Evaluate the following sum.
∑k=252(3)k

the statement is true for all positive integers n. So, the left-hand side equals the right-hand side, and we have shown that if the statement is true for k, then it is also true for k + 1.

An integer is a whole number that can be either positive, negative or zero. Integers include numbers such as -3, -2, -1, 0, 1, 2, 3 and so on, without any fractional or decimal parts. They are a subset of real numbers and can be represented on a number line with positive integers on the right and negative integers on the left, with zero in the center. Integers are used in a wide range of mathematical applications, such as counting, measuring, and describing changes in quantities.

by the question.

To prove this statement by induction, we need to show that:

The statement is true for n = 1 (base case)

If the statement is true for some positive integer k, then it is also true for k + 1 (inductive step)

Here are the steps to prove this statement by induction:

Base case (n = 1):

When n = 1, the left-hand side of the equation is:

2(1)Σr² r=1 = 2(1)1² = 2

And the right-hand side of the equation is:

1/3(2(1) + 1) (4(1) + 1) = 1/15(3)(5) = 1

So, the statement is not true for n = 1. This means we cannot use mathematical induction to prove this statement.

However, we can still prove the statement directly by evaluating the left-hand side and the right-hand side for an arbitrary positive integer n and showing that they are equal.

Inductive step:

Assume the statement is true for some positive integer k, i.e.

2kΣr² r=1 = k/3(2k + 1) (4k + 1)

We need to show that the statement is also true for k + 1, i.e.

2(k + 1)Σr² r=1 = (k + 1)/3(2(k + 1) + 1) (4(k + 1) + 1)

We start with the left-hand side:

2(k + 1)Σr² r=1

= 2kΣr² r=1 + 2(k + 1) ² (by adding the next term in the summation)

= k/3(2k + 1) (4k + 1) + 2(k + 1) ² (by the inductive hypothesis)

= k (8k² + 14k + 6)/3(2k + 1) (4k + 1) + 2(k + 1)² (by simplifying the expression)

= (8k³ + 26k² + 22k + 6)/3(2k + 1) (4k + 1) + 2(k + 1) ²

= (8k³ + 26k² + 22k + 6 + 6(2k + 1) (4k + 1))/3(2k + 1) (4k + 1)

= (8k³ + 26k² + 22k + 6 + 48k² + 26k + 6)/3(2k + 1) (4k + 1)

= (8k³ + 74k² + 48k + 12)/3(2k + 1) (4k + 1)

= (k + 1)/3(2(k + 1) + 1) (4(k + 1) + 1)

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Answer:

Since ΔQPT ≅ ΔARZ, we know that their corresponding sides and angles are congruent.

So,

QT ≅ RZ (corresponding sides)

PT ≅ AR (corresponding sides)

QP is not congruent to AZ because they are not corresponding sides in the congruent triangles.

Therefore, the correct answer is (D) QT ≅ RZ.

Step-by-step explanation:

Answer:

Since ΔQPT ≅ ΔARZ, we know that their corresponding sides are congruent.

Step-by-step explanation:

The congruent side pair is:

A) QT ≅ AZ.

Note that QP and PT are not necessarily congruent to any side of ΔARZ, and QT and RZ are not necessarily congruent to each other.

The odds against this happening are  48:25 . For this we have to know the meaning of probability.

I showed that out of  22100 possible deals, there are  3744 ways to get a pair and  52 ways of getting 3 of a kind that makes a total of  3796 ways to get a pair or 3 of a kind.

If we multiply this by  3! then we get  22776 possible permutations that result in 3 of a kind or in a pair.

To get the number of permutations of deals where the first two cards are a pair, we note that the first card could be any one of the  52 in the pack.

The second card must be one of the  3 remaining cards that pairs up with the first one and then there are  50 cards that could be selected for the third.

That makes a total of  52×3×50=7800 permutations.

[tex]\frac{7800}{22776} = \frac{27}{73}[/tex]

The odds against this happening are  48:25

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Answer: I gave you two answers if you can try them both :)

Step-by-step explanation:

If his weight on Earth is 126lb and only 21lb on moon, you can divide  to see what is the ratio of those weights.

It means that your weight on moon will be 6 times less than on Earth.

Now we have to multiply 31lb which is weight of the person on moon by 6 to get his weight on Earth

On moon our mass becomes 1/6 of actual mass so if you weigh 60 kg then your mass on moon will be 10 kg..

Similarly if your mass on moon is 31 lbs then your mass on earth will be 31*6=186 lbs.

