for a given geometric sequence the seventh term is equal to 19/81 and the 12th term is equal to "-57" find the value of the 15th term

_____________________

[tex]\huge\blue{\mid{\fbox{\tt{Answer}}\mid}}[/tex]

[tex]\huge\blue{\mid{\fbox{\tt{Explanation}}\mid}}[/tex]

Multiplication - Multiplication, otherwise known as the inverse of division repeats a certain number's value into what it has been multiplied by. Ex. [tex]4\cdot4=16[/tex]

Fraction - A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, or three-quarters.

Algebra - Algebra is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all mathematics

Reciprocal - In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by [tex]\frac{1}x[/tex] or x⁻¹, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction [tex]\frac{a}b[/tex] is [tex]\frac{b}a[/tex]. For the multiplicative inverse of a real number, divide 1 by the number.

_____________________

Total: 499.75

Step-by-step explanation:

4 × 110.5 = 442

3 × 19.25 = 57.75

442 + 57.75 = 499.75

The required sum of the first twenty terms of the series ∑n=1log 5r are in AP will be 32.37.

There are four basic properties of logarithms:

logₐ(xy) = logₐx + logₐy.

logₐ(x/y) = logₐx - logₐy.

logₐ(xⁿ) = n logₐx.

logₐx = logₓa / logₓb.

We have been given that the terms of the series ∑n=1log 5r are in AP.

As per the given question,

∑n(log(5r)) = log(5·1)+log(5·2)+log(5·3)+...+log(5·n)

n = 20

∑log(5r) = log(5·1)+log(5·2)+log(5·3)+...+log(5·20)

Using the property of logarithmic logₐ(xy) = logₐx + logₐy,

∑log(5r) = (log(5)+log(1)) + (log(5)+ log(2))+(10g(5)+Log(3))+ ...(10g(5)+Log(20))

∑log(5r) = (log(5)+log(5)+log(5)+...+log(5)

∑log(5r) = 20·1og(5)+((log(1)+log(2)+log(3)+...+log(20)

Using property of logarithmic logₐ(xy) = logₐx + logₐy,

∑log(Sr)=20·1og(5)+log(1×2×3×...20)

∑log(5r) = 20-log(5)+log(201)

∑log(5r) = 32.365524703598

Rounded to the nearest hundredth decimal,

∑log(5r) = 32.37

Thus, the required sum of the first twenty terms of the series ∑n=1log 5r in AP will be 32.37.

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If she wants to find the rate at which she burned calories on day 8. C'(8) is: B. 8.75 calories per day

First step is to find the upper and lower values for the burned calories on day 8.

Upper value (C'(8) =  405 - 386 /2

Upper value C'(8) = 19/2

Upper value C'(8) = 9.5

Lower value C'(8) = 386 -370/2

Lower value C'(8) = 16/2

Lower value C'(8) = 8

Now let find C'(8)

C'(8) = 9.5+8/2

C'(8) = 17.5/2

C'(8) = 8.75 calories per day

Therefore the correct option is B.

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The equations in order from least to greatest unit rate are y = 13/6x,y = 11/5x, y = 21/9x and y = 19/8x

On a graph, the unit rate is obtained by calculating the slope of the line. The slope is the change in y divided by the change in x:

Slope = y2-y1 / x2-x1  

Thus, we can any two convenient points on the graph and use it calculate the slope:

Let's pick points (5, 10) and (12, 25)

This implies x1 = 5, y1 = 10 and  x2 = 12, y2 = 25

Slope = y2-y1 / x2-x1

Slope = 25-10 / 12-5 = 15/7

Thus,  the unit rate of the graph is 15/7 = 2.143

Find the equations with unit rates greater than the unit rate of the graph:

y = 19/8x: unit rate = 19/8 =  2.375

y = 27/13x: unit rate = 27/13 = 2.077

y = 21/9x: unit rate = 21/9 = 2.333

y = 11/5x: unit rate = 11/5 = 2.200

y = 15/17x: unit rate = 15/17 = 0.882

y = 31/15x: unit rate = 31/15 = 2.067

y = 13/6x: unit rate = 13/6 = 2.167

The equations with unit rates greater than the unit rate of the graph are y = 19/8x, y = 21/9x, y = 11/5x and y = 13/6x

Therefore, the equations in order from least to greatest unit rate are y = 13/6x,y = 11/5x, y = 21/9x and y = 19/8x. Drag each tile to the correct box accordingly

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The equation that represents circles that have a diameter of 12 units and a center that lies on the y-axis is x²+ (y-3)² = 36

The location of a circle in the Cartesian plane is represented by a circle equation. We can write the equation for a circle if we know the location of the circle's center and how long its radius is. All of the points on the circle's circumference are represented by the circle equation.

