Please help with this math problem!

The series 2n^2 is divergent.

To express the nth partial sum as a telescoping sum, we need to find a pattern in the terms of the series.

The general term of the series is given by an = 2n^2 - ?.

The nth partial sum can be written as:

sn = a1 + a2 + a3 + ... + an

= 2(1)^2 - ? + 2(2)^2 - ? + 2(3)^2 - ? + ... + 2n^2 - ?

We can simplify the above expression by factoring out 2 from each term:

sn = 2(1^2 + 2^2 + 3^2 + ... + n^2) - n?

Using the formula for the sum of squares, we have:

sn = 2(n(n+1)(2n+1)/6) - n?

Simplifying further, we get:

sn = (n^3 + 3n^2 + 2n)/3 - n?

Taking the limit as n approaches infinity, we get:

lim n->∞ sn = lim n->∞ [(n^3 + 3n^2 + 2n)/3 - n?]

Since the term n? grows without bound as n approaches infinity, the limit of sn does not exist.

Therefore, the series 2n^2 - ? is divergent.

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After 16 years, compounded yearly, Lizzie will have $5,995.68 for college in Option A and $5,789.95 in Option B, rounded to the nearest dollar.

For Option A, the interest rate is 3% compounded yearly. Using the formula for compound interest, we have:

FV = PV x (1 + r)ⁿ

where FV is the future value, PV is the present value, r is the annual interest rate as a decimal, and n is the number of years. Plugging in the values, we get:

FV = $3000 x (1 + 0.03)¹⁶ = $5,995.68

For Option B, the interest rate is 2% compounded quarterly. We need to convert the annual interest rate to a quarterly rate by dividing it by 4. Then we use the formula:

FV = PV x (1 + r/n)^(n x t)

where r/n is the quarterly interest rate, n is the number of compounding periods per year, and t is the total number of years. Plugging in the values, we get:

FV = $3000 x (1 + 0.02/4)^(4 x 16) = $5,789.95

Therefore, Lizzie will have $5,995.68 for college in Option A and $5,789.95 in Option B, rounded to the nearest dollar.

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Each side of the square has a length of approximately 12.02.

Without additional information or a diagram, it is difficult to determine what values are missing. However, we can use the Pythagorean theorem to solve for the length of the sides of the square.

Let's assume that TR is a diagonal of the square, and let x be the length of each side. Then, by the Pythagorean theorem,

TR^2 = x^2 + x^2

17^2 = 2x^2

289 = 2x^2

x^2 = 144.5

x ≈ 12.02

We can also solve this as:

Assuming that PQRS is square and TR is a diagonal of the square, we can use the Pythagorean theorem to find the length of each side of the square. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

In this case, if we draw a diagonal TR in the square, it divides the square into two right triangles, each with sides of length x (the length of one side of the square) and TR/√2 (half the diagonal). Applying the Pythagorean theorem to one of these right triangles, we get:

(x)^2 + (TR/√2)^2 = TR^2

Simplifying and solving for x, we get:

x = √(TR^2 - (TR/√2)^2) = TR/√2 = TR * √2 / 2

Plugging in TR = 17, we get:

x = 17 * √2 / 2 ≈ 12.02

Therefore, each side of the square has a length of approximately 12.02. Without additional information or a diagram, we cannot determine any other missing values.

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The total number of gifts is given as follows:

439 gifts.

The Fundamental Counting Theorem states that if there are m ways to do one thing and n ways to do another, then there are m x n ways to do both.

This can be extended to more than two events, where the number of ways to do all the events is the product of the number of ways to do each individual event, according to the equation presented as follows:

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

For a single gift, the number of options is given as follows:

10 + 4 + 7 = 21 gifts.

For two gifts, the number of options is given as follows:

10 x 4 + 10 x 7 + 7 x 4 = 138 gifts.

For three gifts, the number of options is given as follows:

10 x 4 x 7 = 280 gifts.

Hence the total number of gifts is obtained as follows:

280 + 138 + 21 = 439 gifts.

