The final closed formula answers for each part,
(a) an = n^2 + 1
(b) an = n(n + 1)(n + 2)/6
(c) an = 2n + 6
(d) an = n! + (n-1)! + ... + 2! + 1!
(a) The given sequence can be seen as the sequence of partial sums of the sequence of odd numbers: 1, 3, 5, 7, 9, . . . . That is, the nth term of the given sequence is the sum of the first n odd numbers, which is n^2. Therefore, the closed formula for the given sequence is an = n^2 + 1.
(b) The given sequence can be seen as the sequence of partial sums of the sequence of triangular numbers: 1, 3, 6, 10, 15, . . . . That is, the nth term of the given sequence is the sum of the first n triangular numbers, which is n(n + 1)(n + 2)/6. Therefore, the closed formula for the given sequence is an = n(n + 1)(n + 2)/6.
(c) The given sequence can be seen as the sequence of differences between consecutive squares: 1, 5, 9, 16, 21, . . . . That is, the nth term of the given sequence is the difference between the (n+1)th square and the nth square, which is (n + 1)^2 - n^2 = 2n + 1. Therefore, the closed formula for the given sequence is an = 2n + 6.
(d) The given sequence can be seen as the sequence of partial sums of the sequence defined recursively by a1 = 1 and an+1 = an(n + 1) for n ≥ 1. That is, the nth term of the given sequence is the sum of the first n terms of the recursive sequence. It can be shown that the nth term of the recursive sequence is n! (n factorial), and therefore the nth term of the given sequence is the sum of the first n factorials. That is, an = 1 + 1! + 2! + ... + (n-1)! + n!. Therefore, the closed formula for the given sequence is an = n! + (n-1)! + ... + 2! + 1!.
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3.1 Mrs Gilfillan owns a coffee shop. She serves a mixed berry and almond polenta cake that is baked in espresso cups at her coffee shop. She uses the recipe below to make the cake. Mixed Berry and Almond Polenta Calce Makes 15 espresso cups Ingredients 140 g butter 140 g castor sugar 140 g ground almonds 250 g fat-free cottage cheese 75 g mixed frozen berries 25 g polenta 6 eggs separated (keep the yolks for mayonnaise or scrambled egg) Bake at 356 °F until light brown, 30 to 40 minutes. Fat-free cottage cheese is sold in quantities of 125g at R8,99. Calculate the cost of the fat-free cottage cheese required in the recipe.
The cost of the fat-free cottage cheese required in the recipe is R17.98.
How to determine the cost ?
To determine the cost, we first need to calculate the amount of fat-free cottage cheese required in the recipe. The recipe calls for 250g of fat-free cottage cheese, which can be obtained by using 2 units of 125g each. Knowing the cost of ingredients is important for Mrs Gilfillan to price the cake appropriately to cover her costs and make a profit.
What is the cost of the fat-free cottage cheese required to make the Mixed Berry and Almond Polenta Cake recipe that serves 15 espresso cups at Mrs Gilfillan's coffee shop?
The cost of 1 unit of 125g fat-free cottage cheese is R8.99.
Therefore, the cost of 2 units of 125g is calculated as:
R (2 x 8.99) =R 17.98
Hence, the cost of the fat-free cottage cheese required in the recipe is R 17.98.
This cost is in addition to the cost of the other ingredients, such as butter, sugar, ground almonds, mixed frozen berries, and polenta, as well as the cost of labor, overhead, and other expenses involved in making and selling the cake.
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When conducting a survey, which of the following is the most important reason to use a random sample? Correct. Random selection ensures that the sample is unbiased on average, so that the results of the study can be generalized to the population.
Random sampling is crucial when surveying as it ensures that the sample selected is representative of the population.
By randomly selecting participants from the population, the sample is likely to be unbiased on average, which means that the results of the study can be generalized to the entire population. Without random sampling, the results of the study may be skewed or biased towards a certain group, which can lead to incorrect conclusions and poor decision-making. Therefore, it is essential to use random sampling when surveying to obtain accurate and reliable results.
