Father's age is five times than the son's age. If the sum of their age is 60 years, find their present ages.

Please solve this with step by step.​

Answer:

Probability of drawing an odd number.

Number of odd numbers = 5

Number of numbers in the set = 10

So it's a 5 in 10 chance or 1 in 2 chance.

Probability of drawing a multiple of 3.

Multiples of 3 in the set = 3, 6 and 9 = 3 multiples of 3

Number of numbers in the set = 10

So it's a 3 in 10 chance

Answer:

-1 ≥ 7

Step-by-step explanation:

-1 ≥ 7            =         -4 ≥ 28

  -4 ≥ 28

We simplify both sides by 4 and get -1 ≥ 7

"The multiplier for decay is given by the formula:

multiplier = (1 - (decay rate/100))^n

where "decay rate" is the percentage rate of decay per time period (usually per year), and "n" is the number of time periods.

For a decay rate of 1%, the multiplier would be:

multiplier = (1 - (1/100))^n = 0.99^n

For a decay rate of 2%, the multiplier would be:

multiplier = (1 - (2/100))^n = 0.98^n

The value of the multiplier depends on the number of time periods "n". For example, if we want to find the value of an initial quantity after 3 years of 1% decay, we would use the multiplier of 0.99 raised to the power of 3:

multiplier = 0.99^3 = 0.9703

This means that the final quantity would be 97.03% of the initial quantity.

Similarly, if we want to find the value of an initial quantity after 5 years of 2% decay, we would use the multiplier of 0.98 raised to the power of 5:

multiplier = 0.98^5 = 0.9039

This means that the final quantity would be 90.39% of the initial quantity." (ChatGPT, 2023)

(a) A sound with an intensity of 1 watt per square meter has a decibel level of -120.

(b) A sound with an intensity of 10⁻² watt per square meter has a decibel level of -140.

(c) The number of decibels is not 100 times as great as the intensity, even though the intensity in part (a) is 100 times greater than the intensity in part (b).

Using the relationship of intensity of sound and number of decibels, we can calculated the required number of decibels as follows;

β =10 log(I x 10⁻¹²)

(a) Decibel level: plugging in I = 1 watt per square meter into the equation, we get:

β = 10 log(1 x 10⁻¹²) = 10 log(10⁻¹²) = -120 decibels

(b) Decibel level: plugging in I = 10⁻² watt per square meter into the equation, we get:

β = 10 log(10⁻² x 10⁻¹²) = 10 log(10⁻¹⁴) = -140 decibels

(c) Let's calculate the decibel level for the sound in part (a) using the given formula:

β₁ = 10 log(I₁ x 10⁻¹²) = 10 log(100 x I₂ x 10⁻¹²) = 10 log(I₂ x 10⁻¹⁴) + 20

where;

Using the result from part (b), we can substitute I₂ = 10⁻² watt per square meter and get:

β₁ = -140 + 20 = -120 decibels

Comparing this result to the decibel level for the sound in part (a) (which is also -120 decibels), we see that they are the same.

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The correct answer is experimental 22.5% theoretical 18.2%.

The theoretical probability is the likelihood of the event to happen without conducting an experiment,

The experimental probability is the result of the actual experiment, which in this case is the frequency of the letter M when chosen from the bag.  Theoretical probability in this case is the number of M's in the bag divided by the total number of letters in the bag.

In this case, there are nine M's out of a total of 39 letters, giving a theoretical probability of 18.2%. Since the number of M's chosen in the experiment was nine, out of a total of 39 chosen, the experimental probability is 22.5%.

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The correct answer is experimental 22.5% theoretical 18.2%. To calculate the experimental probability of drawing the letter M, we must first identify the total number of items in the bag.

The theoretical probability is the likelihood of the event to happen without conducting an experiment,

The total number of items in the bag is 9 + 12 + 7 + 2 + 4 + 1 + 3 + 2 = 40. Next, we divide the number of items with the letter M (9) by the total number of items to get the experimental probability of drawing the letter M.

Therefore, 9/40 = 0.225

= 22.5%.  

To calculate the theoretical probability of drawing the letter M, we must first identify the total number of letters in the bag.

There are 8 letters in the bag (M, A, T, H, E, I, C, S).

Therefore, the theoretical probability of drawing the letter M is

8/40 = 0.2

≈18.2%

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Therefore, the zeros of the function f(x) are x = 5 and x = -4.

