The sum of the first 11 terms of the geometric sequence aₙ = 2aⁿ - 1 with an initial value of a₁ = 1/8 is -10.71.
Therefore the answer is -10.71.
The geometric sequence aₙ = 2aⁿ - 1 with an initial value of a₁ = 1/8 represents the amount of money saved after n years of investing.
So sum of n terms
Sn = 2a - 1 + 2a² - 1 + ... + 2aⁿ - 1
Sn = 2( a + a² + ... + aⁿ) - n
The sum of the first n terms of a geometric sequence aₙ = arⁿ⁻¹ with first term a₁ and common ratio r is given by the formula:
Sn = a₁(1 - rⁿ)/ (1 - r), that is
[tex]S_{n} = a_{1}\frac{ (1 - r^n)}{1 - r}[/tex]
So a + a² + ... + aⁿ = a(1 - aⁿ)/ (1 - a)
Thus Sn = 2( a + a² + ... + aⁿ) - n
Sn = 2a(1 - aⁿ)/ (1 - a) - n
In this case, a₁ = a = 1/8 and n = 11
So the sum of the first 11 terms of the geometric sequence is:
Sn = 2×1/8×( 1 - (1/8)¹¹)/ (1 - 1/8) - 11
Sn = -10.71
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The figure shown contains 3 squares and 1 right triangle. What is the length of the hypotenuse of the triangle? Explain how you found your answer
the length of the hypotenuse is ???? cm
In order to find the hypotenuse of the triangle
We need to find the other sides of a right triangle
Length of the 1st side of a right triangle is connected with a square with area 144[tex]cm^{2}[/tex]
Hence in order to find the side we
[tex]\sqrt{144} = side[/tex]
[tex]12 = side[/tex]
Length of the second side is connected to a small square with area 25[tex]cm^{2}[/tex]
[tex]\sqrt{25} = side[/tex]
[tex]5 = side[/tex]
Now we can find the hypotenuse of a right triangle with the phytogorean theorem
[tex]a^{2}+ b^{2} = c^{2}[/tex]
[tex]12^{2}+ 5^{2} = c^{2} \\169 = c^{2}[/tex]
[tex]\sqrt{169}= c[/tex]
[tex]13 = c[/tex]
Hence , the hypotenuse of a right triangle is 13cm
What is the domain of the following function?
Therefore , the solution of the given problem of equation comes out to be each diagonal measures 8.4 units in length..
Define equation.An equation is a mathematical representation of two equal variables, one on each side of a "equals" sign. Everyday issues can be resolved with equations. We commonly look for pre algebra help to deal with problems in real life. Pre-algebra contains the fundamental concepts of mathematics.
Here,
Given: 16x + 2 = 36x - 6
8 = 20x
x = 8/20,
simplifying to 2/5, or 0.4
You would set the two equations equal to one another and find x because a rectangle's diagonals are congruent.
Re-enter the value of x into a single of the equations.
=>16(0.4) + 2
=>6.4 + 2
=>8.4
To make sure, confirm with the other diagonal.
=>36(0.4) - 6
=>14.4 - 6
=>8.4
Each diagonal measures 8.4 units in length..
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The complete question is "To find the diagonal x value by the given equation 16x + 2 = 36x - 6 "
raph of the function f(x) = (x+2)(x+6) is shown
-4 2
4
2+
N
-2
-6
2
4 6
X
What is true about the domain and range of the function?
O The domain is all real numbers, and the range is all
real numbers greater than or equal to -4.
O The domain is all real numbers greater than or equal
to
-4, and the range is all real numbers.
O The domain is all real numbers such that -6 ≤x≤-2,
and the range is all real numbers greater than or equal
to-4.
O The domain is all real numbers greater than or equal
to
-4, and the range is all real numbers such that -6 sx
≤-2.
The domain of a function is the set of all input values (x-values) for which the function produces a valid output (y-value). In this case, the function f(x) = (x+2)(x+6) is defined for all real numbers, so the domain is all real numbers.
What do a function example's range and domain mean?
A straightforward function, such as f(x) = x2, can have a domain (what goes in) of just the counting numbers 1, 2, 3, etc., and a range (what comes out) of the set 1, 4, 9, etc. Another function, g(x) = x2, may have an integer domain of..., -3, -2, -1, 0... 1, 2, 3, and its range is..., 1, 4, 9,...