The standard equation of a circle with center at

(a, b) and radius is r.

So, the standard equation circle is

(x -a)² + (y- b)² = r²

Given:

As, we know the standard form for equation of a circle is

(x-a)² + (y-b)² = b =r²

where (a, b) is the center of the circle and r is the radius of the circle.

We have

diameter = 12 units

radius = 12/2 = 6 units

So, the center will be at (0, 3)

Thus, the equation of circle

(x-0)² + (y-3)² = 6²

x²+ (y-3)² = 36

Hence, the Equation of Circle is x²+ (y-3)² = 36.

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A) It is an exponential equation.

B) y = 2^x

C) The rate between days 2 and 4 is 6 minutes per station number.

Here we have the following data:

(1, 2), (2, 4), (3, 8), (4, 16)

What you can see is that the x-value of the points increases equially between consecutive pairs (increases of one unit).

while the y-value does not do that.

First, it increases by 2, then by 4, then by 8.

So this is an exponential relation.

B) A general exponential will be of the form:

y = a*(b)^x

Usin the first two pairs (1, 2) and (2, 4) we can write two equations:

2 = a*b^1 = a*b

4 = a*b^2

So we have a little system of equations:

2 = a*b

4 = a*b*b

If we take the second equation and we divide that by the first one we get:

4/2 = (a*b*b)/(a*b)

2 = b

Replacing that in the first equation we get:

2 = a*2

2/2 = a = 1

The equation is:

y = 2^x

C) The average rate of change will be:

R = (2^4 - 2^2)/(4 - 2) = (16 - 4)/2 = 12/2 = 6

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A) It is an exponential equation.

B) y = 2^x

C) The rate between days 2 and 4 is 6 minutes per station number.

The new temperature is given as -15°F.

A measurement unit is a unit to measure certain quantities.

For example, the mass can be measured in kilograms, length can be measured in meter and time can be measured in seconds.

To find the measurement of a new quantity known units can be used in operation. As to find the unit for speed we use m/s as a unit, where, m is the unit of distance and s is the unit of time.

Given that,

The present temperature is 0°F.

And, the drop in the temperature is 15°.

The new temperature after the drop can be found by subtracting the amount of drop from the initial temperature as follows,

0 - 15°F

=  -15°F

Hence, the new temperature after the drop is  -15°F.

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The completion of the following accounting ratios is as follows:

A. Profit margin on net sales is 31.6%.

B. Acid test is 86.0%.

Accounting ratios enable financial analysts to understand the relative magnitude of one financial statement account to another.

By their nature, accounting ratios compare or represent an entity's financial performance or position in relative terms.

Current assets = $13,000

Inventory = $4,400

Current liabilities = $10,000

Net income = $7,700 Net sales = $24,400

The profit margin on net sales = Net Income ÷ Net Sales x 100

= $7,700 ÷ $24,400

= 31.6%

Acid Test Ratio = (Current Assets - Inventory) ÷ Current Liabilities

= ($13,000 - $4,400) ÷ $10,000

= 0.86

= 86%

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-353.92-(-283.56)-131.29= -20,165

a) The test statistic is: t = 1.85.

b) The p-value is: 0.0648.

c) The null hypothesis is not rejected.

d) There is not sufficient evidence to conclude that the population mean is different of 4.

The standard deviation is known only for the sample, hence the t-distribution is obtain the test statistic.

The test statistic of the t-distribution is given by the equation presented as follows:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

In which the variables of the equation are given as follows:

For this problem, the values of these parameters are as follows:

[tex]\overline{x} = 4.32, \mu = 4, s = 4.24, n = 600[/tex]

Hence the test statistic is obtained as follows:

[tex]t = \frac{\overline{x} - \mu}{\frac{s}{\sqrt{n}}}[/tex]

[tex]t = \frac{4.32 - 4}{\frac{4.24}{\sqrt{600}}}[/tex]

t = 1.85.

The p-value is obtained using a t-distribution calculator, with a two-tailed test, as we are testing if the mean is different a value, with 600 - 1 = 599 df and t = 1.85, hence it is of 0.0648.

Since the p-value is greater than the significance level of 0.01, it is found that the conclusion regarding the null hypothesis is that it is not rejected, meaning that the sample does not give sufficient evidence to conclude that the population mean is different of 4.

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The answer will be 0.33 using quadratic equation.

