I need help with this practice problem If you can, show your work step by step so I can take helpful notes
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The given geometric series is
[tex]120-80+\frac{160}{3}-\frac{320}{9}+\cdots[/tex]In a geometric series, there is a common ratio between consecutive terms defined as
[tex]r=\frac{-80_{}}{120_{}}=-\frac{2}{3}[/tex]The sum of the first n terms of a geometric series is given by
[tex]S_n=\frac{a(1-r^n)}{1-r},r<1[/tex]Where a is the first term.
From the given series
a = 120
Hence, the sum of the first 8 terms is
[tex]S_8=\frac{120(1-(-\frac{2}{3})^8)}{1-(-\frac{2}{3})}[/tex]Simplify the brackets
[tex]S_8=\frac{120(1-\frac{2^8}{3^8}^{})}{1+\frac{2}{3}}[/tex]Simplify further
[tex]\begin{gathered} S_8=\frac{120(1-\frac{256}{6561})}{\frac{3+2}{3}} \\ S_8=\frac{120(\frac{6561-256}{6561})}{\frac{5}{3}} \\ S_8=\frac{120(\frac{6305}{6561})}{\frac{5}{3}} \\ S_8=\frac{120\times6305}{6561}\div\frac{5}{3} \\ S_8=\frac{120\times6305}{6561}\times\frac{3}{5} \\ S_8=\frac{120\times6305}{6561}\times\frac{3}{5} \\ S_8=\frac{8\times6305}{729} \\ S_8=\frac{50440}{729} \end{gathered}[/tex]Therefore, the sum of the first 8 terms is
[tex]\frac{50440}{729}[/tex]Related Questions
a line passes through the points (2,5) and (-6,4). What is it's equation in point-slope form?
Answers
Answer:
y-5=⅛(x-2)
Explanation:
Given the points (2,5) and (-6,4).
To find the equation of the line joining these points in point-slope form, we begin by finding its slope.
[tex]\begin{gathered} \text{Slope,m}=\frac{Change\text{ in y-axis}}{Change\text{ in x-axis}} \\ =\frac{5-4}{2-(-6)} \\ =\frac{1}{2+6} \\ m=\frac{1}{8} \end{gathered}[/tex]Next, we substitute the slope and any of the given points into the point-slope form below:
[tex]y-y_1=m(x-x_1)[/tex]We use the point (2,5).
• x1=2, y1=5
[tex]y-5=\frac{1}{8}(x-2)[/tex]The equation in point-slope form is y-5=⅛(x-2).
find the measure of each segment
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The value of each line segment is found as 26 units.
What is defined as the mid point?A midpoint is a point in the center of a line connecting two points. The two points of reference are the line's endpoints, and the midpoint is located between the two. The midpoint splits the line connecting such two points in half. Furthermore, a line drawn to bisect the line connecting these two points passes through midpoint.For the given question.
The line is given as DE and the mid point of line is D.
Thus,
CD = DE .....eq 1
The value of each term are given as;
CD = 2x + 7
DE = 4(x - 3)
Put the value in equation 1.
2x + 7 = 4(x - 3)
2x + 7 = 4x - 12
Bring variables and constants on the different sides.
4x - 2x = 7 + 12
2x = 19
x = 9.5
Put the value in each side;
CD = 2×9.5 + 7 = 26 unitsDE = 4(9.5 - 3) = 26 unitsThus, the value of each line segment is found as 26 units.
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HELP ASAP MATH PRE CALC
Answers
The values for the dimensions of the open box are L = (30 - x)inches, W = (30 - x)inches, and H = (x)inches.
The cube as a three dimensional shapes.A cube is 3- dimensional shape with 6 equal sides, 6 faces, and 6 vertices. Each face of a cube is a square. In there dimension, the cube's sides are; the L = length, W = width, and H = height.
From question, squares of equal sides x are cut out of each corner of the metal sheet, hence the dimension for the height of the box is equal to x.
So; H = (x) inches,
L = (30 - x)inches, and
W = (30 - x)inches.
Therefore, the dimensions of the box that can maximize the volume of the metal sheet of 30inches by 30inches are L = (30 - x)inches, W = (30 - x)inches, and H = (x)inches.