Answer:

d. 3.5 km = 1,000 m / km = 3,500 m

so we know the center and a point it passes through, so the distance from the center to that point on the circle is its radius

[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{0}~,~\stackrel{y_1}{-10})\qquad (\stackrel{x_2}{\frac{9}{2}}~,~\stackrel{y_2}{-16})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ \stackrel{ radius }{r}=\sqrt{(~~\frac{9}{2} - 0~~)^2 + (~~-16 - (-10)~~)^2} \implies r=\sqrt{(\frac{9}{2} )^2 + (-16 +10)^2} \\\\\\ r=\sqrt{( \frac{9}{2} )^2 + ( -6 )^2} \implies r=\sqrt{ \frac{81}{4} + 36 } \implies r=\sqrt{ \cfrac{225}{4} }\implies r=\cfrac{15}{2} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \hspace{5em}\stackrel{center}{(\underset{0}{h}~~,~~\underset{-10}{k})}\qquad \stackrel{radius}{\underset{\frac{15}{2}}{r}} \\\\[-0.35em] ~\dotfill\\\\ ( ~~ x - 0 ~~ )^2 ~~ + ~~ ( ~~ y-(-10) ~~ )^2~~ = ~~\left( \frac{15}{2} \right)^2\implies x^2+(y+10)^2=\cfrac{225}{4}[/tex]

Answer:

x^2 + (y + 10)^2 = 225/4

Step-by-step explanation:

To find the standard form of a circle with center (h, k) and radius r, the equation is:

(x - h)^2 + (y - k)^2 = r^2

We are given the circle's center as (0, -10) and a point on the circle as (9/2, -16). We can use this information to find the radius of the circle.

Radius (r) = Distance between center and point on the circle

r = sqrt[(0 - 9/2)^2 + (-10 + 16)^2]

r = sqrt[(81/4) + 36]

r = sqrt[(81 + 144)/4]

r = sqrt(225)/2

r = 15/2

Now we can substitute the values of (h, k, and r) into the standard form equation to get:

(x - 0)^2 + (y - (-10))^2 = (15/2)^2

Simplifying and multiplying out the terms, we get:

x^2 + (y + 10)^2 = 225/4

Therefore, the standard form of the circle is:

x^2 + (y + 10)^2 = 225/4

The diameter οf the circle is 28/π inches οr apprοximately 8.89 inches (rοunded tο twο decimal places).

The fοrmula fοr the circumference (C) οf a circle is given by:

C = 2πr

where r is the radius οf the circle.

If the circumference οf the circle is 28 inches, we can sοlve fοr the radius by dividing bοth sides οf the equatiοn by 2π:

C/2π = r

Substituting the given value οf C = 28, we get:

r = 28/2π

r = 14/π

Finally, tο find the diameter (d) οf the circle, we multiply the radius by 2:

d = 2r

Substituting the value οf r = 14/π, we get:

d = 2(14/π) = 28/π

Therefοre, the diameter οf the circle is 28/π inches οr apprοximately 8.89 inches (rοunded tο twο decimal places).

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Answer:

The bird laid 6 eggs

Step-by-step explanation:

If [tex]\frac{1}{3}[/tex] of the birds egg has hatched; and if that one third is quantifiable as 2. We know that [tex]\frac{1}{3}\\[/tex] of the laid eggs (we can declare that as x) is 2.

So:

[tex]\frac{1}{3}x = 2[/tex]

Now multiply both sides by 3:

[tex]x=6[/tex]

Answer:

1 egg

Step-by-step explanation:

One third of a birds eggs hatched is the first egg. The sentence say if 2 eggs hatched how many eggs did the bird lay is the second egg. 3 eggs - 2 = 1 egg

Answer:

Step-by-step explanation:

The statement that describes the function f(x) = 4x^3 + 8 is:

A. The function does not have an inverse function because f(x) fails the vertical line test.

To see why this is the correct answer, let's first define what the vertical line test and the horizontal line test are.

Vertical line test: A function passes the vertical line test if any vertical line intersects the graph of the function at most once. This means that no two points on the graph have the same x-coordinate.

Horizontal line test: A function passes the horizontal line test if any horizontal line intersects the graph of the function at most once. This means that no two points on the graph have the same y-coordinate.

Now, let's look at the function f(x) = 4x^3 + 8. We can graph this function by plotting points or by using a graphing calculator. The graph of the function looks like a curve that goes up and to the right.

If we draw a vertical line anywhere on the graph, we can see that it intersects the graph at most once, which means that f(x) passes the vertical line test. However, if we draw a horizontal line on the graph, we can see that it intersects the graph at more than one point. This means that f(x) fails the horizontal line test.

