1)² - 8(-1) - 14 = -1 + 3 + 8 - 14 = -4

This expression also evaluates to a non-zero value, so x = -1 is not a root. Finally, we can test x = -2:

(-2)³ + 3(-2)² - 8(-2) - 14 = -8 + 12 + 16 - 14 = 6

This expression evaluates to zero, so x = -2 is a root. Using long division or synthetic division, we can divide the cubic polynomial by x + 2 to get:

x³ + 3x² - 8x - 14 = (x + 2)(x² + x - 7)

The quadratic factor x² + x - 7 can be factored using the quadratic formula, giving us:

x² + x - 7 = [ -1 ± √(1 + 4*7) ] / 2

= [ -1 ± 3√(7) ] / 2

Therefore, the three intersection points are:

(-2, 0)

( [ -1 - 3√(7) ] / 2, 4[ -1 - 3√(7) ] / 2 - 11 )

( [ -1 + 3√(7) ] / 2, 4[ -1 + 3√(7) ] / 2 - 11 )

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Answer:

14.4

Step-by-step explanation:

percentage for corn =30

total no. of acres of the land=48

=30 × 48

100

=14.4

The equation that shows how to find a percentage is:

part/whole = percent/100

This equation can be used to solve for any of the three variables, given the values of the other two.

None of the given numbers make the equation 8/y² + 2 true

What is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas. It involves the study of variables, expressions, equations, and functions.

To solve this problem, we can substitute each of the given numbers (0, 1, 2, 3, 4) for y in the equation 8/y² + 2 and see if the equation is true.

Substituting y=0 would make the denominator of the fraction zero, which is undefined, so y=0 is not a valid choice.

Substituting y=1 would give us:

8/1² + 2 = 8 + 2 = 10

So, 1 is not the answer.

Substituting y=2 would give us:

8/2² + 2 = 8/4 + 2 = 2 + 2 = 4

So, 2 is not the answer.

Substituting y=3 would give us:

8/3² + 2 = 8/9 + 2 = 0.888 + 2 = 2.888

So, 3 is not the answer.

Substituting y=4 would give us:

8/4² + 2 = 8/16 + 2 = 0.5 + 2 = 2.5

So, 4 is not the answer.

Therefore, none of the given numbers make the equation 8/y² + 2 true.

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The length of line segment AB is equal to 8 units.

The magnitude of m∠BAQ is equal to 58°.

In Mathematics and Geometry, a perpendicular bisector can be defined as a line that bisects or divides a line segment exactly into two (2) equal halves and forms an angle that has a magnitude of 90 degrees at the point of intersection.

This ultimately implies that, the length of line segment AB can be calculated by using the following mathematical equation;

RP = 2AB

AB = RP/2

AB = 16/2

AB = 8 units.

Since A and B are the midpoints of PQ and RQ respectively, we have the following angles;

m∠P + m∠Q + m∠R = 180° (sum of all interior angles of ∆PQR)

58° + 38° + m∠R = 180°

m∠R = 180° - (58° + 38°)

m∠R = 84°

Since PR || AB, we have;

m∠R = m∠ABQ = 84° (corresponding angles).

m∠Q + m∠ABQ + m∠BAQ = 180° (sum of all interior angles of ∆ABQ).

m∠BAQ = 180° - (84° + 38°)

m∠BAQ = 180° - 122°

m∠BAQ = 58°.

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Complete Question:

In the ΔPQR, A and B are the midpoints of PQ and RQ respectively. If RP = 16, m∠P = 58°, and m∠Q = 38°, obtain AB and m∠BAQ.

An image of polygon VWYZ has a line of reflection across the x-axis. So option A is correct.

In geometry, the line of reflection is the line over which a figure is reflected. When a point is reflected over a line, it moves the same distance on the opposite side of the line. The line of reflection is perpendicular to the figure at the point of reflection and is equidistant from the figure and its reflection.