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Justin purchased his dream car worth 18500 on a finance for 4 years. He was offered 6% interest rate.assuming no other chargers and no tax,we want to find his monthly installment.
Answer:
To calculate the monthly installment for Justin's car loan, we can use the formula for the present value of an annuity:
PV = (PMT / r) x [1 - 1 / (1 + r)^n]
where PV is the present value of the loan, PMT is the monthly payment, r is the interest rate per period (in this case, per month), and n is the number of periods (in this case, the number of months in 4 years, which is 48).
We know that PV is the amount of the car loan, which is N$18500. We also know that r is 6% per year, which is equivalent to 0.5% per month (since there are 12 months in a year). Finally, we know that n is 48.
Substituting these values into the formula, we get:
18500 = (PMT / 0.005) x [1 - 1 / (1 + 0.005)^48]
Simplifying this expression:
18500 = (PMT / 0.005) x 36.4358
PMT = 18500 / 36.4358 / 0.005
PMT ≈ N$402.87
Therefore, Justin's monthly installment for his car loan is approximately N$402.87.
divide 1350 into 5:4 and then do the same for 261
Answer:
a) 750 and 600,
b) 145 and 116.
Step-by-step explanation:
A) For 1350 divided to two as 5 : 4, we need to add 5 and 4 first = 9:
Then, with the following, calculate the amount for 5:
[tex]\frac{1350}{9}[/tex] x 5
150 x 5
= 750 for 5, then for 4:
150 x 4
= 600 for 4.
B) For 261 divided with into 5 : 4, we need to add 5 and 4 = 9:
Then, with the following, calculate the amount for 5:
[tex]\frac{261}{9}[/tex] x 5
29 x 5
= 145 for 5. Finally, for 4:
29 x 4
= 116 for 4.
What’s 9 X 3/9 answer…………………………………….
Answer:
9 x 3/9
= 39 ⋅ 3= 2727/9
27 ÷ 9
= 3Step-by-step explanation:
You're welcome.
What percentage of 23 miles is 27 miles
taking 23(origin amount) as the 100%, what is 27 off of it in percentage?
[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 23 & 100\\ 27& x \end{array} \implies \cfrac{23}{27}~~=~~\cfrac{100}{x} \\\\\\ 23x=2700\implies x=\cfrac{2700}{23}\implies x\approx 117.39[/tex]
expand and simplify.
(m–3)(m+2)
Answer:
m^2-m-6
Step-by-step explanation:
(m-3)(m+2)
m^2 +2m-3m-6
m^2-m-6
Answer:
m² - m - 6
Step-by-step explanation:
(m - 3)(m + 2)
each term in the second factor is multiplied by each term in the first factor, that is
m(m + 2) - 3(m + 2) ← distribute parenthesis
= m² + 2m - 3m - 6 ← collect like terms
= m² - m - 6
Find the range and standard deviation of the set of data.
11 8 5 11 25
Answer: The range is 20, Standard Deviation, σ: 6.8702256149271
Step-by-step explanation:
I hope it helped!
<3
create a formula with structured references to calculate the percentage of the Sticker Price in column E. Columns C and D have the sticker price and sale price, respectively.
The formula to calculate the percentage of the Sticker Price in column E using structured references would be:
=IFERROR((1-[Sale Price]/[Sticker Price])*100,"% Discount")
Define the term percentage?A number or quantity can be expressed as a fraction of 100 using a percentage.It is often denoted by the symbol % (percent).
Assuming that the Sticker Price is located in column C and the Sale Price is located in column D, the formula to calculate the percentage of the Sticker Price in column E using structured references would be:
=IFERROR((1-[Sale Price]/[Sticker Price])*100,"% Discount")
In this formula, the percentage of the Sticker Price is calculated by subtracting the Sale Price from the Sticker Price, dividing the difference by the Sticker Price, and then multiply by 100 to convert the result into percentage. The IFERROR function is used to handle any errors that may occur if the Sale Price is missing or is equal to or greater than the Sticker Price. If an error occurs, the formula will display the text "% Discount".