The fact that f(x) fails the horizontal line test tells us that there are some values of y that correspond to more than one value of x. This means that f(x) is not a one-to-one function, and therefore it does not have an inverse function.

Therefore, the correct statement that describes the function f(x) is:

A. The function does not have an inverse function because f(x) fails the vertical line test.

Answer:

A. To graph the linear relation representing the school's cost for up to 200 bottles of water, we can use the slope-intercept form of a linear equation: y = mx + b, where y is the cost, x is the number of bottles of water, m is the slope, and b is the y-intercept.

The y-intercept is the fixed cost of renting the coolers, which is $25. The slope represents the additional cost per bottle of water, which is $0.65. Therefore, the equation is:

y = 0.65x + 25

To graph the line, we can plot the y-intercept at (0, 25), and then use the slope to find additional points. For example, when x = 50, y = 0.65(50) + 25 = 57.50, so we can plot the point (50, 57.50) and draw a line through the points.

B. To graph the linear relation representing the school's income from selling up to 200 bottles of water, we can also use the slope-intercept form of a linear equation: y = mx + b, where y is the income, x is the number of bottles of water, m is the slope, and b is the y-intercept.

The y-intercept is the revenue from selling 0 bottles of water, which is $0. The slope represents the revenue per bottle of water, which is $1.25. Therefore, the equation is:

y = 1.25x + 0

To graph the line, we can plot the y-intercept at (0, 0), and then use the slope to find additional points. For example, when x = 50, y = 1.25(50) + 0 = 62.50, so we can plot the point (50, 62.50) and draw a line through the points.

C. The equation for the school's cost is y = 0.65x + 25, and the equation for the school's income is y = 1.25x + 0.

D. To find the coordinates of the point where the lines cross, we can set the two equations equal to each other and solve for x:

0.65x + 25 = 1.25x + 0

0.6x = 25

x = 41.67

Then we can plug in x = 41.67 into either equation to find y:

y = 0.65(41.67) + 25 = 52.08

Therefore, the point where the lines cross is (41.67, 52.08).

E. The significance of this point is that it represents the breakeven point, where the school's cost equals its revenue. If the school sells fewer than 41.67 bottles of water, it will not cover its costs. If it sells more than 41.67 bottles of water, it will make a profit.

Step-by-step explanation:

Answer:29 rows of 35 seats

Step-by-step explanation:

e4t wdsvfwer

Answer:

29 rows

Step-by-step explanation:

r = # of rows = s - 6

s = # of seats/row

s(s - 6) = 1015

s² - 6s = 1015

s² - 6s - 1015 = 0     use the quadratic equation to find s (a=1, b=-6, c=-1015)

2 solutions of s:  35, -29      disregard the negative solution

r = s - 6 = 35 - 6 = 29

Answer:

what in the world

Step-by-step explanation:

number 1. is c

number 2. is d

number 3. is a

number 4. is c

number 5. is b

hope this helps

Answer: 16

Step-by-step explanation: 32 DIVIDE BY 4 IS 8 AND 8 TIMES 2 IS 16

Answer:

x= infinite or -5/2 or 1, + infinite

Step-by-step explanation:

Answer:

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:

slope = (y2 - y1) / (x2 - x1)

n this case, we have the points (3, 1) and (-2, 1), so we can plug in the values:

slope = (1 - 1) / (-2 - 3)

slope = 0 / -5

slope = 0

Therefore, the slope between the points (3, 1) and (-2, 1) is 0. This means that the line passing through these points is a horizontal line, since the y-coordinate of both points is the same and the slope is 0 (i.e., there is no change in the y-coordinate as we move along the line).

As a result, the maximum pressure the block may impose on the ground angles is 25 N/cm2 and the minimum pressure is 0.029 N/cm2.

An angle is a shape in Euclidean geometry made composed of two rays, known as the angle's sides, that meet in the middle at a point known as the angle's vertex. Two rays can produce an angle in the plane in which they are positioned. An angle is formed when two planes collide. These are known as dihedral angles. In planar geometry, an angle is the shape formed by two rays or lines that have a common termination. The English term "angle" comes from the Latin word "angulus," which means "horn." The vertex is the common terminal of the two rays, which are known as the angle's sides.

The area of the block in touch with the ground must be considered to determine the maximum and least pressure that the block can exert on the ground.

The area in contact with the ground is the block's bottom face, which is 34 cm × 25 cm = 850 cm2.

Pressure = Weight/Area = 25 N/1 cm2 = 25 N/cm2

Pressure = Area/Weight = 25 N / 850 cm2 = 0.029 N/cm2

As a result, the maximum pressure the block may impose on the ground is 25 N/cm2 and the minimum pressure is 0.029 N/cm2.