The range of a function is the set of all possible output values (y-values) produced by the function. The function f(x) = (x+2)(x+6) is a polynomial, and all polynomials of the form (ax^2 + bx + c) with real coefficients (a,b,c) have range is all real numbers. The range of the function f(x) = (x+2)(x+6) is also all real numbers.
So the answer is
The domain is all real numbers, and the range is all
real numbers.
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Nachelle just accepted a job at a new company where she will make an annual salary of $57000. Nachelle was told that for each year she stays with the company, she will be given a salary raise of $4000. How much would Nachelle make as a salary after 4 years working for the company? What would be her salary after tt years?
a) Using the slope-intercept equation, after 4 years of working at the new company, Nachelle's salary will be $73,000 per annum.
b) After t years of working at the new company, Nachelle's salary will be $57,000 + $4,000(t).
What is the slope-intercept equation?The slope-intercept equation is in the form of y = mx + b.
The slope intercept describes the equation of a straight line, where m is the slope of the line, y and x are the y and x coordinates, and b is its y-intercept.
Initial starting salary per year = $57,000
Annual raise in salary = $4,000
Period = 4 years
Salary after 4 years = $73,000 [$57,000 + $4,000(4)]
Salary after t years = $57,000 + $4,000(t)
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In which of the following intervals does the trigonometric inequality sec(x)
The Trigonometric inequality Sec (x)< cot(x) will hold true in π/2 < x < π.
How can a trigonometric inequality be expressed?
The intervals where the trigonometric inequality sec (x) cot (x) always holds true are what we are looking for.
This can also be expressed as: 1/cos (x) < 1/tan (x)
Now, this is only possible in the fourth quadrant, where cos x is positive and tan (x) is negative, where: π/2 < x < π.
The values for sin, cos, and tan are all positive in the first quadrant.
Only positive values for sin are present in the second quadrant.
The tan values in the third quadrant are all positive.
Only positive values for cos are present in the fourth quadrant.
Therefore the inequality Sec(x) < cot(x) will hold true in π/2 < x < π.
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How do you find the area of the pre image and the image?
Coordinates for the pre-image are X(3,18) Y(18,18) and Z(18,9) and the image coordinates are X’(1,6) Y’(6,6) Z’ (6,3)
and what is the relationship between the two areas?
Answer: To find the area of the pre-image and the image, you can use the coordinates of the vertices of the shapes to calculate the lengths of the sides and use the formula for the area of a triangle:
Area = (base * height)/2
For the pre-image, the coordinates are X(3,18) Y(18,18) and Z(18,9). The base of the triangle is the distance between X and Z, which is 18 - 3 = 15. The height of the triangle is the distance between Y and Z, which is 18 - 9 = 9. The area of the pre-image is therefore:
Area = (15 * 9)/2 = 67.5
For the image, the coordinates are X’(1,6) Y’(6,6) Z’ (6,3). The base of the triangle is the distance between X’ and Z’, which is 6 - 1 = 5. The height of the triangle is the distance between Y’ and Z’, which is 6 - 3 = 3. The area of the image is therefore:
Area = (5 * 3)/2 = 7.5
The relationship between the two areas is that the area of the image is smaller than the area of the pre-image. This is because the image has been scaled down in size relative to the pre-image. The amount by which the image has been scaled down can be calculated by dividing the area of the image by the area of the pre-image:
Scale factor = area of image / area of pre-image = 7.5 / 67.5 = 0.11
This means that the image has been scaled down by a factor of 0.11 relative to the pre-image.
Step-by-step explanation:
Answer:
preimage area: 67.5 square unitsimage area: 7.5 square unitsimage area is 1/9 of preimage area, the square of the scale factorStep-by-step explanation:
You want to know the areas and their relationship of the preimage with coordinates X(3,18), Y(18,18), and Z(18,9), and the image with coordinates X’(1,6), Y’(6,6), Z’ (6,3).
TrianglesWhen you plot the points on a graph, you see they define right triangles. Each has sides aligned with the x- and y-axes, so it is easy to find their dimensions by counting grid squares or subtracting coordinates.
The preimage triangle is 18-3 = 15 units in the x-direction, and 18-9 = 9 units in the y-direction.
The image triangle is 6-1 = 5 units in the x-direction, and 6-3 = 3 units in the y-direction.
We notice that both the coordinates and the dimensions of the image triangle are 1/3 those of the preimage triangle.
AreasThe area of each triangle is given by the formula ...