Note that the structured references [Sale Price] and [Sticker Price] refer to the column headers rather than the cell references. This makes the formula more dynamic and easier to read, especially when working with large datasets.
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A car travelling north at 48 km/hr is approaching an intersection. A truck travelling East at 60 km/hr is moving away from the same intersection. How is the distance between the car and the truck changing when the car is 9 m from the intersection and the truck is 40 m from the intersection?
The rate at which the distance between the car and the truck is changing when the car is 9 m from the intersection and the truck is 40 m from the intersection is about 19.187 m/s
What is the rate of change of a function?The rate of change of a function is an indication of how fast the function's output changes per unit change in the input of the function.
Let y represent the distance of the car from the intersection, and let x represent the distance of the truck from the intersection, we get;
The distance between the car and the truck, d, can be obtained from Pythagorean Theorem as follows;
d² = y² + x²
2·d·d/dt = 2·y·dy/dt + 2·x·dx/dt
dy/dt = The speed of the car = 48 km/h = 40/3 m/s
dx/dt = The speed of the truck = 60 km/h = 50/3 m/s
y = The distance of the car from the intersection = 9 m
x = The distance of the truck from the intersection = 40 m
d² = 9² + 40² = 1681
d = √(1681) = 41
d = 41 m
Therefore;
2×41×d/dt = 2×9 × 40/3 + 2 × 40 × 50/3 = 4720/3
d/dt = 4720/3/(2 × 41) = 2360/123 ≈ 19.187
d/dt ≈ 19.187 m/s
The rate of change distance between the car and the truck d/dt is about 19.187 m/s
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Volume: Count the Cubes
**HELP!**
If you help me it’ll be worth 30 points!
Answer:
1.5 2.4 3.8
Step-by-step explanation:
1. 5
2. 4
3. 8
4. 6
5. 8
6. 15
7. 8
8. 12
9. 6
ab and bc are perpendicular lines find the value of x of 25
Answer:
If the time is 3:45 how many minutes is it slow or fast
How many proper subsets are in {2,4,6,8...100}
Answer:
159 proper subsets.
Step-by-step explanation:
Given a set {2, 4, 6, 8...100}, how many proper subsets are there?
First, find how many subsets there are in 2 - 10:
That's 16.
Then because there are 10 10s in 100, multiply by 10:
16 x 10 = 160
Finally, because it says proper subsets, subtract by 1:
160 - 1 = 159 proper subsets.
Therefore, there are 159 proper subsets in {2, 4, 6, 8...100}
do the compression values seem resonable given the 3 2 1 mix design, 0.50 w/c ratio and 7 days of curing
The mix design and w/c ratio used in the concrete mix are commonly used for M15 grade concrete, and achieving a compressive strength of 18 MPa after 7 days of curing may be reasonable depending on the target strength and other design specifications.
The actual compressive strength of 18 MPa achieved after 7 days of curing may be lower or higher than the target strength, depending on the design specifications and requirements.
However, the mix design of 3:2:1 (cement:sand:aggregate) with a water-cement (w/c) ratio of 0.50 is a commonly used mix proportion for M15 grade concrete. With proper curing and compaction, this mix design should be capable of achieving a compressive strength of at least 15 MPa at 28 days of curing.
The compressive strength is typically measured after 28 days of curing, not just 7 days. Therefore, it may be premature to conclude whether the achieved compressive strength of 18 MPa is reasonable without testing the concrete at 28 days of curing.
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_____The given question is incomplete, the complete question is given below:
do the compression values seem resonable given the 3:2:1. mix design, 0.50 w/c ratio and 7 days of curing. M 15 grade concrete
Cement = 3 Part
Sand = 2 Part
Aggregate = 1 Part
Total dry volume of Material Required = 1.57 cu.m
actual compressive strength of the concrete = 18 MPa
Todd Jefferson can buy 2 gallons of ice cream for $3.87, 3 gallons for $4.10, or 1 gallon for $1.98. What is the unit price of the best buy, rounded to the nearest tenth of a cent?
help plssss explainnn!!