A polynomial is a mathematical expression consisting of variables and coefficients, combined using the operations of addition, subtraction, multiplication, and non-negative integer exponents. It can have one or more terms, and the degree of a polynomial is the highest power of the variable in the expression.

Here,

To find the zeros of the function f(x) = x² - x - 20, we need to solve for x when f(x) = 0:

x² - x - 20 = 0

We can factor the left side of this equation as:

(x - 5)(x + 4) = 0

Using the zero product property, we know that the product of two factors is zero if and only if at least one of the factors is zero. Therefore, we can set each factor equal to zero and solve for x:

x - 5 = 0 or x + 4 = 0

Solving for x in each equation gives us:

x = 5 or x = -4

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Answer:

69 and 69

Step-by-step explanation:

68 + 1 = 69

69 + 0 = 69

look how much people i hav helped

The sides of the triangle to the nearest hundredth are: a ≈ 8.99, b ≈ 7.96

and c ≈ 9.44.

To solve the triangle, we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all three sides.

Using this law, we can find the length of side a as follows:

sin A / a = sin B / b

sin 56° / a = sin 57° / 9

a = 9 * sin 56° / sin 57°

a ≈ 8.99

Similarly, we can find the length of side c as follows:

sin C / c = sin B / b sin (180° - A - B) / c = sin 57° / 9

sin 67° / c = sin 57° / 9

c = 9 * sin 67° / sin 57°

c ≈ 9.44

Now, we can use the Law of Cosines to find the length of the remaining side, side b:

b 2 = a 2 + c 2 - 2ac cos B

b 2 = (8.99) 2 + (9.44) 2 - 2(8.99)(9.44) cos 57°

b ≈ 7.96

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The headline should be informative and accurately reflect the data presented in the chart.

A chart is a visual representation of data or information that can be used to help people understand complex information quickly and easily. Charts can take many different forms, such as bar charts, line charts, scatter plots, pie charts, and more. They are often used in business, science, and journalism to present data in a way that is clear and easy to interpret. By using charts, people can quickly see patterns and trends in data that might be difficult to discern from a simple list of numbers or words.

Here,

However, a good headline for a chart should accurately and concisely summarize the main findings or trends depicted in the chart. It should also capture the reader's attention and make them want to read more. Depending on the chart, some possible headlines could be:

"New study finds rising trend in global temperatures over past decade"

"Stock market reaches all-time high, driven by tech sector growth"

"Poll shows majority of Americans support stricter gun control laws"

"Obesity rates continue to climb in US, particularly in southern states"

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If the diagonal of the rectangle is 26 and its perimeter is 68, its dimensions are either 24 by 10 or 10 by 24.

Let's denote the length and width of the rectangle as L and W, respectively.

We know that the diagonal of the rectangle is 26, which we can express using the Pythagorean theorem:

[tex]L^2[/tex]+ [tex]W^2[/tex]=[tex]26^2[/tex]

We also know that the perimeter of the rectangle is 68:

2L + 2W = 68

We can simplify the second equation by dividing both sides by 2:

L + W = 34

We can use this simplified equation to express one variable in terms of the other:

L = 34 - W

This is what we get when we enter it into the first equation:

[tex](34 - W)^2[/tex] + [tex]W^2[/tex] = [tex]26^2[/tex]

Expanding the square and simplifying, we get:

[tex]2W^2[/tex] - 68W + 420 = 0

Dividing both sides by 2, we get:

[tex]W^2[/tex]- 34W + 210 = 0

The quadratic formula can be used to find W:

W = (-b ± t[tex]\sqrt{(b^2 - 4ac)}[/tex] )/ 2a

where a = 1, b = -34, and c = 210. Plugging in these values, we get:

W = (34 ± [tex]\sqrt{34^2 - 4(1)(210))}[/tex]) / 2(1)

W = (34 ± [tex]\sqrt{196}[/tex]) / 2

W= (34 ± 14)/2

W = 17 ± 7

As the breadth cannot be negative, we can disregard the negative solution. Therefore, we have:

W = 24 or W = 10

If W = 24, then L = 10 (using L = 34 - W). If W = 10, then L = 24. Therefore, the dimensions of the rectangle are either 24 by 10 or 10 by 24.

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The missing values are given as shown in the table

A cylinder is a three-dimensional solid figure which has two identical circular bases joined by a curvilinear surface at a particular distance from the center which is the height of the cylinder.