A = 1/2bh
Using the dimensions we found above, the areas are ...
preimage area: 1/2(15)(9) = 135/2 = 67.5 . . . . square units
image area: 1/2(5)(3) = 15/2 = 7.5 . . . . square units
Relationship
We have observed that the image dimensions are 1/3 those of the preimage. The image area is 7.5/67.5 = 1/9 of that of the preimage. This value is the square of the scale factor of the dimensions,
need some help with the screenshot
The graph of the function f(x) = 2x² + 2x - 2 is plotted and attached
How to plot the graph of the equationThe graphs of the function is plotted by making a table of value for a number of the x values and the corresponding y values as done below
Using the function the table is created as f(x) = 2x² + 2x - 2
For x = -4, y = 2 * (-4)² + 2 * (-4) - 2 = 22
For x = -2, y = 2 * (--2)² + 2 * (-2) - 2 = 4
For x = 0, y = 2 * (0)² + 2 * (0) - 2 = -2
For x = 2, y = 2 * (2)² + 2 * (2) - 2 = 10
For x = 4, y = 2 * (4)² + 2 * (4) - 2 = 38
The table of values
x y
-4 22
-2 4
0 -2
2 10
4 38
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m(x)=−2(x−3)(x−9) solve for y-intercept
Answer:
(0, -54)
Step-by-step explanation:
We find the y-intercept by plugging in 0 for x:
[tex]y=-2(x-3)(x-9)\\y=-2(0-3)(0-9)\\y=-2*-3*-9\\y=6*-9\\y=-54[/tex]
Help needed please and thank you
Answer:
see image
Step-by-step explanation:
Right at the point (1,2) there's a nice point. Going up and right along the line there's another nice point at (5,5)
To get from one point to the next, you can draw a stair-step line that goes up3 and over4. The slope is up3/over4.
m = 3/4
The slope is 3/4. See image.
Solve the following system of inequalities graphically on the set of axes below. State the coordinates of a point in the solution set. y < c - 1 y > x -4
The graph of the inequality system is attached. The coordinate of a point in the solution set is (-1, 2).
In order to graph the system of inequalities graphically, consider the equation of each inequality.
y = x – 1
y = x - 4
Choosing a few values of x, plot the graph for each equation. Shade the area bounded by the line based on the inequality. If the inequality is less than, shade the area right to the line and if the inequality is greater than, shade the area left to the line. Hence, y < x – 1 is represented by red area and y > x – 4 is represented by blue area. The overlapping shaded area contains the coordinates that represent the solution of the inequality system. The coordinate of a point in the solution set is (-1, 2).
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Please help- thank you!!
Answer: u = -3
Step-by-step explanation:
To solve the equation, we will isolate the u variable by simplifying and using inverse operations.
Given:
4(3u + 5) = 2(2u - 2)
Distribute the 4 and the 2:
12y + 20 = 4u - 4
Subtract 20 from both sides of the equation:
12y = 4u - 24
Subtract 4u from both sides of the equation:
8u = -24
Divide both sides of the equation by 8:
u = -3
The admission fee to the carnival is $4. 50 for children and $7. 25 for adults. If a large group purchased 28 tickets and paid $159 how many adult tickets did they purchase?
The number of adult tickets purchased is 12.
What is the arithmetic operation?
The four fundamental operations of arithmetic are addition, subtraction, multiplication, and division of two or more quantities. Included in them is the study of numbers, especially the order of operations, which is important for all other areas of mathematics, including algebra, data management, and geometry. The rules of arithmetic operations are required in order to answer the problem.
Let no. of adult = x
No. of children = 28 -x
So, 7. 25 * x + (28 - x) *4. 50 = 159
⇒2.75x +126=159
⇒2.75x =33
⇒x=33/2.75
x=12
Hence, the number of adult tickets purchased is 12.
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50PTS 50PTS don’t explain just answer.
Answer:
[tex] \sf \: x = 7[/tex]
Step-by-step explanation:
Now we have to,
→ find the required value of x.
Forming the equation,
→ 2x + 5 = 3x - 2
Then the value of x will be,
→ 2x + 5 = 3x - 2
→ 2x - 3x = -2 - 5
→ -x = -7
→ [ x = 7 ]
Hence, the value of x is 7.
What is line segment in maths class 7?
A line which is confined between two endpoints is known as a line segment. In other words, we can say that a line segment is a portion of a line that is connected by two points.
The length of the line segment does not vary and the distance between them does not change. The units in which a line can be calculated in centimetres, millimetres, meters, inches, or feet.
The difference between a line segment and a line is that the line segment had two endpoints while the line extends indefinitely. When only one side of the line has an endpoint, it is called a ray or a half-open line segment.