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Find the solution of this system of linearequations. Separate the x- and y- values with acomma. Enclose them in a pair of parantheses. System of equations4x + 8y = 838x + 7y = 76- 8x - 16y = -1668x + 7y = 76
Answers
Given,
System of equation is,
[tex]\begin{gathered} 4x+8y=83\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots(i) \\ 8x+7y=76\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots(ii) \end{gathered}[/tex]Taking the equation (i) as,
[tex]\begin{gathered} 4x+8y=83 \\ 4x=83-8y \\ x=\frac{83-8y}{4} \end{gathered}[/tex]Substituting the value of x in equation (ii) then,
[tex]\begin{gathered} 8x+7y=76 \\ 8(\frac{83-8y}{4})+7y=76 \\ 664-64y+28y=304 \\ 36y=360 \\ y=10 \end{gathered}[/tex]Substituting the value of y in above equation then,
[tex]\begin{gathered} x=\frac{83-8\times10}{4} \\ x=\frac{3}{4} \end{gathered}[/tex]Hence, the value of x is 3/4 and y is 10. (3/4, 10)
System of equation is,
[tex]\begin{gathered} -8x-16y=-166\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots(i) \\ 8x+7y=76\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots(ii) \end{gathered}[/tex]Taking the equation (i) as,
[tex]\begin{gathered} -8x-16y=-166 \\ 8x+16y=166 \\ 4x+8y=83 \\ 4x=83-8y \\ x=\frac{83-8y}{4} \end{gathered}[/tex]Substituting the value of x in equation (ii) then,
[tex]\begin{gathered} 8x+7y=76 \\ 8(\frac{83-8y}{4})+7y=76 \\ 664-64y+28y=304 \\ 36y=360 \\ y=10 \end{gathered}[/tex]Substituting the value of y in above equation then,
[tex]\begin{gathered} x=\frac{83-8\times10}{4} \\ x=\frac{3}{4} \end{gathered}[/tex]Hence, the value of x is 3/4 and y is 10. (3/4, 10)
in two or more complete sentences, compare the slopes of the two functions. in your comparison, include which function has the greatest slope.
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Slope of g (x) : ZERO. The slope of any horizontal line is zero, 0.
Slope of f (x) :
let's take the points ( -4, 7) and (-2, 5) from the table
[tex]m=\frac{y_2-y_1}{x_2-x_1}=\frac{5-7}{-2-(-4)}=\frac{-2}{-2+4}=\frac{-2}{2}=-1[/tex]Answer: The slope of g (x ) is zero since it is a horizontal line while the slope of f (x) is -1. The slope of g(x) i greater than the slope of f(x).
translate the inequality into a sentence. ten subtracted from the product of 9 and a number is at most -17. use x for unknown number
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Help on math question precalculus Match the description with the correct base for the logarithm.-LOG without a subscript has a base of -Ln has a base of Choices =10,e
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The formal way of writing a logarithm is the following:
[tex]\log _ab[/tex]Where "a" is the base of the logarithm and "b2 is the argument.
If "a = 10", then the base is not written, like this:
[tex]\log _{10}b=\log b[/tex]In the case that the base is the constant number "e" then the logarithm is called a "natural logarithm" and it is written as follows:
[tex]\log _eb=\ln b[/tex]a) Reflection, then translationb) Rotation, then translationc) Reflection, then rotationd) Rotation, then reflection
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We have the following:
Therefore we can conclude that from step 1 to step 2, it is a rotation because it moves on its own axis and from step 2 to 3, it is a reflection
Mrs. algebra ordered some small and medium pizzas for her daughter‘s birthday party. The small pizzas cost $5.75 each and the medium pizzas cost $8.00 each. She bought three more medium pizzas than small pizzas and her total order came to $51.50 How many pizzas of each did Mrs. Algebra order? Write an equation and solve.
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we have the following:
x is small pizzas
y is medium pizzas
[tex]\begin{gathered} 5.75\cdot x+8\cdot y=51.5 \\ y=x+3 \\ 5.75\cdot x+8\cdot(x+3)=51.5 \\ 5.75x+8x+24=51.5 \\ 13.75x=51.5-24 \\ x=\frac{27.5}{13.75} \\ x=2 \end{gathered}[/tex]therefore, the answer is:
2 small pizzas and 5 (2+3) medium pizzas
1. Write the value of the digit in the hundreds place and the value of the digit in the tens place in 440. What is the relationship between the values of those two digits? The ___ in the in the hundreds place has a value _____ times as great as the____in the ____ place.
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The ___ in the in the hundreds place has a value _____ times as great as the____in the ____ place.
• We have 440
,• 400 + 40
,• Four hundreds + four tens
The relationship between the values of these two digits is that they are the same, but the four in the hundreds place has a value ten times as great as the four in the tens place.