What is quadratic equation

it's a second-degree quadratic equation which is an algebraic equation in x. Ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant term, is the quadratic equation in its standard form. A non-zero term (a 0) for the coefficient of x2 is a prerequisite for an equation to be a quadratic equation. The x2 term is written first, then the x term, and finally the constant term is written when constructing a quadratic equation in standard form. In most cases, the numerical values of letters a, b, and c are expressed as integral values rather than fractions or decimals.

The equation given 3k² + k -3=0

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

k =1/3

k=0.33

Hence the value of k is 0.33.

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The simplified power functions are a) f(x) = [tex]\frac{-3}{2} x^-5[/tex] and b)g(x)=5[tex]x^\frac{1}{3}[/tex] .

What is power function ?

A function having a single term that is the sum of a real number, a coefficient, and a variable raised to a fixed real number is called a power function. Consider functions for area or volume as examples (a number that multiplies a variable raised to an exponent is known as a coefficient). A power function is a function where y = x n, where n is any real constant number. In reality, many of our parent functions, including linear and quadratic functions, are power functions.

Here ,

a) f(x) = [tex]\frac{-3}{2x^5}[/tex]

To write in the form of f(x)= [tex]kx^p[/tex]

=> f(x) = [tex]\frac{-3}{2}* \frac{1}{x^5}[/tex]

=> f(x) = [tex]\frac{-3}{2} x^-5[/tex]

b) g(x) = 5[tex]\sqrt[3]{x}[/tex]

Here we know that [tex]\sqrt[n]{a} = a^\frac{1}{n}[/tex] , then

=>g(x)=5[tex]x^\frac{1}{3}[/tex]

Hence the simplified power functions are a) f(x) = [tex]\frac{-3}{2} x^-5[/tex] and b)g(x)=5[tex]x^\frac{1}{3}[/tex] .

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The required image of points (-4, 6) is (-6, -4) as per the given condition,

Functions are the relationship between sets of values. e g y=f(x), for every value of x there is its exists in a set of y. x is the independent variable while Y is the dependent variable.

Here,

As the image of the point (5, -1) is mapped as, (1, 5),
Which implies a function that (x, y) mapped as (-y , x)
So for (-4, 6) mapped as (-6, -4)

Thus, the required image of points (-4, 6) is (-6, -4) as per the given condition,

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Answer:

[tex]x\leq -7 or x > -2[/tex]

Step-by-step explanation:

-8x+14 ≥ 60 or - 4x + 50 < 58
     -8x ≥ 56 or         - 4x < 8
        [tex]x \leq -7[/tex] or            [tex]x > -2[/tex]

Yes, I agree with Jorge.

Jorge's reasoning states that if two sides of the triangles are similar with a common measure of angle then one triangle can fit/overlap with the other triangle.

Jorge could improve his proof by stating his explanation in terms of mathematics.

He could have used the SAS similarity to show that the given triangles are similar.

What is the similarity test of two triangles?

If the ratio of the lengths of two sides of one triangle equals the ratio of the lengths of two sides of another triangle, and the included angles are equal, the two triangles are similar.

In the given figures,  the length of side AB is equal to the length of side DE.

                        Side AB = Side DE ----------(I)

Similarly, the length of side BC is equal to the length of side EF.

                        Side BC = Side EF ----------(II)

And the measure of angle  ABC is equal to the measure of angle DEF.

                              ∠ ABC = ∠ DEF -------(III)

From (I), (II), and (III), and by using the SAS test we can say that

                               ΔABC ~ ΔDEF    ----------- SAS test

Hence,

Yes, I agree with Jorge.

Jorge's reasoning states that if two sides of the triangles are similar with a common measure of angle then one triangle can fit/overlap with the other triangle.

Jorge could improve his proof by stating his explanation in terms of mathematics.

He could have used the SAS similarity to show that the given triangles are similar.

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Answer: 6x - 4 = 68

Step-by-step explanation:

           Vertical angles are congruent to each other. The two angles given, 6x - 4 and 68, are vertical angles. This means we can set them equal to each other for our equation. Using this logic, the answer is;

                         6x - 4 = 68

X= 20 & Y= 70 in right-angled triangle .

What is right angle triangle?

2X + 140 = 180

2X = 40°

   X = 20°

X + Y = 90°

   Y = 70°

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The value of x is -1 for equation -4x = -x + 3

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

The given equation is -4x = -x + 3

In the given equation minus four times of x equal to minus x plus three

x is the variable

Plus and minus are operators

-4x = -x + 3

Take the variables to one sides

Add +x on both sides

-4x+x=-x+x+3

-3x=3

Divide both sides by -3

x=-1

Hence, the value of x is -1 for equation -4x = -x + 3

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The number of months Amy can take the Gymnastic = 2 months.