We should understand that the area of a cylinder is given by

Area = 2λr(r+h)

and the volume of the cylinder to be;

volume - λr²h

by substitution we get the values as

diameter        radius         base area     height     `cylinder volume

units                units             square         units          cubic units

                                                units

4                         2                 150.9            10                   62.9

6                         3                151.0               7                    63π

8                          4               25π                  0.875           80π

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Answer: A all you have to do is look where the line is going

Answer:

C

Step-by-step explanation:

under a reflection in the line y = - x

a point (x, y ) → (- y, - x ) , then

(- 1, 6 ) → (- 6, - (-1) ) → (- 6, 1 )

(- 3, 2 ) → (- 2, - (- 3) ) → (- 2, 3 )

The inverse of the function f(x) = 2(x - 4)³ + 7 is g(x) = ∛[(x - 7) / 2] + 4.

A mathematical notion known as a function explains the connection between two sets of values known as the domain and the range. A function takes a value from the domain as an input, applies some logic to it, and returns a value in the range. Just one output value in the range corresponds to each input value in the domain. Functions can be described verbally or with mathematical symbols. They are employed throughout the mathematical discipline as well as in the sciences of physics, engineering, economics, and computer science.

The given function is f(x) = 2(x - 4)³ + 7.

Swap the variables, that is substitute x = y and vice versa as follows:

x = 2(y - 4)³ + 7

x - 7 = 2(y - 4)³

(x - 7) / 2 = (y - 4)³

Taking the cube root of both sides gives:

y - 4 = ∛[(x - 7) / 2]

y = ∛[(x - 7) / 2] + 4

Therefore, the inverse of f(x) = 2(x - 4)³ + 7 is g(x) = ∛[(x - 7) / 2] + 4.

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Answer:

-32

Step-by-step explanation:

since 4x+y equals 7, multiply -5 by 7

Answer:

40x - 5y = -40x - 5(4x + y) = -40x - 20x - 5(7) = -60x - 35

Step-by-step explanation:

To find the value of -5(4x+y), we first need to simplify the expression inside the parentheses:

-5(4x + y) = -5(4x) - 5(y) = -20x - 5y

Now, we can substitute the value of y from the given equation:

-20x - 5y = -20x - 5(4x + y) = -20x - 20x - 5y = -40x - 5y

Since we know that 4x + y = 7, we can substitute this into the above equation:

-40x - 5y = -40x - 5(4x + y) = -40x - 20x - 5(7) = -60x - 35

Answer:

i think 150 is the answer

Step-by-step explanation:

150 is a 3 digit numbers (as written in question (a)) and is a multiple of 5 and 3

150/5 = 30

150/3 = 50

In response to the given question, we can state that With d = 4 metres, polynomials the deck dimensions are 16 metres by 128 metres, and the deck area is 2048 square metres.

A polynomial is a mathematical statement composed of equations and uncertainty that exclusively uses additions, addition and subtraction, multiplications, and real number powers of variables. The form x2 4x + 7 indicates a single determinate x algebraic. A polynomial expression in mathematics is made up of determinants (also known as freshly made) and equations that may be added, deducted, multiplied, then raised to minus integer powers of semi. A polynomial is an algebraic statement that includes variables and coefficients. An expressions can really only incorporate the operations add, subtraction, duplication, and non-negative integer factors. These expressions are referred to as polynomials.

substituting d = 4 into the formulas given in the problem.

Then, using the polynomial you discovered in Part A, we can calculate the breadth of the deck when d = 4.

8d2 + 20d = 8(42) + 20(4) = 128 width

With d = 4 metres, the breadth of the deck is 128 metres.

Next, we may utilise the supplied deck length, 4d = 4(4) = 16 metres, to calculate the deck area when d = 4:

16 × 128 = 2048 Area = Length x Width

With d = 4 metres, the deck dimensions are 16 metres by 128 metres, and the deck area is 2048 square metres.

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The dilation of triangle ABC about point P with a scale factor of 12 is an enlargement.

In a dilation, the size of a figure is changed by multiplying its dimensions by a scale factor. If the scale factor is greater than 1, the figure is enlarged. If the scale factor is between 0 and 1, the figure is reduced in size. In this case, the scale factor is 12, which is greater than 1. Therefore, triangle ABC is enlarged to form triangle A'B'C' by a factor of 12. Therefore, the dilation is an enlargement.

In the context of triangles, an enlargement means that all sides and angles of the original triangle are increased by the same factor to create a larger triangle. In this case, the sides and angles of triangle ABC are multiplied by a factor of 12 to create triangle A'B'C'. As a result, the corresponding sides and angles of the two triangles are proportional, but the enlarged triangle is 12 times larger than the original triangle.

