Need Help (25 Points)

It would take approximately 12.5 hours for only pipe A to fill the pool, 168 hours for only pipe B, 48.6 hours for only pipe C, and 3.5 hours for all pipes.

What is equations ?

An equation is a mathematical statement that shows two expressions to be equal to each other. It typically contains one or more variables, and solving the equation involves finding the values of the variables that make the equation true. Equations are used in many areas of mathematics, science, engineering, and other fields to describe relationships between different quantities or to solve problems. Examples of equations include linear equations, quadratic equations, and differential equations.

Let's represent the amount of work done by each pipe per hour:

Pipe A: a

Pipe B: b

Pipe C: c

From the problem, we know that:

a + b = 1/9.5 (1) (pipes A and B take 9.5 hours to fill the pool)

b + c = 1/6 (2) (pipes B and C take 6 hours to fill the pool)

a + c = 1/10 (3) (pipes A and C take 10 hours to fill the pool)

We can solve this system of equations to find the values of a, b, and c:

Adding (1) and (2), we get:

a + 2b + c = 1/9.5 + 1/6

Multiplying both sides by 2, we get:

2a + 4b + 2c = 1/4.75 + 1/

Simplifying, we get:

2a + 4b + 2c = 13/28 (4)

Adding (1) and (3), we get:

2a + b + c = 1/9.5 + 1/10

Multiplying both sides by 20, we get:

40a + 20b + 20c = 2 + 1.9

Simplifying, we get:

40a + 20b + 20c = 3.9 (5)

Now we can solve for a, b, and c using (4) and (5). Multiplying (4) by 5 and subtracting (5) from it, we get:

6b = 1/28

Simplifying, we get:

b = 1/168

Substituting b into (1), we get:

a = 1/9.5 - 1/168

Simplifying, we get:

a = 1/12.32

Substituting b into (2), we get:

c = 1/6 - 1/168

Simplifying, we get:

c = 7/336

Now we can find the time it takes for each pipe to fill the pool individually:

Only pipe A: 1/a = 12.32 hours ≈ 12.5 hours

Only pipe B: 1/b = 168 hours

Only pipe C: 1/c = 48 hours ≈ 48.6 hours

All pipes: 1/(a+b+c) = 3.46 hours ≈ 3.5 hours (using the values we found for a, b, and c)

Therefore, it would take approximately 12.5 hours for only pipe A to fill the pool, 168 hours for only pipe B, 48.6 hours for only pipe C, and 3.5 hours for all pipes.

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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 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:

y = 4x + 1/2

Step-by-step explanation:

The equation is y = mx + b

m = the slope

b = y-intercept

We know

m = 4

b = 1/2

So, the equation is y = 4x + 1/2

Answer:

  (c) two hundred seventy-five and one-eighth cm² = 275 1/8 cm²

Step-by-step explanation:

You want the surface area of the rectangular prism represented by the given net.

The first attachment shows the missing dimensions. The height of the vertical rectangle in the center of the net is a total of 2×(5 3/4 + 4) cm.

The area is the sum of the areas of the rectangular faces of the prism. It is the sum of the individual rectangles shown in the net diagram.

This area can be computed nicely by adding the vertical dimensions and multiplying by the width of the central rectangle. Then, added to that, is the area of the two "wings" on either side of that central rectangle.

In each case, the area of a rectangle is the product of its length and width.

The second attachment shows the calculation of the total area of the figure.

The surface area of the prism is 275 1/8 square centimeters.

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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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

Both fireworks will explode at time 0.55 seconds after Firework B launches.

Distance is a unit of measurement for the separation between two places. This scalar number is typically expressed in terms of metres, feet, or miles. Depending on the situation, other distance formulae can be used, such as the geometry distance formula or the physics average speed formula. Displacement, a vector quantity that considers the direction of motion, is distinct from distance. Distance is the overall length of an object's route, whereas displacement is the change in an object's location from its beginning position to its end position.

Let us suppose time = t.

Given that, Firework B is launched 0.25 s before Firework A.

Time taken for Firework A to explode after Firework B launches is t - 0.25.

The distance traveled by each firework is given by:

Distance of Firework A = 320t

Distance of Firework B = 220(t - 0.25) = 220t - 55

Since both fireworks explode at the same height, their distances traveled must be equal. Therefore:

320t = 220t - 55

100t = 55

t = 0.55 seconds

Hence, both fireworks will explode 0.55 seconds after Firework B launches.

