It is proven that "the sum of the squares of four consecutive integers is an even integer" is the solution part.
To prove that the sum of the squares of four consecutive integers is an even integer, we need to show that it can be expressed as twice another integer.
Let's assume the four consecutive integers are n, n+1, n+2, and n+3.
The sum of their squares can be expressed as:
[tex]n^2 + (n+1)^2 + (n+2)^2 + (n+3)^2[/tex]
Expanding the squares, we have:
[tex]n^2 + (n^2 + 2n + 1) + (n^2 + 4n + 4) + (n^2 + 6n + 9)[/tex]
Combining like terms, we simplify this expression to:
[tex]4n^2 + 12n + 14[/tex]
Now, let's consider this expression [tex]4n^2 + 12n + 14[/tex].
We can factor out 2 from each term, resulting in:
[tex]2(2n^2 + 6n + 7)[/tex]
Since [tex]2n^2 + 6n + 7[/tex] is an integer, let's denote it as k, where k is an integer.
Therefore, the expression becomes:
2k.
We have successfully expressed the sum of the squares of four consecutive integers as twice another integer (2k), which proves that it is an even integer.
Hence, the sum of the squares of four consecutive integers is an even integer.
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I'm trying to solve a equation of 3x-2=1 In the book, it says that 3 -2=-1 turns into 3x=3. I don't get it?! Shouldn't it be 3x=-1?
Answer: The book is right, it would break down to 3x = 3.
3x - 2 = 1 so add the 2 to the other side.
3x = 3 and then you can divide both sides by 3 to get x = 1.
Sorry if I misread what you were asking
Answer: The answer should be 3x = 3.
Step-by-step explanation:
Here's why:
3x -2 = 1
First one must isolate the 3x in order to solve the problem. So, then one must add 2 to both sides, since the 2 is negative. (So if it is a -2, add it to the other side because remember, you are doing it on both sides and you have to get it to the other side).
So doing that, you get:
3x = +2 +1
3x = 3
If you want to simplify further:
3x = 3
x = 1
Three spherical balls with radius r are contained in a rectangular box. two of the balls are each touching 5 sides of the rectangular box and the middle ball. the middle ball also touches four sides of the rectangular box. What is the volume of the space between the balls and the rectangular box? This is a SAT question and is no calculator. Show all the work Answer is 4r^3(6-pi)
Answer:
The volume of the space between the balls and the rectangular box is [tex]4r^{3}(6 - \pi)[/tex]
Step-by-step explanation:
The attachment below shows the description of the rectangular bow and the three spherical balls.
From the description,
Two of the balls are each touching 5 sides of the rectangular box, say the 5 sides touched by one of the balls are sides 1,2,3,4, and 5; then the other ball will touch sides 2,3,4,5, and 6). The middle ball also touches four sides of the rectangular box, These four sides touched by the middle ball will be sides 2,3,4, and 5.This means the balls are tightly fitted into the rectangular box.