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If Kayla can pay off up to $175 monthly for the purchase of a new television for $487.89, the interest she will pay at 23.5% compounded monthly is D. $18.87.

The compound interest payable on the credit card for the purchase of a new television can be computed using an online finance calculator as follows:

N (# of periods) = 3 months

I/Y (Interest per year) = 23.5%

PV (Present Value) = $487.89

PMT (Periodic Payment) = $-175

Results:

FV = $18.23

Sum of all periodic payments = $525.00

Total Interest = $18.87

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The total surface area of the foam pit can be calculated by finding the area of each face and adding them together.

Since the pit is a rectangular prism, it has six faces: the top, bottom, front, back, left, and right. The area of each face can be calculated using the formula for the area of a rectangle, which is length times width.

To calculate the total surface area of a rectangular prism with an open top, we need to add the areas of all six faces together.

The area of each face can be calculated using the formula for the area of a rectangle (length times width).

The top of the foam pit is open, so we don't need to include it in our calculation.

After finding the area of each face, we simply add them all together to get the total surface area.

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√6a + 3 < a + 2,  we have proven that √6a + 3 < a + 2 for all a > 1 using the Mean Value Theorem.

To use the Mean Value Theorem to prove √6a + 3 < a + 2 for all a > 1, we first note that the function f(x) = √6x + 3 is continuous on the interval [1,a].

Next, we need to find a point c in the interval (1,a) such that the slope of the line connecting (1,f(1)) and (a,f(a)) is equal to the slope of the tangent line to f(x) at c.

The slope of the line connecting (1,f(1)) and (a,f(a)) is given by:

(f(a) - f(1)) / (a - 1) = (√6a + 3 - √9) / (a - 1) = (√6a) / (a - 1)

To find the slope of the tangent line to f(x) at c, we first find the derivative of f(x):

f'(x) = (1/2) * (6x + 3)^(-1/2) * 6 = 3 / √(6x + 3)

Then, we evaluate f'(c) to get the slope of the tangent line at c:

f'(c) = 3 / √(6c + 3)

Now, by the Mean Value Theorem, there exists a point c in (1,a) such that:

f'(c) = (√6a) / (a - 1)

Setting these two expressions for f'(c) equal to each other, we get:

3 / √(6c + 3) = (√6a) / (a - 1)

Solving for c, we get:

c = (a + 2) / 6

(Note that c is indeed in (1,a) since a > 1.)

Now, we can evaluate f(a) and f(1) and use the Mean Value Theorem to show that:

√6a + 3 - √9 < (√6a) / (a - 1) * (a - 1)

Simplifying, we get:

√6a + 3 < a + 2

as desired.

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After formula for the volume of a cylinder, the increase in water level results in approximately 260 more cubic feet of water in the tank.

The current water level is 10 feet, which is 120 inches. When the water level increases by 2 inches, the new water level will be 122 inches.

The radius of the tank is half of the diameter, which is 30 feet or 360 inches.

The current volume of water in the tank can be calculated using the formula for the volume of a cylinder: V = πr²h, where r is the radius and h is the height of the water level.

V = π(360²)(120) ≈ 15,465,920 cubic inches

When the water level increases by 2 inches, the new height of the water level is 122 inches.

The new volume of water in the tank can be calculated using the same formula:

V = π(360²)(122) ≈ 15,914,693 cubic inches

The difference in volume between the two levels is:

15,914,693 - 15,465,920 = 448,773 cubic inches

To convert cubic inches to cubic feet, we divide by 1728:

448,773 ÷ 1728 ≈ 259.6 cubic feet

Rounding to the nearest cubic foot, we get:

260 cubic feet

Therefore, the increase in water level results in approximately 260 more cubic feet of water in the tank.

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(Explanation below)

x=2

x = 2 is the solution of the equation

Two or more expressions with an Equal sign is called as Equation.