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Answer:

D. (-1,3)

Step-by-step explanation:

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

Answer:

135 square meters

Step-by-step explanation:
Area = Length x Breadth

15 m x 9 m = 135 square meters.

Answer:

135m

Step-by-step explanation:

l×b = Area

15×9 = 135m

Answer:

Step-by-step explanation:To prove that the roots of the equation x² + (1-k)x + k-3 = 0 are real for all real values of k, we need to show that the discriminant of the equation is non-negative for all values of k.

The discriminant of a quadratic equation ax² + bx + c = 0 is given by b² - 4ac. If the discriminant is positive, then the equation has two distinct real roots; if it is zero, then the equation has one real root (a repeated root); and if it is negative, then the equation has no real roots.

So, in this case, the discriminant of the equation is:

(1-k)² - 4(1)(k-3)

= 1 - 2k + k² - 4k + 12

= k² - 6k + 13

We need to show that k² - 6k + 13 ≥ 0 for all real values of k.

To do this, we can complete the square:

k² - 6k + 13

= (k - 3)² + 4

Since the square of any real number is non-negative, we have (k-3)² ≥ 0 for all k, which means that (k-3)² + 4 ≥ 4.

Therefore, k² - 6k + 13 ≥ 4 for all real values of k, which means that the discriminant of the quadratic equation x² + (1-k)x + k-3 = 0 is non-negative for all real values of k. Hence, the roots of the equation are real for all real values of k.

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:

see the work below

Step-by-step explanation:

20.)    x = √32²+32² = √2048 = 45.25

y = 32(sin60) = 27.7

z = 32(cos60) = 16

22.)   x = 7√3 / cos60 = 24.25 = 14√3

y = 14√3 / tan30 = 42

sin30 = 14√3 / (z+7√3)

z + 7√3 = 14√3 / sin30

z = 14√3/√sin30 - 7√3 = 36.37 = 21√3

24.) z = 18(tan30) = 10.4

h = √10.4² + 18² = √432 = 20.785

x = y

2x² = 20.785²

x = √20.785²/2 = 14.7

y = 14.7

Answer:

Question 20:

Question 22:

Question 24:

Step-by-step explanation:

A 45-45-90 triangle is a special right triangle where the measures of its sides are in the ratio 1 : 1 : √2.  Therefore, the formula for the ratio of the sides is b : b : b√2 where:

A 30-60-90 triangle is a special right triangle where the measures of its sides are in the ratio  1 : √3 : 2.  Therefore, the formula for the ratio of the sides is c: c√3 : 2c where:

Side x is the hypotenuse of a 45-45-90 triangle with congruent legs measuring 32 units. Therefore b = 32.

[tex]\implies x=b\sqrt{2}=32\sqrt{2}\; \sf units[/tex]

Side y is the side opposite the 60° angle in a 30-60-90 triangle with a hypotenuse of 32 units. Therefore, 2c = 32 so c = 16.

[tex]\implies y = c\sqrt{3}=16\sqrt{3}\; \sf units[/tex]

Side y is the side opposite the 30° angle in the same 30-60-90 triangle.

[tex]\implies z=c = 16\; \sf units[/tex]

Side x is the hypotenuse of a 30-60-90 triangle with the leg opposite the 30° angle measuring 7√3 units. Therefore c = 7√3.

[tex]\implies x=2c=2 \cdot 7 \sqrt{3}=14\sqrt{3}\; \sf units[/tex]

Therefore, other leg of the same triangle (opposite the 60° angle) measures c√3 = 7√3 · √3 = 21 units.

Side y is the hypotenuse of a 30-60-90 triangle with the leg opposite the 30° angle measuring 21 units. Therefore c = 21.

[tex]\implies y=2c=2 \cdot 21 = 42\; \sf units[/tex]

Side z is the leg of the same 30-60-90 triangle opposite the 60° angle.

[tex]\implies z=c\sqrt{3}=21\sqrt{3}\; \sf units[/tex]

Side z is the side opposite the 30° angle in a 30-60-90 triangle with the other leg (opposite the 60° angle) measuring 18 units. Therefore, c√3 = 18, so c = 18/√3 = 6√3 units.