Answer:
[tex]xy^8[/tex]
Step-by-step explanation:
Notice if you have the same base you can ADD the exponent, for example:
[tex]x^{-6} x^{7} =x^{-6+7}=x^{1 }=x[/tex]
[tex]y^{6} y^{2} =y^{6+2}=y^{8 }\\[/tex]
so the answer is
[tex]xy^8[/tex]
5. What is the difference between (-3)² and 3²
Answer:
the difference is the first one, (-3)^2, is negative while the other is positive
Step-by-step explanation:
k is what or is it none for the equation?
The piecewise function will be continuous in its domain if k = 49/48
How to find the value of k such that the function is continuous?Here we have a piecewise function and we want to find the value of k suc that the function is continuous on the domain.
To get that, both piecese of the function must ahve the same value in the jump, then we will get:
f(7) = f(7)
Replacing the functions we will get:
k·7² = 7·7 + k
Let's solve that equation for k.
49k = 49 + k
49k - k = 49
48k = 49
k = 49/48
That must be the value of k such that the function is continuous in all its domain.
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which best describes the resulting three-dimensional figure?
The resulting three-dimensional figure is a cone with base radius of 17 inches.
What is a cone?A cone is a three-dimensional geometric shape that has a circular base and a curved surface that tapers to a point, called the apex. The base of the cone is a circle, and the height of the cone is the perpendicular distance from the apex to the center of the base. The surface of the cone is made up of all the points that are equidistant from the apex and lie on the base.
According to the given information :Consider an isosceles triangle with 82 inches of circumference and a leg length of 24 inches.
This means that 82 - 2(24) = 82 - 48 = 34 inches is the length of the other side.
The triangle is rotated along a line that passes through points B and D, with point D serving as the intersection of lines AC.
When this happens, a cone will be created in three dimensions with a base radius equal to the length of line AD (i.e. half of the length of line AC being the other side of the isosceles triangle).
The resulting cone's base radius is 1/2 (34) = 17 inches.
Hence, the resulting three-dimensional figure is a cone with base radius of 17 inches.
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Find the general anti-gradient, f, for the following: (i.e. f was the original function, and you are given the gradient of the function.) (a) Vf(x,y) = (6x + 4yº, –2 + 12xy?) (b) Vf(x,y) = ( 95 - 2e-20, - sako + 4y)
(a) The general anti-gradient for ∇f(x,y) = (6x + 4y^3, –2 + 12xy^2) is f(x,y) = 3x^2 + 4xy^3 – 2y + C
(b) The general anti-gradient for ∇f(x,y) = (6x/y^2 -2e^-2x, 6x^2/y^3+4y) is f(x,y) = 3x^2/y^2 + e^-2x + 2y^2 + C
(a) To find the general anti-gradient f for ∇f(x,y) = (6x + 4y^3, –2 + 12xy^2), we need to integrate each component of the gradient with respect to its corresponding variable.
Thus, we have
f(x,y) = ∫(6x + 4y^3) dx + C1
f(x,y) = 3x^2 + 4xy^3 + C1
f(x,y) = ∫(–2 + 12xy^2) dy + C2
f(x,y) = –2y + 4x y^3 + C2
where C1 and C2 are constants of integration
Therefore, the general anti-gradient for ∇f(x,y) = (6x + 4y^3, –2 + 12xy^2) is
f(x,y) = 3x^2 + 4xy^3 – 2y + C
where C is an arbitrary constant.
(b) To find the general anti-gradient f for ∇f(x,y) = (6x/y^2 -2e^-2x, 6x^2/y^3+4y), we need to integrate each component of the gradient with respect to its corresponding variable
Thus, we have
f(x,y) = ∫(6x/y^2 -2e^-2x) dx + C1
f(x,y) = 3x^2/y^2 + e^-2x + C1
f(x,y) = ∫(6x^2/y^3+4y) dy + C2
f(x,y) = 3x^2/y^2 + 2y^2 + C2
where C1 and C2 are constants of integration.