A polygon is a closed two-dimensional shape with straight sides. The sides are line segments that intersect only at their endpoints, called vertices. Polygons can have any number of sides, but they must have at least three. Common examples of polygons include triangles, squares, rectangles, pentagons, hexagons, and octagons.

The formula for the line of reflection in a Cartesian coordinate system is:

x = a

where a is the equation of the line of reflection.

This formula represents a vertical line that passes through the point (a, 0) and reflects points across the line of reflection to their corresponding images on the other side.

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Answer:

reflection across x=2

Step-by-step explanation:

the picture should be able to explain itself! :)))

Answer: the length of TR is 24.5.

Step-by-step explanation: 9x + 1 = 13x - 11

22 = 4x

x = 5.5

Presently that we know x, ready to discover the values of PR and QS:

PR = 9x + 1 = 9(5.5) + 1 = 49

QS = 13x - 11 = 13(5.5) - 11 = 56

Since TR is half of the PR bisector, ready to discover it by separating PR by 2:

TR = PR/2 = 49/2 = 24.5

The two middle terms, -11w and +11w, cancel each other out, leaving only the first and last terms. This is why there is no middle term with a variable in the factorization of the difference of squares pattern.

When expanding a binomial expression in the form of (a + b)ⁿ, the middle term can be found using the following formula:

Middle term coefficient = nC(k), where k = (n+1)/2 if n is odd, and k = n/2 or (n/2 + 1) if n is even. The middle term coefficient is then multiplied by the product of a raised to the power of (n-k) and b raised to the power of k.

The difference of squares pattern is a special algebraic pattern that arises when we factor a polynomial that is the difference between two perfect squares. For example,

x² - y² = (x+y)(x-y)

When we apply this pattern to the expression w - 121, we can rewrite it as:

w² - 11²

And, we can use the difference of squares pattern to factor it as:

(w + 11)(w - 11)

Notice that there is no middle term with a variable in this factorization. This is because when we multiply (w + 11)(w - 11), the middle term cancels out.

To see why this happens, let's expand the product:

(w + 11)(w - 11) = w² - 11w + 11w - 121

The two middle terms, -11w and +11w, cancel each other out, leaving only the first and last terms. This is why there is no middle term with a variable in the factorization of the difference of squares pattern.

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0 is the value of (p o q) (0) in function .

What are functions in mathematics?

Function, in mathematics, a formula, rule, or law that defines the relationship between one variable (the independent variable) and another (the dependent variable).

                                  Functions are ubiquitous in mathematics and essential in the natural sciences for formulating physical relationships. A function is a relationship between inputs, with each input associated with exactly one output. Each function has a domain and a codomain or scope. 

P(0) = 0

Q(0) = - 1

(P o q)(0)  =  P(0) * Q(0)

         = 0 * - 1

          = 0

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To calculate the variance of complaints received per week, we first need to find the mean or expected value of the number of complaints. This can be calculated by multiplying the number of complaints in each category by its corresponding probability, and then adding up the results.

Expected value = (0 x 0.13) + (1 x 0.21) + (2 x 0.44) + (3 x 0.07) + (4 x 0.06) + (5 x 0.09) = 2.54

Next, we can use the following formula to calculate the variance:

Variance = Σ[(x - µ)²P(x)]

where Σ = sum, x = number of complaints, µ = expected value, and P(x) = probability.

Using this formula and the probabilities given in the table, we get:

Variance = (0 - 2.54)²(0.13) + (1 - 2.54)²(0.21) + (2 - 2.54)²(0.44) + (3 - 2.54)²(0.07) + (4 - 2.54)²(0.06) + (5 - 2.54)²(0.09) = 1.4856

Rounding this answer to the nearest hundredth, we get a variance of 1.49.

In conclusion, the variance of complaints received per week is 1.49, meaning that the number of complaints is quite spread out around the mean, which is 2.54. This could indicate that the department store has some variability in its service quality, as the number of complaints can vary widely from week to week.

Higher rates of depression and anxiety are one negative mental health outcome associated with social media use

The paragraph discusses the role of social media in modern life and its impact on mental health. It notes that social media has become a ubiquitous part of life that allows people to connect with others from around the world, including friends, family, and strangers.