Amy wants to start taking gymnastics.

The fee structure of the gymnastics is divided into two parts.

There is a one time fee payment of 35$ and there is a monthly payment that costs 85$ per month.

So the first month fee of the gymnastics = 35 + 85 = 120$

Savings of Amy = 225$

If Amy pays the 1st month fee ,

then the she will be left with the savings  = 225$ - 120$ = 105 $

It means Amy can join the gymnastics for one more month .

Hence the number of months Amy can take the Gymnastic = 2 months .

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[tex]{ \huge{ \boxed{ \red{ \sf{x = 7}}}}}[/tex]

Step-by-step explanation:-

[tex]{ \sf{126 = 2 \times 3 \times 3 \times k}}[/tex]

[tex]{ \sf{126 = 6 \times 3 \times k}}[/tex]

[tex]{ \sf{126 = 18 \times k}}[/tex]

[tex]{ \sf{126 = 18k}}[/tex]

Divide both the sides by 18 in order to find the value of k. Then,

[tex]{ \sf{ \frac{126}{18} = \frac{18}{18} k}}[/tex]

[tex]{ \boxed{ \boxed{ \purple{ \sf{x = 7}}}}}[/tex]

Answer: 0.000089

Step-by-step explanation: 0.000089 is already in standard form, if you want 0.000089 in the scientific form then that would be...

8.9 x 10⁻⁵

Answer:

A. [tex]\log_9{3}=x[/tex]

Step-by-step explanation:

[tex]y=e^x\rightarrow\log_e{y}=x[/tex]

Therefore,

[tex]3=9^x\rightarrow\log_9{3}=x[/tex]

The length of wall in real life is 4 feet.

Given that on an architect's blueprint, 1 inch corresponds to 4 feet.

In this architect's plan, a wall represented by a line 1 inches long on the blue print.

So, the length of wall in real life is = 4 feet.

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Answer:

4ft

Step-by-step explanation:

If 1 inch corresponds to 4 ft on the blueprint. The ratio is 1 inch on the blue print for every 4 ft in real life.

The money available for private investment is $250 billion dollar

What is demand curve ?

The demand curve is a graphical representation of the connection between the cost of a commodity or service and the amount that is demanded over time. The price will often be shown on the left vertical axis in a representation, and the amount needed will typically be shown on the horizontal axis.

Refer to the attachment for graph:

Once investments and government borrowing are added together, the total is = $400

if borrowing from the government is= $150

private investment

=( investment and and government borrowing) - (government borrowing )

= $400 - $150 = $250

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The simple interest that Nike will receive after 2 years is $500.

The principal amount =$5000

Rate = 5% = 0.05

Time = 2 years

Now simple interest = PRT = 5000 × 0.05 × 2 = 500

Hence the simple interest earned by Nike is $500

Calculating the amount of interest that will be owed on a sum of money at a certain rate and for a specific period of time is possible using simple interest.

Simple interest does not change the principal amount in the same way as compound interest, which adds the interest of one year's principal to the next year's principal to calculate interest.

A loan is a sum of money that a person obtains from a bank or another type of financial institution in order to pay bills. A few types of loans are personal, student, auto, and mortgage loans.

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The Minimum value is y(min) = 48 , x = 5 and The maximum value is   [tex]y = -2(x -5)^2+48[/tex]

Given,

In the question:

The given Function is :

[tex]y = -2x^2 + 20x -2[/tex]

To find the maximum or minimum value of the function:

Now, According to the question:

For maximum value:

The given Function is :

[tex]y = -2x^2 + 20x -2[/tex]

Take out the leading coefficient is :

[tex]y = -2(x^2 + 10x) -2[/tex]

Complete the square:

[tex]y = -2(x^2 + (2(-5))x +(-5)^2+(-2-(-2).(-5)^2)[/tex]

Express the function in vertex form:

[tex]y = -2(x -5)^2+48[/tex]

For Minimum Value:

y(min) = 48

x = 5

Hence, The Minimum value is y(min) = 48 , x = 5 and The maximum value is   [tex]y = -2(x -5)^2+48[/tex].

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The amount that is outstanding when Brett tallies up the transactions in his check.is $9.19.

From the information, Brett tallies up the transactions in his check register and comes up with a total balance of $248.53, but his bank statement says that his balance is $257.72.

The amount that is outstanding will be the difference between the amount given. This will be:

= Bank statement - Total balance

= $257.72 - $248.53

= $9.19

This is the outstanding amount.

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