[tex]\implies z=c = 6\sqrt{3}\; \sf units[/tex]

Therefore, the hypotenuse of the same triangle measures 2c = 12√3 units.

Sides x and y are the congruent legs of a 45-45-90 triangle with hypotenuse measuring 12√3 units. Therefore b√2 = 12√3, so b = 6√6.

[tex]\implies x=6 \sqrt{6}\; \sf units[/tex]

[tex]\implies y=6 \sqrt{6}\; \sf units[/tex]

The requried, we would expect 8 of the next 16 muffins sold to be bran muffins.

Out of a total of 50 muffins sold, 25 were bran muffins. That means that the proportion of bran muffins to total muffins sold is:

25 / 50 = 1/2

To find how many of the next 16 muffins sold would be bran muffins, we can multiply the proportion of bran muffins by the total number of muffins:

(1/2) * 16 = 8

Therefore, we would expect 8 of the next 16 muffins sold to be bran muffins.

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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:

x=4

Step-by-step explanation:

2x+(-3) =5

2x-3+3=5+3

2x/2=8/2

x=4

[tex]\cfrac{cos^2(a)}{sin(a)-1}\implies \cfrac{1-sin^2(a)}{sin(a)-1}\implies \cfrac{\stackrel{ \textit{difference of squares} }{1^2-sin^2(a)}}{sin(a)-1} \\\\\\ \cfrac{[1-sin(a)][1+sin(a)]}{sin(a)-1}\implies \cfrac{~~\begin{matrix} [1-sin(a)] \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~[1+sin(a)]}{-[~~\begin{matrix} 1-sin(a) \\[-0.7em]\cline{1-1}\\[-5pt]\end{matrix}~~]}\implies -[1+sin(a)][/tex]

Answer:-1

Step-by-step explanation:

[tex]\frac{cos^{2}a}{sina-1} \\cos^{2}a=\frac{1+cos2a}{2}\\\frac{\frac{1+cos2a}{2}}{sina-1}=\frac{1+cos2a}{2(sina-1)}=\frac{1+cos2a}{2sina-2}=\frac{1+cos2a}{-(2-2sina)}\\1-2sina=cos2a\\\frac{1+cos2a}{-(1+(1-2sina))}=\frac{1+cos2a}{-(1+cos2a)}=-1[/tex]

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: Z=-5

Step-by-step explanation:

we want to group Z on only one side of the equation

-18=Z-3+2Z

-18-Z-2Z=-3 (subtract Z and 2Z from both sides, to remove Z from one side and to move it to the other side)

-3Z=-3+18 (regroup Z and add 18 on both sides, to have Z on only one side)

-3Z=15 (add -3+18)

3Z=-15 (multiply both sides by -1 to get a positive Z)

Z=-5 (divide both sides by 3)

Answer: 16

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

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: To solve this expression, we need to simplify it using the order of operations (PEMDAS) and perform the indicated operations:

2.3x^2 + 69 / 465

First, we need to simplify the fraction 69/465. Both the numerator and denominator have a common factor of 3, so we can simplify the fraction by dividing both by 3:

69 / 465 = 23 / 155

Now we can substitute this simplified fraction back into the original expression:

2.3x^2 + 23 / 155

The expression is now in simplified form, and there are no more operations we can perform. So this is the final answer.

Alternatively, if the expression was meant to be solved for x, we would need an equation (e.g., 2.3x^2 + 69 / 465 = some value) in order to solve for x. As it stands, we cannot solve for x with the given expression.

Step-by-step explanation:

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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The approximate value of x which the number of bacteria would approximately be the same include the following: A. 2.9 Days.

In Mathematics, an exponential function can be represented or modeled by using the following mathematical equation:

[tex]f(x) = a(b)^x[/tex]

Where:

Based on the information provided about the two bacteria cultures, we can logically deduce the following exponential functions;

[tex]y = 100(1.02)^x \\\\y = 80(1.05)^{2x}[/tex]

By equating the two exponential functions above, we have:

[tex]100(1.02)^x = 80(1.05)^{2x}\\\\\frac{1.02^x}{1.05^{2x}} =80/100\\\\\frac{1.02^x}{1.1025^{x}} =0.8\\\\0.9252^x = 0.8[/tex]

By taking the logarithm of both sides, we have:

[tex]Log_{0.9252}(0.9252^x) = Log_{0.9252}(0.8)[/tex]

x = 2.9 days.

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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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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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