However, the paragraph goes on to mention that social media use has been linked to negative mental health outcomes such as depression, anxiety, and stress.

These negative outcomes may be due to the pressure individuals feel to present a perfect image of themselves online, leading to feelings of inadequacy and self-doubt.

Despite these negative outcomes, the paragraph notes that social media can also be valuable for promoting social support and connection, especially among marginalized communities.

Furthermore, it can be a source of education and awareness, allowing people to learn about important social and political issues and engage with them.

Hence,

Higher rates of depression and anxiety are one negative mental health outcome associated with social media use

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According to the question the volume of soil in Pedro's planter box is 56 cubic feet (ft³).

Volume is a measure of the amount of space that a three-dimensional object occupies or contains. It is measured in units of cubic centimeters (cm3), cubic meters (m3) or liters. Volume is an important concept in many areas of mathematics, including geometry, calculus and physics. It is also a key concept in many other disciplines, including engineering, architecture and the sciences. Volume is an important property of a solid object, and it is often used to quantify the size or capacity of an object. Volume can be calculated by multiplying the length, width and height of an object. It can also be calculated using the formula for the volume of a cylinder, cone, sphere or other shapes.

To calculate this, we need to multiply the length of the planter box (7 ft) by the width (4 ft) by the height of the soil (2 ft). This gives us the following equation: 7 ft x 4 ft x 2 ft
= 56 ft³.

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Answer:3) 50 feet and 10 feet

Step-by-step explanation: Multiply every single choice

Answer:

4)

Step-by-step explanation:

Let x1 = length; x2 = width

+) x1 × x2 = 500

+) 2(x1 + x2) = 90 => x1 + x2 = 45

=> x1 and x2 are the roots of equation: [tex]x^2-45x+500=0[/tex]

=> x1 = 25; x2 = 20

Answer:

Step-by-step explanation:

a lot (90,000) (i think)

Therefore, Fatima's monthly payment would be $460.23.

Interest is the amount of money that is charged by a lender to a borrower for the use of money or a loan. It is usually calculated as a percentage of the principal amount (the amount of money borrowed) and can be either simple or compound, depending on the type of loan. Interest is typically paid by the borrower to the lender over a set period of time, such as monthly, quarterly, or annually. The interest rate can be fixed or variable, and is determined by a variety of factors including the borrower's creditworthiness, the amount of the loan, and the length of the repayment period.

Here,

To calculate the monthly payment for a car loan, we need to use the formula:

M = P [ i(1 + i)ⁿ] / [ (1 + i)ⁿ – 1]

Where:

M = monthly payment

P = principal (the amount of the loan)

i = monthly interest rate (annual interest rate divided by 12)

n = number of months

First, we need to calculate the monthly interest rate:

i = 4.7% / 12

= 0.003917

Next, we need to calculate the number of months:

n = 5 years x 12 months/year

= 60 months

Now, we can plug in the values and solve for M:

M = 25,000 [ 0.003917(1 + 0.003917)⁶⁰ ] / [ (1 + 0.003917)⁶⁰ – 1]

M = $460.23 (rounded to the nearest cent)

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The statement "The product of a number and ten decreased by eight" is represented by the correct equation D. 10n - 8.

A number's product is the outcome of multiplying it by another number.

Addition, subtraction, multiplication, division, exponents, and square roots are some other examples of mathematical expressions.

A mathematical action known as an exponent symbolizes the repeated multiplication of the same integer. It is represented by a tiny number above and to the right of a base number.

For instance, 23 signifies that the number 2 has been multiplied three times, for a result of 8.

The sentence "The product of a number and ten decreased by eight" is correctly expressed as D. 10n - 8.

With this formula, a number is multiplied by 10 first, and the result is then reduced by 8.

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Answer: 30 and 5

Step-by-step explanation:

We can use the procedures below to calculate what proportion of the original principal the overall amount of interest represents. This total amount of interest received corresponds to about 5.34% of the initial investment.