Each of the balls has a radius r
Hence, The volume of one of the balls is given by the volume of a sphere
[tex]Volume of a sphere = \frac{4}{3} \pi r^{3} \\[/tex]
The volume occupied by one of the balls is [tex]\frac{4}{3} \pi r^{3} \\[/tex]
∴ The volume occupied by the three spherical balls will be
3 × [tex]\frac{4}{3} \pi r^{3} \\[/tex]
= [tex]4\pi r^{3}[/tex]
The volume occupied by the three spherical balls [tex]4\pi r^{3}[/tex]
For the rectangular box,
The volume of a rectangular box = [tex]l w h[/tex]
Where [tex]l\\[/tex] is the length
[tex]w[/tex] is the width and
[tex]h[/tex] is the height
Since the balls are tightly packed,
The width of the rectangular box will be the diameter of the balls
diameter of the balls = 2r
∴ [tex]w[/tex] = 2r
The height of the rectangular box will also be the diameter of the balls
∴ [tex]h[/tex] = 2r
The length of the rectangular box will be 3 times the diameter of the balls
∴[tex]l[/tex] = 3 × 2r = 6r
Hence,
The volume of a rectangular box = 6r × 2r × 2r
= 24r³
The volume of the space between the balls and the rectangular box is given by
Volume of the space between the balls and the rectangular box =
volume of the rectangular box - volume occupied by the three spherical balls
Volume of the space between the balls and the rectangular box= 24r³ - 4πr³
= 24r³ - 4πr³
= 4r³(6 - π)
A family visits a car show to research information on vehicles they might consider purchasing. Brochures for each vehicle provide helpful information for the vehicle. Which of the following is an example of a discrete quantitative variable that might be included in the brochure? A. number of seats B. type of transmission C. presence of a sunroof D. fuel efficiency (in mpg)
Answer:
Option A:
Number of seats
Step-by-step explanation:
A discrete quantitative variable is a variable that can be enumerated. This means that they are in units in which numbers can be assigned to and can be counted.
The number of seats present in the car can be counted. This feature can also be evaluated based on its numeral value, rather than its quality. In a simple form, the buyers feel that the more the number of seats present in the car, the more people it can carry. Hence, the family would love to buy a car with a good number of seats in it.
The other features in the options are rather continuous, qualitative, or boolean. Some of them are continuous because they cannot be counted e.g fuel efficiency. The others such as the presence of a sunroof can be seen as a boolean variable. (it can either be true or false)
Type of the transmission is a qualitative variable
Answer: fuel efficiency
Step-by-step explanation:
took the exam review on edg
Given the following estimated regression, please give the correct magnitude of the coefficients, and provide a one sentence interpretation of the relationship between Y and X as talked about in class Wage = 15.00 + 0.28-YrsEdu. Where Wage is currently in dollars, and YrsEdu. are in years. A) Suppose we want Wages to be in thousands of dollars instead of dollars. B) Suppose we want Yrsedu. to be in terms of weeks instead of years. C) Suppose we want Wages to be in hundreds of dollars AND YrsEdu. in days (given that a year of education is only 180 days).
Answer:
Following are the answer to the given choices:
Step-by-step explanation:
In potion A:
The model, that will be formed can be defined as follows:
[tex]\widehat{W age} = \hat b_{0} + \hat b_{1} x Y_{rs} \epsilon done \\\\ \hat B_1 = \frac{\sum_{i=1}^{n} (W_{age}i - \widehat{W age} \times (Y_{rs} Edui - \widehat{Y_{rs}\epsilon})}{\int S_{yrs}\epsilon dre}\\[/tex]