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Convert one three fifths to a percent find that percent of 30 Divide your answer by 25% of 16
Answer:
40%
Step-by-step explanation:
1 and 3/5 as a fraction is 8/5 which as a percent is 160%.
25% of 16 is 4
160%/4 = 40%
What is an equivalent expression example?
A given set of expressions are known as equivalent expression if and only if the result after evaluating those expression is the same number.
Example,
3 + 2 = 5
4 + 1 = 5
5 + 0 = 5
These three are equivalent as because the sum of all of them is same , i.e., 5.
An equivalent expression is an expression which consists of variables, coefficients, constants, and mathematical functions such as addition, subtraction, multiplication, and division.
In general, if two objects are identical, they are then called as identical.
These expressions are also known as duplicates of each other though they look different but they are same.
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Mr Davis is creating a spice mixture for a recipe.2/5 of the spice mixture was oregano. 1/5 of the mixture was basil. The rest of the mixture was chilli powder
Answer:2/5
Step-by-step explanation:
2/5+1/5 equals 3/5, so if the rest of the mixture is chili powder, it would be 2/5
At a banquet hall there are 25 tables that seat 4 people and 30 tables that seat 6 people. There are no other tables in the banquet hall. What is the ratio of the number of tables that seat 4 people to the total number of tables in the banquet hall?
Answer:
1:11
Step-by-step explanation:
We know
There are 25 tables that seat 4 people
First, we find the total number of tables by taking
25 + 30 = 55 tables
Then we find the ratio of tables that seat 4 people to the total number of tables by
5 / 55 = 1/11 = 1:11
So, the ratio is 1:11
Write and solve an inequality for the problem.
It cost $660 to put on the school play. How many tickets must be sold at $6 a piece in order to make a profit?
Answer:
more than 110
Step-by-step explanation:
You want to know the number of $6 tickets that must be sold to pay for the $660 cost of a school play and make a profit.
ProfitThe profit is the difference between revenue and cost. The revenue from the sale of n tickets will be 6n. We want the profit to be positive, so we want ...
6n -660 > 0 . . . . . . positive profit
n -110 > 0 . . . . . . divide by 6
n > 110 . . . . . . add 110
More than 110 tickets must be sold to make a profit.
__
Additional comment
Sale of exactly 110 tickets will cover the cost, but there will be no profit.
Mariana spent $166.50 on games played at a laser tag arena
last year. If the laser tag arena charges $4.50 per game, how
many games did Mariana play at the arena last year?
Answer: 37
Step-by-step explanation:
divide the total that she spent over how much each game is (166.50/4.50). this will give you the number of games that she played.
If x = 6 and y = − 2 , evaluate the following expression: 2 x y
Answer:
-24
Step-by-step explanation:
2(6)(-2)
What is a logarithmic curve?
A logarithmic curve is a mathematical function.
A logarithmic curve, also known as a logarithmic function, is a type of mathematical function that describes a relationship between a variable x and a variable y in which the logarithm of y is a linear function of x.
The general form of a logarithmic function is y = logb(x), where b is the base of the logarithm, and x and y are real numbers. The base of the logarithm can be any positive number, but the most common bases used are 10 (common logarithm) and e (natural logarithm).
The graph of a logarithmic function is a curve that starts at negative infinity and increases gradually as x increases. The slope of the curve becomes closer to zero as x increases, and it is undefined for x=0 or x<0.
Logarithmic functions are often used in many areas of science, engineering, and mathematics to model real-world phenomena that exhibit exponential growth or decay, such as population growth, radioactive decay, sound levels and more.
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help help help help use slope form to write the equation of a line that passes through the point (7,-16) with slope 3.
required equation is, y=3x-37.
What is slope?In mathematics, the slope or gradient of a line is a number that describes both the direction and the steepness of the line.
Here, given that,
we have to find,
the equation of a line that passes through the point (7,-16) with slope 3.
we know,
y = mx+c
i.e. -16 = 3*7+c
so, c= -37
hence, required equation is, y=3x-37.
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Ramiro aked 60 random people from hi chool how they ue the library. Ue the information from Ramiro’ repreentative ample to make prediction about the chool-wide reult. The chool ha 540 tudent
Yes it is possible to make the prediction about Ramiro's School but not completely.
It is possible to use the information from Ramiro's sample to make predictions about the school-wide results, but it is important to note that the sample may not be fully representative of the entire population of students at the school. There may be sampling biases or other factors that affect the results.