Divide the principal even by rate of interest, the time period, and other factors to arrive at simple interest. Simple return Equal principal + interests + hours is the marketing formula.The most typical technique to figure out interest is to use a portion of the principal sum. He would only pay his share of the 100% interest, for example, if somebody borrows $100 from the a partner and pledges to repay the loan with 5% interest. $x (0.05) = $5. When, you must pay interest.when you lend money after borrowing it and adding interest. Interest is often determined as an indicator of the overall of the loan total. The interest rate of the loan is the name given to this percentage.

Determine the total interest that was earned in Step 1.

It states that $33.75 was earned in interest overall.

Step 2: Determine the interest generated before to taxes.

The sum of the interest earned after tax can be determined by dividing the entire sum of the interest by (1 + rate of taxation), where the rate of tax is given as a decimal. Daniela was subject to a tax of 15% on the investment earnings. The tax rate in this instance is 15%, which really is equal to 0.15.

Interest gained before taxes is equal as $33.75 / (1 Plus 0.15), that results in a value of $29.35.

3. Determine the initial principal.

The $550 that Daniela first put into the CD is referred to as the original primary.

Compute the proportion of the initial principle in step four.

By dividing the sum of interest generated after tax by the original principal and multiplying the result by 100 as express it as a percentage, one can determine what proportion of the original principal the entire amount of interest represents.

(Interest paid before taxation / Original principal) / 100 equals the percentage of the original principal.

= ($possess / $550) w x 100

≈ 5.34%

Hence, the total interest earned is equivalent to roughly 5.34% of the initial capital.

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Answer: d=10

Step-by-step explanation:

Answer:

In order to determine whether 387 is a term of a sequence, we need to know the rule or formula for generating the sequence. Without this information, it is not possible to determine whether 387 is a term of the sequence or not.

If we assume that the sequence is an arithmetic sequence, where each term is obtained by adding a fixed constant to the previous term, we can use the following formula to determine whether 387 is a term of the sequence:

an = a1 + (n-1)d

where a1 is the first term of the sequence, d is the common difference between consecutive terms, and n is the term we are trying to find.

If we substitute the values for the first few terms of the sequence, we can check whether 387 is a term or not. For example, if the first few terms of the sequence are:

a1 = 3

a2 = 8

a3 = 13

a4 = 18

and so on, with a common difference of 5 between consecutive terms, we can use the formula to find the value of the 129th term of the sequence:

a129 = a1 + (129-1)d

a129 = 3 + 128(5)

a129 = 643

Since 387 is not equal to 643, it is not a term of this sequence. However, without knowing the rule or formula for generating the sequence, it is impossible to say for certain whether 387 is a term or not.

Answer:

a) To solve the differential equation y''-2y'-3y= e^4x, we first find the characteristic equation:

r^2 - 2r - 3 = 0

Factoring, we get:

(r - 3)(r + 1) = 0

So the roots are r = 3 and r = -1.

The general solution to the homogeneous equation y'' - 2y' - 3y = 0 is:

y_h = c1e^3x + c2e^(-x)

To find the particular solution, we use the method of undetermined coefficients. Since e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = Ae^4x

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

y_p' = 4Ae^4x

y_p'' = 16Ae^4x

Substituting these into the original differential equation, we get:

16Ae^4x - 8Ae^4x - 3Ae^4x = e^4x

Simplifying, we get:

5Ae^4x = e^4x

So:

A = 1/5

Therefore, the particular solution is:

y_p = (1/5)*e^4x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^3x + c2e^(-x) + (1/5)*e^4x

b) To solve the differential equation y'' + y' - 2y = 3xe^x, we first find the characteristic equation:

r^2 + r - 2 = 0

Factoring, we get:

(r + 2)(r - 1) = 0

So the roots are r = -2 and r = 1.