[tex]\ the \ new \ W_{age} = 10^{-3} \times W_{age} \\\\\hat {B_{1 new}} = 10^{-3} \hat {B}\\\\ \hat{B_{0}} = \widehat{W age} -\hat {B_1} \bar{Y_{rs}\epsilon dus} \\\\\hat{B_{0 \ new}} = \widehat{W age} \times 10^{-3} - 10^{-3} \hat {B_1} Y_{rs}\epsilon dus \\[/tex]
[tex]= 10^{-3} \hat {B_0}\\[/tex]
[tex]\hat B_{0 \ new} = 15 \times 10^{-3} \\\hat B_{1 \ new} = 0.28 \times 10^{-3} \\[/tex]
In potion B:
[tex]\ The \ new \ Y_{rs}\epsilon dus = \frac{Y_{rs} \epsilon dus}{7} \\\hat b_{1 \ new} = 7 \hat b_1\\\hat b_{ 0 \ new } = \widehat{W_{age}} - 7 \hat b_1 \times \frac{\bar Y_{rs}\epsilon dus}{7} \\\hat b_{0 \ new} = \hat B_{0}\\\hat b_{0} = 15.00 \\\hat b_{1} = 0.04 \\[/tex]
In potion C:
[tex]\ Y_{rs}\epsilon dus \ new = \frac{Y_{rs} \epsilon dus}{180} \\\\\widehat{W_{age} new} = \widehat{W_{age}} \times 10^{-2} \\\\\hat b_{1 \ new} = \hat b_{1} \times 10^{-2} \times 180 = \hat b_1 \times 1.8 \\\\\ hat b_{ 0 \ new } = 10^{-2} \widehat{W_{age}} - \hat b_1 \times 1.8 \times \frac{\bar Y_{rs}\epsilon dus}{180} \\\\[/tex]
[tex]= 10^{-2} \ \hat B_{0}\\[/tex]
[tex]\hat b_{0 \ new} = 0.15 \\\hat b_{1 \ new} = 0.504 \\[/tex]
Please find the perimeter of both shapes if x = 4.5
Answer:
Rectangle: 27
Triangle: 39
Step-by-step explanation:
To find the perimeter of a rectangle use the expression P = 2(l + w)
Before substituting 4.5 for x simplify the equation
2((-3 + x) + 1/2(4x + 6)) = P
First simplify in the parenthesis
2(-3 + x + 2x + 3) = P
Now combine like terms, then multiply 2 and 3x
2(3x) = P
P = 6x
Lastly substitute 4.5 for x to find the perimeter
P = 6(4.5)
P = 27
Now do the same thing that we did with the rectangle but with a triangle
P = a + b + c
(3x - 4) + (3x - 4) + 2/3(6x + 3) = P
10x - 6 = P
10(4.5) - 6 = P
P = 39
What fraction of students are boys if 5 of the 8 students are boys
Answer:
Step-by-step explanation:
5/8
Answer: 5/8
Step-by-step explanation:
Well if 5 of the 8 students in total are boys then the fraction will be 5/8 because is the number of boys out of the total people.
Solve the equation below, and check the solution. 0.3x = 9 The solution set is { }
I know the answer is x = 30, but I do not know how to give my answer in the form of a solution set??
Answer:
{30}
Step-by-step explanation:
The solution set can be written a couple of ways:
Roster: lists the elements of the set:
x = {30}
Set builder notation: defines the elements of the set according to some rule or rules
{x : x = 30}
The latter form can have variations that define the domain of x on the left, such as
[tex]\{x\in\mathbb{Z}:x=30\}[/tex]
This is pronounced, "x is an element of the set of integers such that x is equal to 30." This sort of notation is more commonly used when the set is specified by an inequality or a functional relationship. You may see the colon (:) replaced by a vertical bar (|).
_____
Comment on the above notations
I don't like the notation x = {30} because the solution to the equation is a value, not a set of values. I prefer to write it as x ∈ {30}. "x is an element of the set containing 30." The "is an element of" symbol (∈) is not always available, so the equal sign is often used instead.
Could you simplify this question? 4xy-6xy+5y-9x
The solution of expression is,
⇒ - 2xy + 5y - 9x
We have to given that,
An expression to solve,
⇒ 4xy - 6xy + 5y - 9x
Now, We can simplify the expression by combining like terms as,
⇒ 4xy - 6xy + 5y - 9x
⇒ - 2xy + 5y - 9x
Therefore, The solution of expression is,
⇒ - 2xy + 5y - 9x
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which of the binomials below is a factor of this trinomial
x^2-13+42
Answer:
[tex] \boxed{ \bold{ \boxed{ \sf{(x - 7)(x - 6)}}}}[/tex]
Step-by-step explanation:
[tex] \sf{ {x}^{2} - 13x + 42 }[/tex]
Here, we have to find the two numbers that adds to 13 and multiplies to 42.