The given equation is [tex](3X-5)^(^1^/^4^) + 3 = 4[/tex]

We have to find the value of x

Subtracting 3 from both sides:

[tex](3X-5)^(^1^/^4^) = 1[/tex]

Raising both sides to the fourth power:

3X - 5 = 1^4

3X - 5 = 1

Adding 5 to both sides:

3X = 6

Dividing by 3:

X = 2

Therefore, x = 2 is the solution of the equation

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This value will represent the range within which the middle 50% of Emilia's daily miles are distributed

Hi! The box plot represents the miles Emilia ran after school for 21 days. To find the IQR (Interquartile Range), you can use the box plot by identifying the values of the first quartile (Q1) and the third quartile (Q3).

The IQR is calculated by subtracting Q1 from Q3 (IQR = Q3 - Q1). By examining the box plot, locate Q1 and Q3 on the plot, and perform the subtraction to find the IQR.

This value will represent the range within which the middle 50% of Emilia's daily miles are distributed.

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The total distance travelled by dust in the given time of 4.0 minutes at the spin rate of 500rpm is equal to 26,400cm.

Spin rate of the cd is equal to 500rpm

distance of piece of dust from the center = 2.1 cm

Distance traveled by a point on the CD that is 2.1 cm from the center in one revolution =  circumference of the circle with a radius of 2.1 cm,

C = 2πr

   = 2π(2.1 cm)

   ≈ 13.2 cm

Distance traveled by the dust in one revolution is 13.2 cm.

Calculate the number of revolutions that the CD makes in 4.0 minutes.

  1 minute = 60 seconds

⇒ 4.0 minutes = 4.0 x 60

⇒4.0 minutes = 240 seconds

The CD rotates at a rate of 500 revolutions per minute,

In 4.0 minutes it will rotate,

500 x 4.0

= 2000 times.

The total distance traveled by the dust in 4.0 minutes is,

distance traveled per revolution x number of revolutions

= 13.2 cm/rev x 2000 rev

= 26,400 cm

Therefore, the total distance traveled by the dust in 4.0 minutes is 26,400 cm.

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The relation a into b represent  all possible combinations of elements in a and b as ordered pairs.

The relation "a into b" refers to the cartesian product of the sets a and b, which is denoted by a × b. The cartesian product of two sets is a set of all possible ordered pairs, where the first element of each ordered pair comes from the first set, and the second element comes from the second set.

In this case, a = {0, 1, 2} and b = {-1, 1, 2}. So, their cartesian product a × b is the set of all possible ordered pairs (a, b), where a is an element of a and b is an element of b. Therefore, we have:

a × b = {(0, -1), (0, 1), (0, 2), (1, -1), (1, 1), (1, 2), (2, -1), (2, 1), (2, 2)}

This means that the relation a into b consists of all possible combinations of elements in a and b as ordered pairs.

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The inequality that can be used to determine the number of outfits Jason can purchase while staying within his budget is 68.54 o ≤ 274.16.

Solving the inequality gives:

o ≤ 4

The total cost of items that Jason spends on :

= 217. 34 + 36. 32 + 12. 18

= $ 265. 85

The amount that Jason has left from the $ 540 is:

= 540 - 265. 85

= $ 274. 16

If each outfit costs $ 68. 54 then the inequality that would help stay in budget is:

68.54 o ≤ 274.16

Solving this, gives:

o ≤ 274. 16 / 68.54

o ≤ 4

In conclusion, Jason can purchase up to 4 biking outfits while staying within his budget.

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The value of m∠MGA between the tangent and the diameter is 90 degrees.

A line that touches the circle at one point is known as a tangent to a circle.

Therefore, MU is tangent to the circle O at the point G. The diameter of the

circle is GA.

Therefore, let's find the angle m∠MGA in the circle.

If a tangent and a diameter meet at the point of tangency, then they are perpendicular to one another.  Therefore, the point where they meets form a right angle(90 degrees).

Hence,

m∠MGA  = 90 degrees

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you will have to walk approximately 17.2 miles back home.

A right triangle is a triangle in which one of the angles measures 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.  the length of the hypotenuse. Right triangles have many important applications in mathematics, science, and engineering, particularly in trigonometry, which is the study of the relationships between the sides and angles of triangles.