Please provide a deep explanation with examples so I can understand and learn, thank you
Answers
Since the package of 500 sheets has dimensions of
[tex]216\times279\times45[/tex]Since 7000 sheets will need to be put in
[tex]\frac{7000}{500}=14[/tex]14 similar package
Since the dimensions of the case are
[tex]216\times279\times270[/tex]The length and the width of the package are the same as the length and the width of the case
Then we will use the heights of them to find how many package we can put in the case
[tex]\frac{270}{45}=6[/tex]That means we can fill the case with 6 packages
Since we have 14 packages, then we will need 6 + 6 + 2
3 cases 2 full and one has 2 packages only
1
Pratap Puri rowed 26 miles down a river in 2 hours, but the return trip took him 6; hours. Find the rate Pratap can row
in still water and find the rate of the current. Let x=rate Pratap can row in still water and y = rate of the current.
What is the rate that Pratap rows in still water?
Pratap can row at a rate of
(Type an integer or a decimal.)
in still water.
Answers
The speed of current will be "4.5 mph" and the rate Pratap can row in still water will be "8.5 mph".
What does "speed" mean in mathematics?
Speed is what it means. the speed of a change in an object's location in any direction. Speed is defined as the ratio of distance to the amount of time it took to cover that distance. Speed is a scalar quantity because it just has a direction and no magnitude.Given:
Distance "26 miles" in time "2 hours".
Let,
Speed of water = y
Pratap speed when rowing in still water = x
As we know,
Speed = distance/time
then,
x + y = 26/2
x + y = 13
x = 13 - y
In return trip took him time "6.5 hours",
x - y = 26/6.5
x - y = 4
By substituting the value of "x", we get
13 - y - y = 4
13 - 2y = 4
2y = 13 - 4
2y = 9
y = 9/4 = 4.5 mph (Rate of the current)
By substituting the value of "y", we get
x = 13 - a
x = 13 - 4. 5 = 8.5 mph (Pratap can row in still water)
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[tex]3 - \frac{1}{2} = 3 + n[/tex]what is nthank you
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y+2=−3(x−4)y, plus, 2, equals, minus, 3, left parenthesis, x, minus, 4, right parenthesis Complete the missing value in the solution to the equation.
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The required equation is 11y = 3x + 2.
What is equation?
An algebraic equation with simply a constant and a first-order (linear) term, such as y=mx+b, where m is the slope and b is the y-intercept, is known as a linear equation.
Given y+2=−3(x−4)y .....................(1)
Simplifying (1) and we get
y+2=−3x + 12y
=> 3x + y - 12y + 2 = 0
=> 3x -11y + 2 = 0
=> 3x - 11y = -2
=> 11y - 3x = 2
=> 11y = 3x + 2
Therefore, the required equation is 11y = 3x + 2.
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40.0 Reyna runs a textile company that manufactures T-shirts. The profit, p, made by the company is modeled by the function p=s2+95-142, where s is the number of T-shirts sold. How many T-shirts should be sold to earn a profit of more than $2,000?
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But cannot be negative, hence s= 42. This implies that 42 shirts will be sold to make a profit of exactly $2000.
To earn a profit of more than $2000, then s must be greater than 42
This makes the answer to be s > 42
The correct answer is the second option
Please help, will give brainliest!!!!
i am asked to find the range of this, (of the possible third angle)
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Answer:
rage=<C-<B
=101°-70°
=30°
1-Findes the length indicated.2- Find the angle indicated.3-Find the distance between each pair of points.
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1.
LM = LN - MN
LM = 22 - 5 (Replacing)
LM= 17 (Subtracting)
2.
m∠EDW= m∠EDC - m∠WDC
m∠EDW= 106° - 40° (Replacing)
m∠EDW= 66° (Subtracting)
3.
Using the formula for the distance between two points we have:
[tex]\begin{gathered} d=\sqrt[]{(x2-x1)^2+(y2-y1)^2} \\ x1=8,x2=-6,y1=3,y2=3 \\ d=\sqrt[]{(8-(-6))^2+(3-3)^2}\text{ (Replacing)} \\ d=\sqrt[]{(8+6)^2+(0)^2}\text{ (Subtracting)} \\ d=\sqrt[]{(14)^2^{}}\text{ (Adding)} \\ d=14\text{ (Raising 14 to the power of 2 and taking the square root)} \\ d=14 \\ \text{ The distance between these points is 14} \end{gathered}[/tex]Using the formula for the midpoint we have:
[tex]\begin{gathered} (\frac{x1+x2}{2},\frac{y1+y2}{2}) \\ x1=8,x2=-6,y1=3,y2=3 \\ (\frac{8+(-6)}{2},\frac{3+3}{2}) \\ (\frac{2}{2},\frac{6}{2})\text{ (Subtracting and adding)} \\ (1,\text{ 3) (Dividing)} \\ \text{The midpoint is (1,3)} \end{gathered}[/tex]H.O.T. FOCUS ON HIGHER ORDER THINKING 20. Communicate Mathematical Ideas Explain how to graph the inequality 8≥ y.