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Answer:

t = 1

Step-by-step explanation:

I will assume that there is a parenthesis missing:  the equation should read 3t-18=4(-3- 3/4t) and not 3t-18=4(-3- 3/4t

3t-18=4(-3- 3/4t)

3t-18= -12- 12/4t   [Remove parentheses by multiplying each term by 4]

3t-6= - 3/t    [Simplify]

3t^2-6t = -3  [Multiply both sides by t]

3t^2-6t +3 = 0

3(t^2-2t+1)=0

3(t-1)*(t-1) = 0

Two solutions, both the same value:  

(t-1) = 0

t = 1

===

Does t=1 result in 0 for the equation?

3t-18=4(-3- 3/4t)

3(1)-18=4(-3- 3/4*(1))

3-18 = 4(-3-3/4)

-15 = 4(-3.75)

-15 = -15  YES

The ladder slipped down the wall by approximately 296 cm to the nearest 1 cm.

We can use the Pythagorean theorem to solve this problem.

Let the distance the ladder slips down the wall be represented by x (in cm).

Then, at the start of the day, we have a right triangle formed by the wall, the ground, and the ladder, with the ladder being the hypotenuse.

The length of the ladder is 3m = 300cm, and the distance from the foot of the ladder to the wall is 50cm.

Therefore, we have:

(300)² = x² + (50)²

Simplifying this equation, we get:

90000 = x² + 2500

Subtracting 2500 from both sides, we get:

87500 = x²

Taking the square root of both sides, we get:

x = √87500

x = 295.8 cm

x ≈ 296 cm

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Using linear equations 5x + 8y = 115 and 3x + 5y = 70,  the cost of each table is $15 and the cost of each chair is $5.

What is a linear equation, exactly?

A linear equation is a mathematical expression that describes a straight line relationship between two variables. It can be written in the form y = mx + b, where x and y are variables, m is the slope of the line, and b is the y-intercept.

Now,

Let x = cost of each table and y= cost of each chair. Then, we can set up the following system of linear equations based on the given information:

5x + 8y = 115 (equation 1)

3x + 5y = 70 (equation 2)

Equation 1 represents the cost of 5 small tables with 8 chairs, and equation 2 represents the cost of 3 tables with 5 chairs.

Now,

Solve equation 2 for x:

3x + 5y = 70

3x = 70 - 5y

x = (70 - 5y)/3

Substitute x into equation 1:

5x + 8y = 115

5[(70 - 5y)/3] + 8y = 115

Simplify and solve for y:

350/3 - 25/3 y + 8y = 115

23/3 y = 115 - 350/3

y = 5

Substitute y = 5 into equation 2 and solve for x:

3x + 5y = 70

3x + 5(5) = 70

3x = 45

x = 15

Therefore,

               the cost of each table is $15 and the cost of each chair is $5.

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The population of a city with 400,000 population growing at a rate of 4% per year can be calculated using the following formula:
P​(t) = P​0 ​ (1 + r)​t
Where P​0 is the initial population, r is the growth rate, and t is the number of years the growth is applied.
In this example, the initial population is 400,000, the growth rate is 0.04, and t is 17 years. Applying this formula, we get:
P(17) = 400,000 (1 + 0.04)17
P(17) = 615,288
Therefore, the population of this city in 17 years will be 615,288 if the growth rate of 4% per year continues.

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Answer:

The correct option is A. x = -1, x = 3, x = ±√5.

We found the zero x = -1 through synthetic division, and then we factored the cubic polynomial using the Rational Root Theorem and synthetic division to obtain (x + 1)(x^3 - 3x^2 - 6x + 15). We found that the remaining zeros of the polynomial function are the roots of the quadratic factor x^2 - 3x - 5, which are x = (3 ± √29))/2.

Therefore, the zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15 are x = -1, x = 3, x = (3 + √(29))/2, and x = (3 - √(29))/2, which simplifies to x = (3 ± √(5))/2.

Option A lists all of these zeros, so it is the correct option. Options B and C do not list all of the zeros of the polynomial function.

STEPS: Here are the steps to find all the zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15:

Write the polynomial function in descending order of degree: f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15.

Use the Rational Root Theorem to generate a list of possible rational zeros: ±1, ±3, ±5, ±15.