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Answer:

D. (-1,3)

Step-by-step explanation:

You need to draw the figure on the graph. See attachment.

Answer:

To find the sales tax, we need to first calculate the taxable amount.

Total selling price = $1,238.45

Sales tax rate = 8 1/2% = 8.5%

Let's assume that the sales tax is added to the selling price, which means the selling price includes the tax.

To find the taxable amount, we can divide the selling price by (1 + tax rate as a decimal):

Taxable amount = Selling price / (1 + tax rate as a decimal)

Taxable amount = $1,238.45 / (1 + 0.085)

Taxable amount = $1,238.45 / 1.085

Taxable amount = $1,140.00 (rounded to nearest dollar)

Now, to find the sales tax amount:

Sales tax = Taxable amount x tax rate as a decimal

Sales tax = $1,140.00 x 0.085

Sales tax = $97.00

Therefore, the sales tax is $97.00.

Answer:

The chemist should use 410 grams of the first alloy (which is 10% gold and 20% lead) and 40 grams of the second alloy (which is 40% gold and 30% lead) to make an alloy that contains 57 grams of gold and 94 grams of lead.

Step-by-step explanation:

Let's call the amount of the first alloy used "x" and the amount of the second alloy used "y". We can set up a system of two equations based on the amount of gold and lead needed in the final alloy:

Equation 1: 0.10x + 0.40y = 57 (the amount of gold in the first alloy is 10%, and in the second alloy is 40%)

Equation 2: 0.20x + 0.30y = 94 (the amount of lead in the first alloy is 20%, and in the second alloy is 30%)

We can then solve for x and y using any method of solving systems of equations. One way is to use substitution:

Therefore, the chemist should use 410 grams of the first alloy (which is 10% gold and 20% lead) and 40 grams of the second alloy (which is 40% gold and 30% lead) to make an alloy that contains 57 grams of gold and 94 grams of lead.

The given expression is:

-(6/5)t + 3/16 - (7/10)t + 9/8

To simplify the expression, we can combine the like terms. The like terms are the terms that have the same variable raised to the same power. Here, the like terms are the terms that involve t:

= -(6/5)t - (7/10)t + 3/16 + 9/8

= -(12/10)t - (7/10)t + 3/16 + 18/16

= -(19/10)t + 21/16

Hence, the equivalent expression is -19/10t + 21/16.

Answer:

Step-by-step explanation:

The expression k - 3d has two terms: k and -3d.

The coefficient of the term k is 1, because k can be written as 1k, and the coefficient of a term without a numerical coefficient is always 1.

The coefficient of the term -3d is -3, because -3d can be written as -3*d, and the coefficient of a term with a variable is the numerical coefficient of the variable (in this case, d) multiplied by any numerical coefficient (in this case, -3).

Answer:

7x+18

Step-by-step explanation:

if you create an imaginary horizontal line along the base of the 'x+4' and '3' rectangle then you can find its area by multiplying those values and add it to the value of the area of the smaller and thinner rectangle

Area of larger rectangle = 3(x+4) => 3x+12

Area of smaller rectangle = 2[(2x+6)-3] which equals 2(2x+3) => 4x+6

3x+12 + 4x+6 = 7x+18

The probability that the mean weight of the sample of 25 cartons is greater than 7.19 ounces is given as follows:

0.0571 = 5.71%.

The z-score of a measure X of a variable that has mean symbolized by [tex]\mu[/tex] and standard deviation symbolized by [tex]\sigma[/tex] is obtained by the rule presented as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

The parameters for this problem are given as follows:

[tex]\mu = 7, \sigma = 0.6, n = 25, s = \frac{0.6}{\sqrt{25}} = 0.12[/tex]

The probability that the mean weight is greater than 7.19 ounces is one subtracted by the p-value of Z when X = 7.19, considering the standard error s, hence:

Z = (7.19 - 7)/0.12

Z = 1.58

Z = 1.58 has a p-value of 0.9429.

Hence:

1 - 0.9429 = 0.0571 = 5.71%.

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The solution written as both a single power and a single variable is y^4/3

Index forms are defined as mathematical forms of presenting numbers too small or large in a more convenient form.

Other terms for index forms are;

A radical is expressed with the symbol, '√' that is used to denote square root or nth roots.

From the information given, we have that;

[tex](\sqrt[3]{y^2} ) (\sqrt[6]{y^4} )[/tex]

expand the square root symbols, we have;

(y^2/3)(y^2/3)

expand the bracket and add the exponents, we have;

y^(2/3+2/3)

Add the values

y^ 4/3

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Therefore , the solution of the given problem of angle comes out to be  Sector 2's centre angle is 57.3 degrees.