Therefore, the general anti-gradient for ∇f(x,y) = (6x/y^2 -2e^-2x, 6x^2/y^3+4y) is
f(x,y) = 3x^2/y^2 + e^-2x + 2y^2 + C
where C is an arbitrary constant.
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The given question is incomplete, the complete question is:
Find the general anti-gradient, f, for the following: (i.e. f was the original function, and you are given the gradient of the function.) (a) ∇f(x,y) = (6x + 4y^3, –2 + 12xy^2) (b) ∇f(x,y) = (6x/y^2 -2e^-2x, 6x^2/y^3+4y)
FAST
Write two expressions to represent the following situation. Then, find the answer.
When Nathan went to bed, the outside temperature was 28°F. When he woke up the next morning, the temperature had decreased to -13°F. By how many degrees did the temperature change during the overnight hours
The temperature changed by 41°F during the overnight hours, decreasing from 28°F to -13°F.
How is an expression determined?The difference in temperature from 28°F to -13°F can be shown in one of two ways:
Temperature change equals final temperature - The starting temperatureTemperature change: (-13°F) - (28°F)Temperature change: -41°FTemperature change = final temperature minus initial temperature; temperature change = -13°F minus 28°F; temperature change = -41°F; temperature change = 41°F.To calculate the change in temperature, the first exponent equation subtracts the initial temperature from the end temperature. The change in this instance is negative and points to a drop in temperature.
The absolute value function is employed in the second statement to guarantee a positive result regardless of whether the temperature rose or fell. The result is the same as the first expression in this instance, but with a positive sign.
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A rectangular picture is 7 inches by 5 inches and has a border of uniform width surrounding it. If the area of the picture and the border is 80 square inches find the width of the border
The width of the rectangle border is 3 in.
What is area of rectangle?
The area a rectangle occupies is the space it takes up inside the limitations of its four sides. The dimensions of a rectangle determine its area. In essence, the area of a rectangle is equal to the sum of its length and breadth.
Here We know that ,
Area of the rectangle = length* width square unit.
Here area of the picture and border is 80 [tex]in^2[/tex] Then,
=> (7+w)*(5+w)=80
=> [tex]35+5w+7w+w^2=80[/tex]
=> [tex]12w+w^2=45[/tex]
=> [tex]w^2+12w-45=0[/tex]
=> [tex]w^2+15w-3w-45=0[/tex]
=> w(w+15)-3(w+15)=0
=> (w-3)(w+15)=0
=> w=3 , w=-15( negative is neglected)
Hence the width of the rectangle border is 3 in.
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instead of using the values {1, 2, 3, 4, 5, 6} on dice, suppose a pair of dice have the following: {1, 2, 2, 3, 3, 4} on one die and {1, 3, 4, 5, 6, 8} on the other. find the probability of rolling a sum of 6 with these dice. be sure to reduce
Answer:
your answer to the lowest terms
The probability of rolling a sum of 6 with these dice is 1/12.
Suppose we want to choose 2 colors, without replacement, from the 5 colors red, blue, green, purple, and yellow. (If necessary, consult a list of formulas.) (a) How many ways can this be done, if the order the choices is relevant? (b) How many ways can this be done, if the order of the choices is not relevant?
(a) There are 20 ways to choose 2 colors without replacement when the order is relevant. (b) There are 10 ways to choose 2 colors without replacement when the order is not relevant.
(a) If the order of the choices is relevant, then we need to use the permutation formula. The number of ways to choose 2 colors from 5, without replacement and with order considered, is given by P(5,2) = 5!/(5-2)! = 5x4 = 20.
(b) If the order of the choices is not relevant, then we need to use the combination formula. The number of ways to choose 2 colors from 5, without replacement and with order not considered, is given by C(5,2) = 5!/(2!3!) = 10.