Use synthetic division to test each possible zero. We start with x = -1:

-1 │ 1 -2 -8 10 15

│ -1 3 -5 -5

└───────────────

1 -3 -5 5 10

x = -1 is a zero of the polynomial function. We can write f(x) as:

f(x) = (x + 1)(x^3 - 3x^2 - 5x + 10)

Use the Rational Root Theorem and synthetic division to factor the cubic equation x^3 - 3x^2 - 5x + 10:

3 │ 1 -3 -5 10

│ 3 0 -15

└─────────────

1 0 -5 -5

x = 3 is not a zero of the polynomial function.

-3 │ 1 -3 -5 10

│ -3 24 -57

└────────────

1 -6 19 -47

x = -3 is not a zero of the polynomial function.

The only remaining possible rational zeros are ±1/1 and ±5/1, but testing these values using synthetic division does not yield any more zeros.

Solve for the remaining zeros of the polynomial function by factoring the quadratic equation x^2 - 3x - 5 using the quadratic formula or factoring by grouping:

x = (3 ± √(29))/2

These are the remaining zeros of the polynomial function.

Therefore, the zeros of the polynomial function f(x) = x^4 - 2x^3 - 8x^2 + 10x + 15 are x = -1, x = 3, x = (3 + √(29))/2, and x = (3 - √(29))/2, which simplifies to x = (3 ± √(5))/2.

Option A lists all of these zeros, so it is the correct option.

Hope this helps! I'm sorry if it doesn't! :]

If the length of rectangle is 5.25, then the area of rectangle will be less than 25 square meters.

What is area of rectangle?

Area of rectangle is the region covered by the rectangle in a two-dimensional plane. A rectangle is a type of quadrilateral, a 2d shape that has four sides and four vertices.

We know the perimeter of rectangle is 20 meters. So according to formula of Perimeter of rectangle = 2 (L + B) we can conclude that sum of length and width will be 10 meters.

We are given length as 1, 3, 5, 7, and 9.

So width of the rectangle will be 9, 7, 5, 3 and 1.

Area of Rectangle = L x B

So area of rectangle will be 9, 21, 25, 21, and 9.

If the length of rectangle is 5.25, then the area of rectangle will be less than 25 square meters.

Values are changing in a linear way and not in an exponential way.

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The standard matrix is:

| 0      -1 |
| -1      0 |

The standard matrix of the linear transformation T: R2 to R2 that reflects each vector through the line x1=-x2 can be found by following these steps:

1. Start with the standard basis vectors for R2, which are (1,0) and (0,1).

2. Apply the transformation T to each of these vectors to find the new vectors. For the first vector, (1,0), we have x1=1 and x2=0. The reflection through the line x1=-x2 gives us a new vector with x1=-x2=-0=0 and x2=-x1=-1. So the new vector is (0,-1).

3. For the second vector, (0,1), we have x1=0 and x2=1. The reflection through the line x1=-x2 gives us a new vector with x1=-x2=-1 and x2=-x1=-0=0. So the new vector is (-1,0).

4. The standard matrix of the transformation T is the matrix whose columns are the new vectors we found. So the standard matrix is:
```
| 0 -1 |
| -1  0 |
```
This is the standard matrix of the linear transformation T: R2 to R2 that reflects each vector through the line x1=-x2.

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In the following question, if Chin paid $16. 24 for 7 bags of flour that were on sale for 10% off. Each bag contained 2 pounds of flour. The regular price for 1 pound of flour is $1.29.

Chin paid $16.24 for 7 bags of flour that were on sale for 10% off.

Let's first find the total cost of the 7 bags of flour before the discount:

Total cost before discount = $16.24 ÷ 0.9 = $18.04

We know that each bag contained 2 pounds of flour, so the total amount of flour Chin bought was:

7 bags x 2 pounds/bag = 14 pounds

So the cost of 1 pound of flour is:

$18.04 ÷ 14 pounds = $1.29 per pound

Therefore, the regular price for 1 pound of flour would be $1.29.

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In the triangle, the values are p=10.3, [tex]\angle Q=19.5\textdegree[/tex] and [tex]\angle R = 112.5\textdegree[/tex].

The functiοns οf an angle in a triangle are knοwn as trigοnοmetric functiοns, cοmmοnly referred tο as circular functiοns. In οther wοrds, these trig functiοns prοvide the relatiοnship between a triangle's angles and sides. There are five fundamental trigοnοmetric functiοns: sine, cοsine, tangent, cοtangent, secant, and cοsecant.

Here in the given triangle PQR, [tex]\angle P = 48\textdegree[/tex] , q=5 and r=13.