The general solution to the homogeneous equation y'' + y' - 2y = 0 is:

y_h = c1e^(-2x) + c2e^x

To find the particular solution, we use the method of undetermined coefficients. Since 3xe^x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax + B)e^x

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

y_p' = Ae^x + (Ax + B)e^x

y_p'' = 2Ae^x + (Ax + B)e^x

Substituting these into the original differential equation, we get:

2Ae^x + (Ax + B)e^x + Ae^x + (Ax + B)e^x - 2(Ax + B)e^x = 3xe^x

Simplifying, we get:

3Ae^x = 3xe^x

So:

A = 1

Therefore, the particular solution is:

y_p = (x + B)e^x

Taking the derivative of y_p, we get:

y_p' = (x + 2 + B)e^x

Substituting back into the original differential equation, we get:

(x + 2 + B)e^x + (x + B)e^x - 2(x + B)e^x = 3xe^x

Simplifying, we get:

-xe^x - Be^x = 0

So:

B = -x

Therefore, the particular solution is:

y_p = xe^x

The general solution to the non-homogeneous equation is:

y = y_h + y_p

y = c1e^(-2x) + c2e^x + xe^x

c) To solve the differential equation y" - 9y' + 20y = x^2*e^4x, we first find the characteristic equation:

r^2 - 9r + 20 = 0

Factoring, we get:

(r - 5)(r - 4) = 0

So the roots are r = 5 and r = 4.

The general solution to the homogeneous equation y" - 9y' + 20y = 0 is:

y_h = c1e^4x + c2e^5x

To find the particular solution, we use the method of undetermined coefficients. Since x^2*e^4x is a solution to the homogeneous equation, we try a particular solution of the form:

y_p = (Ax^2 + Bx + C)e^4x

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

y_p' = (2Ax + B)e^4x + 4Axe^4x

y_p'' = 2Ae^4x +

The weight of the diamonds in each lot was 1/10 of a pound, or 0.1 pound.

What is arithmetic sequence?

An arithmetic sequence is a sequence of numbers in which each term after the first is found by adding a fixed constant number, called the common difference, to the preceding term. For example, the sequence 2, 5, 8, 11, 14, ... is an arithmetic sequence with a common difference of 3, since each term after the first is found by adding 3 to the preceding term.

The nth term of an arithmetic sequence can be found using the formula:

an = a1 + (n-1)d.

The merchant received 7/10 of a pound of diamonds and divided them into 7 equal lots. To find the weight of each lot, we need to divide 7/10 by 7:

(7/10) ÷ 7 = (7/10) × (1/7) = 1/10

Therefore, the weight of the diamonds in each lot was 1/10 of a pound, or 0.1 pound.

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The width of the rectangle is 4 units.

What is rectangle?

Rectangle is a four sided polygon or specifically a particular type of parallelogram having two opposite sides are equal and one angle is right angle that is 90°. It has four vertices and two diagonals intersect each other.

Given that,

The length of a rectangle is 2 units more than the width.

Let, the width of the rectangle is w units.

Then the length of the rectangle is w+2 units.

Area of any rectangle is length × width

                                        = (w+2)×w square units

Given that,

The area of the rectangle is   24 square units.

Equating both  the values we get,

                    w(w+2)= 24

We have to solve the equation for w.

    Multiplying the bracket term with w we get,

w²+ 2w = 24

⇒ w² + 2w- 24=0

⇒ (w+6)(w-4)=0

so either w= -6 or w=4

As width cannot be negative so w= 4.

Hence, the width of the rectangle is 4 units.

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Answer:

the value of $s$ is 13.

Step-by-step explanation:

We are given that $r=9$ and $4r+3s=75$. We can use the value of $r$ to find $s$.

Substituting $r=9$ into the second equation, we get:

$4(9) + 3s = 75$

Simplifying the left-hand side, we get:

$36 + 3s = 75$

Subtracting 36 from both sides, we get:

$3s = 39$

Dividing both sides by 3, we get:

$s = 13$

Therefore, the value of $s$ is 13.

Answer: y=6x-1

Step-by-step explanation:

y=6x-1

x represents the time luke has been riding(hours) and the y represents the distance he has travelled (miles)