You have formed a right triangle with legs of length 10 and 14. To find the length of the hypotenuse, which is the distance back to your starting point, we can use the Pythagorean theorem:

[tex]c^2 = a^2 + b^2[/tex]

[tex]c^2 = 10^2 + 14^2[/tex]

[tex]c^2 = 100 + 196[/tex]

[tex]c^2 = 296[/tex]

c = sqrt(296)

c ≈ 17.2

Therefore, you will have to walk approximately 17.2 miles back home.

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The total number of yellow cups and blue cups paint that Jordy need is 10 t and (20/3) t, under the condition that he created the shade of green paint by mixing 1/2 t of yellow paint with 1/3 t of blue paint.

Here we have to apply the principles regarding basic multiplication of fraction.

Now to get 20 cups of green paint, Jordy requires 10 cups of yellow paint and 20/3 cups of blue paint.
Then the process of evaluating the number of blue paint cups needed is
Jordy created the shade of green paint by mixing 1/2 t of yellow paint with 1/3 t of blue paint.
To get 1 unit of green paint, Jordy needs 1/2 t of yellow paint and 1/3 t of blue paint.
To get 20 units of green paint,
Jordy needs
20 ×(1/2) t = 10 t of yellow paint
20 ×(1/3) t = (20/3) t of blue paint.
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Chantal's design will not work as it does not meet the length requirements for the desired amount of slack and vertical change.

The rule of 5% slack states that the cable should have 5% of the cable's length in slack, so for a 130 feet cable, there would need to be 6.5 feet of slack. Since Chantal's design only has 10 feet of vertical change, that would not leave enough slack to meet the rule.

Also, the 6-8 feet of vertical change per 100 feet of horizontal change rule states that the vertical change should be between 6-8 feet for every 100 feet of horizontal change. In this case, the horizontal change is 130 feet, so the vertical change should be between 7.8-10.4 feet. Since Chantal's design has only 10 feet of vertical change, it does not meet this rule either.

Therefore, Chantal's design will not work as it does not meet the length requirements for the desired amount of slack and vertical change.

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Applying the Equidistant Chords Theorem, the value of x in the circle given above is calculated as: b. 50.

The Equidistant Chords Theorem states that If two chords in a circle, or in congruent circles, are equally distant from the center, then the chords are congruent.

The image given reveals that the two chords are equidistant from the center of the circle, therefore:

x = 2(25)

x = 50.

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The completed table with residuals rounded to hundredths place:

| Hours | Retweets | Predicted Value | Residual |

| 1     | 65       |-79.86          |-144.86   |

| 2     |90        |-58.96          |-31.04    |

|3      |162       |-20.16          |-141.84   |

|4      |224       |17.64           |-206.64   |

|5      |337       |75.54           |-262.54   |

|6      |466       |133.44          |-332.44   |

|7      |780       |191.34          |-409.34   |

|8      |1087      |249.24          |-238.24   |

We can evaluate the predicted value by staging the given hours in the function

f(x) = 138.9x - 218.76.

for instance,  hours = 1:

f(1) = (138.9 x 1) - 218.76

= -79.86

likewise, we can find predicted values for all hours.

Residual = Actual Value - Predicted Value

For instance, for hours = 1:

Residual = Actual Value - Predicted Value

       = 65 - (-79.86)

       = 144.86

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If we are drawing without replacement, the probability is approximately 27.7%, which is closest to option B: 27%.

The probability of drawing a yellow marble on the next draw depends on whether we are drawing with or without replacement.

If we are drawing without replacement, then the probability of drawing a yellow marble is 13 out of the remaining 47 marbles, since we have already drawn 35 white marbles and 13 yellow marbles out of the 48 total marbles.

If we are drawing with replacement, then the probability of drawing a yellow marble on the next draw is still 1/3, or approximately 33.3%.

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The actual length of the truck is 15.6 meters and the actual width is 3.24 meters.