The numbers are 7 and 6
[tex] \sf{ {x}^{2} - (7 + 6)x + 42}[/tex]
⇒[tex] \sf{ {x}^{2} - 7x - 6x + 42}[/tex]
Factor out x from the expression
⇒[tex] \sf{x(x - 7) - 6x + 42}[/tex]
Factor out 6 from the expression
⇒[tex] \sf{x(x - 7) - 6(x - 7)}[/tex]
Factor out x - 7 from the expression
⇒[tex] \sf{(x - 7)(x - 6)}[/tex]
Hope I helped!
Best regards!!
Answer:
x-7 For AP EX peeps
Step-by-step explanation:
8 limes/$2.00 You have $5. How much change would you receive after purchasing 12 limes?
Answer:
2 dollars
Step-by-step explanation:
if 8 limes costs 2 dollars, then 4 limes cost 1 dollar
you purchased 12 limes so since you already know what 8 limes coms, you could subtract that
12-8=4
now you need to find out how much for 4 limes which would be half the price of 8 limes.. which should be 1 dollar
so $2.00+$1.00= $3.00
finally 5 dollars minus 3 would result in a change of 2 dollars
Simplify the expression(2x²y)^3
Answer:
8x^6y^3
Step-by-step explanation:
(2x2y)3
=8x6y3
Point J is on line segment I K ‾ IK . Given J K = x + 6 , JK=x+6, I J = 9 , IJ=9, and I K = 2 x , IK=2x, determine the numerical length of J K ‾ . JK .
Answer:21
Step-by-step explanation:
given f(x) = 3x -1 & g(x)= -x + 6, find f(-2) + g(5)
Answer:
When added together, the sum of the given functions have a value of -6.
Step-by-step explanation:
When we see f(-2), it means that our value of x for the f(x) function is -2. When we see g(5), it means that our value of x for the g(x) function is 5. So, let's use this information and plug it into our equation.
f(-2) + f(5) = (3(-2) - 1) + (-(5) + 6)
Multiply 3 by -2 and distribute the negative to 5.
f(-2) + f(5) = (-6 - 1) + (-5 + 6)
Subtract 1 from -6 and add 6 to -5.
f(-2) + f(5) = (-7) + (1)
Add 1 to -7.
f(-2) + f(5) = -6
So, when added, the sum of the functions is -6.
8.01 + 24.192 need answer please
Answer: 32.202
or 32.20
or 32.2
please help me with this question i’ll mark you as brainliest
Answer:
11^3/5
Step-by-step explanation:
Fractional exponents work like so:
[tex]x^{\frac{a}{b}} = \sqrt[b]{x^a}[/tex]
Essentially, the numerator (a) of the fraction is the power and the denominator (b) of the fraction is the root.
[tex]x^\frac{power}{root}[/tex]
We are given:
[tex]\sqrt[5]{11^3}[/tex]
The power is 3 and the root is 5. Let's substitute the values in.
[tex]x^\frac{power}{root}[/tex]
[tex]power=3\\root=5[/tex]
[tex]x^\frac{3}{5}[/tex]
x is the base. In this case, the base is 11.
[tex]11^\frac{3}{5}[/tex]
Therefore, 11^3/5 is the correct answer.
3 folders cost $2.91 dollar sign . Which equation would help determine the cost of 2 folders? WRITE THE LETTER IN THE SPACE BELOW. see the file plz
Answer:
none of the above
Step-by-step explanation:
the equation is supposed to be 3=2.91, 2=x, I hope you get it.