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Given the inequality:
8 ≥ y
Let's graph the inequality.
To graph the inequality, take the following steps:
Step 1.
Rewrite the inequality for y and slip the inequality.
[tex]y\le8[/tex]Step 2.
Draw a solid horizontal line at y = 8.
Since the y is less than or equal to 8, shade the region below the boundary line.
Thus, we have the graph of the inequality below:
Alan is putting money into a savings account. He starts with $550 in the savings account, and each week he had $70. Let S represent the total amount of money in the savings account in dollars, and let W represent the number of weeks Allen has been adding money. Write an equation relating S to W. Then use this equation to find the total amount of money in the savings account after 18 weeks.
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Equation: S = $70·W + $550
Amount of manoey after 18 weeks: $1810
To solve this, we have two variables, the amount of weeks (W) and savings (S)
Each week, $70 dolars are added to the account. Then we can write this as: $70·W.
Now there is an initial amount of $550. Then we must add that mount to the previous calculation: $70·W + $550
This give us the savings on each week. THen THe complete equation is S = $70·W + $550
Now, to know the savings after 18 weeks, we can replace W = 18 and solve:
[tex]\begin{gathered} S=$70\cdot W+$550 \\ S=70\cdot18+550 \\ S=1260+550=1810 \end{gathered}[/tex]Thus, the savings after 18 weeks is $1810
QuestionAreli invested a principal of $950 in her bank account with an interest rate of 3%. How much interest did she earn in 5years?
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$ 142.5
Explanation
to solve this we can use the formula:
[tex]i=\text{Prt}[/tex]where i is the interest, P is the principal, r is the rate ( in decimals) and t is the time
then
Step 1
Let
[tex]\begin{gathered} i=i \\ P=950 \\ r=3\text{ \% =0.03} \\ \text{Time= 5 years} \end{gathered}[/tex]replace in the formula
[tex]\begin{gathered} i=950\cdot0.03\cdot5 \\ i=142.5 \end{gathered}[/tex]hence, the answer is
$ 142.5
I hope this helps you
The 3D object above is sliced parallel to the base. What shape is formed? triangle octagon rectangle hexagon
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When a 3D object is sliced such that the top is parallel to the base, then the top and the base formed same shape.
The shape formed at the base;
It is a six-sided shape. A six-sided polygon is called HEXAGON
Allen's goal is to have between 1,500 and 1,600 bottles in his collection. Write and solve a compound inequality to determine the number of weeks it will take Allen to reach his goal.
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The compound inequality is 1500 < 300 + 25x < 1600, the solution is 48 < x < 52 and the number of weeks to reach his goal 48 to 52 weeks
How to determine the compound inequality?The given parameters from the question are
Existing collection = 300
Rate = 25 bottles each week
Represent the number of weeks by x and the total number of bottles with y
So, we have the following equation
y = Existing collection + Rate * x
This gives
y = 300 + 25x
Also, we have
The goal is to have between 1,500 and 1,600 bottles in his collection.
This means that
1500 < Total < 1600
So, we have
1500 < 300 + 25x < 1600
Evaluate the like terms
1200 < 25x < 1300
Divide by 25
48 < x < 52
Hence, the number of weeks is 48 to 52 weeks
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Possible question
Allen wants to add to his existing collection of 300 bottles. Starting today, he will collect 25 bottles each week.
Allen's goal is to have between 1,500 and 1,600 bottles in his collection. Write and solve a compound inequality to determine the number of weeks it will take Allen to reach his goal.
Forproblems 5-10, determine what type of symmetry each figure has. If the figure has line symmetry, determine how many lines of symmetry the figure has. If the figure has rotational symmetry, determine the angle of rotational symmetry and if the figure also has point symmetry. (A figure can have both line and rotational symmetries or neither of these symmetries)
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7. The figure has line and rotational symmetries. There are 2 lines of symmetry. The angle of symmetry is 180°
8. The figure has no symmetry
The graph of F(x), shown below, resembles the graph of G(x) = x^2 but it hasbeen stretched somewhat and shifted. Which of the following could be theequation of F(x)?