The actual length of the truck can be calculated by multiplying the length of the model by the scale factor:

Similarly, the actual width of the truck can be calculated as:

Therefore, the actual length of the truck is 15.6 meters and the actual width is 3.24 meters.

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i hope this helps you.

The probability that we want to get is 0.167

To get this probability, we need to take the quotient between the number of students that preffers athletics and literature and the total number of students in the table.

We have:

Total number = 5 + 3 + 2 +8 = 18

number that preffers athletics and literature = 3

Then the probability is:

P = 3/18 = 0.167

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The angular velocity of the point on the rim of the wheel is 25 radians per second.

The linear velocity of a point on a wheel's rim is determined by:

v = rω

v = linear velocity

r = radius of the wheel

ω = angular velocity.

In this case, the diameter of the wheel is 8 feet, so the radius is 4 feet. The linear velocity is given as 100 feet per second, so we have:

100 = 4ω

Solving for ω, we get:

ω = 25 radians per second

Therefore, the angular velocity of the point on the rim of the wheel is 25 radians per second.

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The experimental probability of the computer generating a 1 is D. 40 %.

First, add up the outcomes to see the total number of times the integers were generated ;

= 36 + 11 + 13 + 12 + 18

= 90

The number of times 1 was generated was 36.

The experimental probability is therefore;

= number of times 1 was generated / total number of outcomes

= 36 / 90

= 0. 4

= 40 %

Therefore, the experimental probability of the computer generating a 1 is 40 %.

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Answer:

m<V=100.5°

u=84.86u

v=154.86u

Step-by-step explanation:

m<V= 180-46.9-32.6=100.5°

sin46.9°/115=sin32.6°/u

sin46.9°u=115sin32.6°

u=115sin32.6°/sin46.9°

u=84.86u

sin46.9°/115=sin100.5°/v

sin46.9°v=115sin100.5°

v=115sin100/5°/sin46.9°

v=154.86

Answer: 1. Multiplying Powers with same Base

For example: x² × x³, 2³ × 2⁵, (-3)² × (-3)⁴

In multiplication of exponents if the bases are same then we need to add the exponents.

Consider the following:

1. 2³ × 2² = (2 × 2 × 2) × (2 × 2) = 23+2

= 2⁵

2. 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 34+2

= 3⁶

3. (-3)³ × (-3)⁴ = [(-3) × (-3) × (-3)] × [(-3) × (-3) × (-3) × (-3)]

                       = (-3)3+4

                       = (-3)⁷

4. m⁵ × m³ = (m × m × m × m × m) × (m × m × m)

                 = m5+3

                 = m⁸

From the above examples, we can generalize that during multiplication when the bases are same then the exponents are added.

aᵐ × aⁿ = am+n

In other words, if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, then

aᵐ × aⁿ = am+n

Similarly, (ab

)ᵐ × (ab

)ⁿ = (ab

)m+n

(ab)m×(ab)n=(ab)m+n

Note:

(i) Exponents can be added only when the bases are same.

(ii) Exponents cannot be added if the bases are not same like

m⁵ × n⁷, 2³ × 3⁴

Step-by-step explanation:

Answer:

2^4

Step-by-step explanation:

doing this type of division is like subtracting the powers, 9-5=4, 2^4.

the opposite applies for multiplaction, it's like addition for the powers,

2^9*2^5=2^14.

Using the fundamental counting principle, there are 20 possible outcomes for the fitness tracker, considering its battery life and color options.

To find the total number of possible outcomes for the given criteria, we can use the fundamental counting principle. This principle states that if there are m ways to perform one task and n ways to perform another task, then there are m x n ways to perform both tasks together.

In this case, we have 4 options for battery life (1 day, 3 days, 5 days, and 7 days) and 4 options for color (silver, green, blue, pink, and black). Using the fundamental counting principle, we can multiply the number of options for battery life by the number of options for color to find the total number of possible outcomes:

4 options for battery life x 5 options for color = 20 possible outcomes.

Therefore, there are 20 possible combinations of battery life and color for a fitness tracker.

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