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In the figure above, what are the measurements of
Check the picture below.
gogoggooggoogogogogogogogogogogogg
What are the different types of early numeration system which have been developed to represent numbers throughout history of mathematics?
Explanation: The evolution of numbers developed differently with disparate versions, which include the Egyptian, Babylonians, Hindu-Arabic, Mayans, Romans, and the modern American number systems.
What is the meaning of "the homotopy classes of paths from x to x in a space X"?
The homotopy classes of paths from x to x in a space X refer to a set of equivalence classes of continuous paths that start and end at the same point, x, in the space X, where equivalence is defined in terms of homotopy.
What is the homotopy about?In other words, for any two paths, there exists a continuous transformation (called a homotopy) between them such that the endpoints remain fixed. Two paths are said to be homotopic if they can be continuously deformed into each other while keeping their endpoints fixed. The set of all paths that are homotopic to each other forms an equivalence class.
The homotopy classes of paths from x to x are important in algebraic topology, as they provide a way to study the topological structure of a space by analyzing the properties of the paths within it. They can also be used to define higher algebraic structures such as the fundamental group and higher homotopy groups.
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What can you say about the profit ratios? Comment on the results obtained according to the time series analysis and the cross-sectional analysis. How would you evaluate job performance?
Profit ratios are used to assess a company's profitability by evaluating its profit in relation to other financial indicators like sales, assets, or equity.
What is the profit ratios?The return on assets ratio is used to evaluate the connection between the assets your business uses and the profits it makes. You calculate it using information from the balance sheet and the income statement. (The ratio is multiplied by 100 to become a percentage.)
But, this does not necessarily imply that this is the profit margin you should aim for, a modest margin is 5%. A healthy margin is 10%, and a large margin is 20%.
Therefore, To evaluate job performance using these methods, you would need to define your research goals and variables, select your cross-section and time period, collect and analyse the relevant data, and draw conclusions based on your findings.
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Find the real solutions of the following equation by graphing.
x³ - 6x²+5x=0
The Solution(s) is/are ? .
The real solutions of the following equation are 5 and 1
[ Graph is attached below ]
Solving equation graphically:
To solve the given equation graphically we need to find the coordinate points that pass through the graph.
This can be done by taking 'x' values and solving for them 'y' values.
Now draw the graph using the above coordinates and find the solution as shown below.
Here we have
x³ - 6x²+ 5x = 0
Let y = x³ - 6x²+ 5x
To draw the graph find the coordinates of points as follows
At x = 0 => y = (0) + (0) + (0) = 0
At x = 1 => y = (1)³ - 6(1)² + 5(1) = 0
At x = -1 => y = (-1)³ - 6(-1)² + 5(-1) = - 12
At x = 2 => y = (2)³ - 6(2) + 5(2) = 6
From the above calculation,
The coordinates of the points to draw the graph are (0, 0), (1, 0), (-1, -12), and (2, 6)
Here the solutions of the graph are the x-coordinate of points where the graph cuts the x-axis
From the figure, the graph will cut the x-axis at 1 and 5
Therefore,
The real solutions of the following equation are 5 and 1
[ Graph is attached below ]
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For which value of FJ must triangle FGJ be similar to triangle FHG ?
For the triangles to be similar the value of FJ = 9.
What are similar triangles?Triangles that resemble one another but may not be precisely the same size are said to be comparable triangles. When two items have the same shape but different sizes, they can be considered to be comparable. This indicates that comparable forms superimpose one another when amplified or demagnified. The term "Similarity" refers to this characteristic of like forms.
Let us suppose the value of FJ = x.
Now, for the triangle that are similar the corresponding sides are in proportion.
That is,
FG/FH = FJ/FG
Here, FH = x + 27
Substituting the values we have:
18/x + 27 = x / 18
18(18) = x (x + 27)
x² + 27x - 324 = 0
Splitting the middle term we have:
(x + 36) (x - 9) = 0
x = -36 and x = 9
Since, length cannot be negative we have the value of x = 9.
Hence, for the triangles to be similar the value of FJ = 9.
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