With that said, based on the information provided, we can calculate the proportion of students in Ramiro's sample who use the library in a certain way, and then use that proportion to estimate the number of students at the school who use the library in that way.
For example, if Ramiro's sample shows that 40 out of the 60 people he surveyed use the library to study, we can estimate that approximately 40/60 = 67% of the school's students use the library to study. We can then use this proportion to estimate that approximately 540 x 67% = 361 students at the school use the library to study.
It's important to note that these predictions are based on sample data and they may not be accurate. To make more accurate predictions, a larger, more representative sample would be needed.
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Pretend you are constructing a histogram for describing the distribution of salaries for individuals who are 40 years or older, but are not yet retired.
Salary age
$57,000. 00 27
$40,200. 00 21
$21,450. 00 44
$21,900. 00 27
$45,000. 00 40
Required:
a. What is on the Y-axis? Explain.
b. What is on the X-axis? Explain.
c. What would be the probable shape of the salary distribution? Explain why?
a) The data on the y-axis: the numbers of people in each salary category.
b) The data on the x-axis: salaries
c) The distribution of salaries is skewed to the right.
We know that a histogram is a graph in which the x-axis is a quantitative variable split into bins, and the y-axis is the frequency.
For given distribution of salaries for individuals, the Y-axis would have the frequency i.e., the numbers of people in each salary category.
And x-axis would have income, because this is out quantitative variable of interest.
consider the following histogram.
We can observe that the graph of distribution of salaries is skewed to the right, because most income data are positively skewed. There would be a substantial number of individuals at or near the lowest salary level, but some people make a tremendously high salary.
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An item is regularly priced at $90. It is now priced at a discount of 40% off the regular price. What is the price now?
Answer:
Price after discount is $54
You save $36
Step-by-step explanation:
The mean of 4 5 6 7 8
Answer:
6Step-by-step explanation:
The mean of 4 5 6 7 8
there are 5 numbers in sequence, you can solve at a glance by looking at the number in the middle, or add the numbers and divide them by their number, in any case the answer will be 6
4 5 6 7 8 (6 is central)
or
(4 + 5 + 6 + 7 + 8) : 5 =
30 : 5 =
6
How to find the legs of a 45 45 90 triangle given the hypotenuse?
If the hypotenuse is given to the 45 45 90 triangles, the leg of the triangle is calculated after dividing the hypotenuse by [tex]\sqrt{2}[/tex].
The 45, 45, and 90 triangle is a type of isosceles triangle where the two legs are the same as each other and other non-right angles are both 45 degrees. Most of the time, Pythagoras' theorem is used for finding out the missing legs and hypotenuses of 45, 45, and 90 triangles.
If the leg of the triangle is equal to a, then:
The second leg is also equal to a;The hypotenuse is a√2;The area is equal to a²/2; andThe perimeter equals a(2 + √2).Read more about the isosceles triangle:
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Darnell and Hector ride their bikes at constant speeds. Darnell leaves Hector's house to bike home. He can bike the 8
miles in 32 minutes. Five minutes after Darnell leaves, Hector realizes that Darnell left his phone. Hector rides to catch
up. He can ride to Darnell's house in 24 minutes. Assuming they bike the same path, will Hector catch up to Darnell
before he gets home?
a. Write the linear equation that represents Darnell's constant speed.
i need this ASAP
Darnell's constant speed, as determined by the solution, is 20 miles per hour.
What is the process of linear algebra?It covers linear transformations more broadly as well as vectors, matrices, and vector spaces. One area of mathematics is linear algebra. In comparison to other areas of mathematics that are frequently driven by innovative ideas and unresolved problems, linear algebra is a discipline that is very well understood.
According to the given information:Distance=8 miles
Time taken is=32 minutes
speed=distance/time
speed=15 miles/hour
Darnel has travelled a distance of 1.5x5/60=0.125 miles in under five minutes.
Hector bicycle speed = distance/time
hector speed of bike= 20 miles/hour
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What is the domain of √?
The domain of √ (square root) is all non-negative real numbers.
The domain of a function represents the set of all input values that can be used in the function without resulting in an undefined or meaningless output.
The square root function, represented by √, is only defined for non-negative real numbers. This is because the square root of any negative number is not a real number, it becomes an imaginary number.
This is because in order for the square root to be defined, the radicand (the value inside the square root symbol) must be greater than or equal to zero. Therefore, any real number greater than or equal to zero can be used as the input for the square root function, making its domain all non-negative real numbers.
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