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Solution
The final answer
Option C
identify all pairs of alternate interior angles. Are the pairs of alternate interior angles equal in measure?
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have two parallel lines L1 and L2 crosses by a secant line m.
It means that the measure of alternate interior angles is equal.
The alternate interior angles are:
∠3 and ∠6
∠4 and ∠5
A food safety guideline is that the mercury in fish should be below one part per million (ppm). listed below are the amounts of mercury found in tuna sushi sampled at different stores in a major city. construct a 98% confidence interval estimate of the mean amount of mercury in the population. does it appear that there’s too much mercury in tuna sushi?0.58 0.82 0.10 0.88 1.32 0.50 0.92
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The amounts of mercury found in tuna sushi sampled at different stores are:
0.58, 0.82, 0.10, 0.88, 1.32, 0.50, 0.92
Number of samples, N = 7
[tex]\begin{gathered} \text{The mean, }\mu\text{ = }\frac{0.58+0.82+0.10+0.88+1.32+0.50+0.92}{7} \\ \mu\text{ = }\frac{5.12}{7} \\ \mu\text{ =}0.73 \end{gathered}[/tex]Standard deviation
[tex]\begin{gathered} \sigma\text{ = }\sqrt[]{\frac{\sum ^{}_{}{(x_1-\mu)^2}}{N}} \\ \sigma\text{ = }\sqrt[]{\frac{(0.58-0.73)^2+(0.82-0.73)^2+(0.10-0.73)^2+(0.88-0.73)^2+(1.32-0.73)^2+(0.50-0.73)^2+(0.92-0.73)^2}{7}} \\ \sigma\text{ =}\sqrt[]{\frac{0.9087}{7}} \\ \sigma\text{ =}\sqrt[]{0.1298} \\ \sigma\text{ = }0.36 \end{gathered}[/tex]The confidence interval is given by the equation:
[tex]\begin{gathered} CI\text{ = }\mu\pm z\frac{\sigma}{\sqrt[]{N}} \\ CI=0.73\pm2.33(\frac{0.36}{\sqrt[]{7}}) \\ CI\text{ = }0.73\pm0.32 \\ CI\text{ = (0.73-0.317})\text{ to (0.73+0.317)} \\ CI\text{ = }0.413\text{ < }\mu<1.047 \end{gathered}[/tex]Which equation could be represented by the number line? O A. -5 + 7 = 2 O.B. -3+(-4)= -7 O C. 4+ (-7)=-3 O D.7+(-6) = 1 SURAT E PREVIOUS
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C. 4 + (-7) = -3
C. 4 + (-7) = -3 could represent by the number line because no other equation has reflected the lines' displacement of 7 grid from a certain point going to the left which is equivalent to -7.
The rest of the choices is not a possible equation to the line.
A. -5 + 7 = Displacement of 7 Grid to the Right from Point -5
B. -3 - 4 = Displacement of 4 Grid to the Left from Point -3
D. 7 - 6 = Displacement of 6 Grid to the Left from Point 7
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Deena works at a customer service call center. She fields an average of 7 calls per hour. Employees are encouraged to field more than 280 calls per week. Deena has already fielded 112 calls this week.
How many more hours, x, does Deena need to work this week to reach the weekly goal of fielded calls if she continutes to field an average of 7 calls per hour? Select the inequality that includes the fewest number of hours Deena can work this week and still reach the weekly goal.
A.
x > 24
B.
x > 40
C.
x > 3
D.
x > 31
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sketch the graph of and identify the axis of symmetry
Answers
Given the following equation:
[tex]y=(x-1)^2+2[/tex]We will sketch the graph and identify the axis of symmetry.
the given function is a quadratic function with a vertex at (1, 2)
the graph of the function will be as follows:
As shown, the graph of the function has an axis of symmetry at x = 1
So, the answer will be option 3) x = 1
Thor is at the lowest point of Asgard whichis 280 feet below sea level (-280 ft.). He flies tothe highest point 520 feet above sea level(520 ft.). How far did he fly? Answer in acomplete sentence.
Answers
We have that:
We want to find how far did he fly. This means, we want to find the distance between -280 ft and 520 ft. (Which is 520 ft + 280 ft = 800 ft)
In order to find it out we substract them (we substract the first measure from the second):
520 - (-280)
Since - (-280) = +280, then
520 - (-280) = 520 + 280
= 800
Answer: he flied 800 ft