Now using law of cosine fοrmula,

=> [tex]p^2=q^2+r^2-2qr.cos\angle P[/tex]

=> [tex]p^2=5^2+13^2-2.5.13.cos 48\textdegree[/tex]

=> [tex]p^2=25+169-130cos48\textdegree[/tex]

=> [tex]p^2=107[/tex]

=> p = 10.3

Now using law of sine then,

=> [tex]\frac{p}{sinP}=\frac{q}{sinQ}[/tex]

=> [tex]sinQ=\frac{q sinP}{p}[/tex]

=> [tex]sin Q= \frac{5sin 48\textdegree}{10.3}[/tex]

=> sin Q = 0.344

=>[tex]\angle Q = 19.5\textdegree[/tex]

Now sum οf all angles in triangle is 180°. Then,

=>[tex]\angle P+\angle Q+\angle R = 180\textdegree[/tex]

=> [tex]\angle R = 180-48-19.5=112.5\textdegree[/tex]

Hence the answers are p=10.3, [tex]\angle Q=19.5\textdegree[/tex] and [tex]\angle R = 112.5\textdegree[/tex].

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The sequences can converge to 0 and also diverge towards infinity.

The two sequences are not mentioned in the question. Please provide the two sequences so that I can give you the first 5 terms of each sequence and determine if each sequence converges or diverges.

To find the first 5 terms of a sequence, you need to use the formula given for the sequence and plug in the values for the first 5 terms. For example, if the sequence is given by a_n = 2n + 1, the first 5 terms would be a_1 = 2(1) + 1 = 3, a_2 = 2(2) + 1 = 5, a_3 = 2(3) + 1 = 7, a_4 = 2(4) + 1 = 9, and a_5 = 2(5) + 1 = 11.

To determine if a sequence converges or diverges, you need to find the limit of the sequence as n approaches infinity. If the limit exists, then the sequence converges to that limit. If the limit does not exist, then the sequence diverges.

For example, the sequence a_n = 1/n converges to 0 as n approaches infinity, while the sequence a_n = n^2 diverges as n approaches infinity.

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Jocelyn's village additional area is (-w² + 6w + 17) m². The combined total area of both is (5w² + 14w + 7) m².

An equation is an expression that shows how two or more numbers and variables are related using mathematical operations of addition, subtraction, multiplication, division, exponents and so on.

Let w represent he width of the town hall.

Area of Jocelyn's village = 2w² + 10w + 12

Area of Lorlesha's village = 3w² + 4w - 5

The difference in their area = (2w² + 10w + 12) - (3w² + 4w - 5) = -w² + 6w + 17

The sum of their area = (2w² + 10w + 12) + (3w² + 4w - 5) = 5w² + 14w + 7

Jocelyn's village additional area is (-w² + 6w + 17) m². The combined total area of both is (5w² + 14w + 7) m².

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The composition of the two translation rules is:

(x, y) → (x - 1, y - 6)

What is a translation rule?

A translation rule is a mathematical description of how to move each point in a geometric shape by a fixed distance in a certain direction. It is used to describe a transformation called a translation, which moves a shape without changing its size, shape, or orientation.

To find the composition of the two translation rules, we apply the second rule first and then apply the first rule to the result.

Let's consider a point (x, y). Applying the second rule (x, y) → (x - 5, y - 5) gives us a new point:

(x - 5, y - 5)

Now we apply the first rule (x, y) → (x + 4, y - 1) to this new point:

(x - 5 + 4, y - 5 - 1)

Simplifying:

(x - 1, y - 6)

Therefore, the composition of the two translation rules is:

(x, y) → (x - 1, y - 6)

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The solution to the given homogenous differential equation dy/dx = y² + x² / x^2 is y² = -x(y³ / 3 + Cy³ - 3Cx)

The given differential equation is: dy/dx = y² + x² / x^2

To solve this, we can first separate the variables by bringing all the y-terms on one side and all the x-terms on the other side:

(1/y²)dy = (x² / x² + y²)dx

Next, we can integrate both sides:

∫(1/y²)dy = ∫(x²/x² + y²)dx

Using the substitution u = y/x, we can simplify the integrals:

∫(1/y²)dy = ∫(1 + u²)dx

-1/y = x + (1/3)u³ + C

where C is the constant of integration.

Substituting back u = y/x, we get:

-1/y = x + (1/3)(y/x)³ + C

Multiplying both sides by -y³, we get:

y² = -x(y³ / 3 + Cy³ - 3Cx)

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