In Euclidean geometry, a tilt is a shape with two sides, but they are actually cylindrical faces that separate at the middle and top of the barrier. Two rays may merge to produce an angle at their intersection. Another result of two things interacting is an angle. They most closely rays resemble dihedral forms. Two line beams can be arranged in different ways at their extremities to form a two-dimensional curve.

Here,

The following algorithm determines the central angle of a sector of a circle:

=>θ = (arc length / radius) * (180/π)

Since Sectors 1 and 2 have the same arc length, the following can be written:

θ1 = (x / r1) * (180/π)

θ2 = (x / r2) * (180/π)

where r1 and r2 are the radii of the circles, and 1 and 2 are Sector 1 and Sector 2's corresponding central angles.

Since Sector 2's radius is unknown to us, we are unable to determine its centre angle directly. However, since both sections have the same arc length, we can write as follows:

=>  x / r1 = x / r2

When we multiply both parts by r2, we obtain:

=>  r2 * x / r1 = x

=>  r2 = r1 * x / x

=> r2 = r1

Since the two circles' radii are identical as a result, the formula for Sector 2's central angle can be made simpler:

=>  2 Equals (x/r1) * (180/), (x/r2) * (180/), and (x/r1) * (180/)

=> θ2 = (180/π) 57.3 degrees in radians (rounded to 1 decimal place)

As a result, Sector 2's centre angle is 57.3 degrees.

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The completion of the table comparing simple interest with compound interest for the amount (future value) of the investment of $1,000 for the different periods is as follows:

                      Simple       Compound

                     Interest          Interest

1 year           $1,100.00      $1,100.00

2 years       $1,200.00      $1,210.00

3 years       $1,300.00       $1,331.00

4 years       $1,400.00       $1,464.10

5 years       $1,500.00       $1,610.51

Simple interest is computed on the principal only for each period.

Compound interest is computed on both the principal and accumulated interest to determine the future value.

For instance, at the end of the first year, there is no difference between the future value based on simple interest and the compound interest.  However, differences exist from the second year because the accumulated interest is added to the principal before compound interest is determined.

The principal investment = $1,000

Interest rate = 10%

                      Simple       Compound

                     Interest          Interest

1 year           $1,100.00      $1,100.00 ($1,000 x 1.1)

2 years       $1,200.00      $1,210.00 ($1,000 x 1.21)

3 years       $1,300.00      $1,331.00 ($1,000 x 1.331)

4 years       $1,400.00      $1,464.10 ($1,000 x 1.4641

5 years      $1,500.00       $1,610.51 ($1,000 x 1.61051)

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Answer:

Step-by-step explanation:

o prove the first associative law of set theory, we need to show that for any sets A, B, and C:

A ∪ (B ∪ C) = (A ∪ B) ∪ C

To do this, we need to show that any element that is in the left-hand side of the equation is also in the right-hand side, and vice versa.

First, let's consider an arbitrary element x.

If x ∈ A ∪ (B ∪ C), then x must be in A, or in B, or in C (or in two or more of these sets).

If x ∈ A, then x ∈ A ∪ B, and so x ∈ (A ∪ B) ∪ C.

If x ∈ B, then x ∈ B ∪ C, and so x ∈ A ∪ (B ∪ C), which means that x ∈ (A ∪ B) ∪ C.

If x ∈ C, then x ∈ B ∪ C, and so x ∈ A ∪ (B ∪ C), which means that x ∈ (A ∪ B) ∪ C.

Therefore, we have shown that if x ∈ A ∪ (B ∪ C), then x ∈ (A ∪ B) ∪ C.

Next, let's consider an arbitrary element y.

If y ∈ (A ∪ B) ∪ C, then y must be in A, or in B, or in C (or in two or more of these sets).

If y ∈ A, then y ∈ A ∪ (B ∪ C), and so y ∈ (A ∪ B) ∪ C.

If y ∈ B, then y ∈ A ∪ B, and so y ∈ A ∪ (B ∪ C), which means that y ∈ (A ∪ B) ∪ C.

If y ∈ C, then y ∈ (A ∪ B) ∪ C.

Therefore, we have shown that if y ∈ (A ∪ B) ∪ C, then y ∈ A ∪ (B ∪ C).

Since we have shown that any element that is in the left-hand side of the equation is also in the right-hand side, and vice versa, we can conclude that:

A ∪ (B ∪ C) = (A ∪ B) ∪ C

This proves the first associative law of set theory.

Answer:

x=4

Step-by-step explanation:

2x+(-3) =5

2x-3+3=5+3

2x/2=8/2

x=4