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y2
This question is incomplete, the complete question is;
Find the volume V of the solid obtained by rotating the region bounded by the given curves about the specified line. y2 = 2x, x = 2y; about the y-axis
Answer:
V = π (512/15)
Step-by-step explanation:
Given that;
region of rotation
y² = 2x, x = 2y
Region is rotated about y-axis as shown in the image
for the point of intersection,
y²/2 = 2y
y² - 4y = 0
y(y-4) = 0
∴ y = 0, y = 4
so the region lies in 0 ≤ y ≤ 4
Now cross section area of washer is
A(y) = π(outer radius)² = π(inner radius)²
A(y) = π(2y)² - π(y²/2)²
A(y) = π(4y²) - π(y⁴/4)
A(y) = π(4y² - (y⁴/4))
now volume of the solid of revolution is
V = ⁴∫₀ A(y) dy
V = ⁴∫₀ π(4y² - (y⁴/4))dy
V = π {4⁴∫₀ y² - 1/4⁴∫₀y⁴ dy }
V = π { 4/3 [y³]₀⁴ - 1/20 [y⁵]₀⁴ }
V = π { 4/3 [4]₀⁴ - 1/20 [4]₀⁴ }
V = π { 4/3 [64]₀⁴ - 1/20 [1024]₀⁴ }
V = π { 256/3 - 1024/20 }
V = π { (5120 - 3072) / 60 }
V = π (512/15)
A community sports league is raising money by making custom shirts to sell at league games. They plan to sell the shirts for $14. Each shirt costs $7 to make. They spent $55 for advertising. Use n to represent the number of shirts they sell. Multiply this by the money they make for each shirt, then subtract the advertising cost. Which expression represents the money that the league raises?
O A. (14 – 7)n – 55
B. 55 – (14 – 7) n
O C. 14 – 7n – 55
O D. 14n – 7 – 55
Answer:
A. (14 – 7)n – 55
Step-by-step explanation:
it cost $14 to make so $14 - $7 to make then you have to advertise it AFTER and it takes more money to advertise so, -55 so the answer would be:
(14 – 7)n – 55
Show that 551 is a rational by writing it in the form a/b, where a and b are integers
Answer:
551/1
Step-by-step explanation:
[tex]551 = \dfrac{551}{1}[/tex]
__
Any integer can be written as a rational number with a denominator of 1.
Is GPA qualitative or quantitative? Also, is it nominal, ordinal, interval or ratio?
While the letter grade to numerical contribution for a single subject is ordinal, the moment you compute a grade point average you already treated it as interval at that moment (otherwise you have no basis on which to assert that A+C = B+B).
Answer:
General rule of thumb: if you can add it, it's quantitative. For example, a G.P.A. of 3.3 and a G.P.A. of 4.0 can be added together (3.3 + 4.0 = 7.3), so that means it's quantitative. While the letter grade to numerical contribution for a single subject is ordinal, the moment you compute a grade point average you already treated it as interval at that moment
Draw a two-card hand from a well-shuffled standard deck of cards. Given that the first card is a King, what is the chance the second card is NOT a King
Answer:
P ≈ 0,94
Step-by-step explanation:
The normal deck of cards consists of 52 cards, (with 4 kings )
If we take a king a deck will have 51 cards ( three kings )
The probability to get a non-king card is
Probability of non-king card (P) = 1 - Probability of getting a king card (P₁)
Probability of getting a king card P₁= 3/51 ≈ 0,059
Then P = 1 - 0,059
P ≈ 0,94
Mr smith and mr stein were driving to a buisness meeting 250 miles from their office. Mr Smith drove the first x miles, then Mr Stein drove the rest of the way. Write an algebric expression for how many miles mr stein drove
What is 5 minus the square of a number.
Answer:
5 - n^2
Step-by-step explanation:
Represent the unknown number by n. Then the desired expression is:
5 - n^2
Expressing the problem as a mathematical equation, the required expression ls n² - 5
Let the number = n ;
The square of the number, n = n²
Subtracting 5 from the squared Value of the number n² ;
The expression becomes ; n² - 5
Therefore, the required expression which expresses 5 minus the square of a number is n² - 5
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Determine whether the function is continuous or discontinuous. Justify your reasoning. If discontinuous, identify points of discontinuity.
Please I am in need of urgent help.
Brainliest to the first answer
[tex]f (x)=\left \{ {{(2-x^2) if x\leq1 } \atop {(x) if x\ \textgreater \ 1}} \right.[/tex]
Simplify completely x^2+4x-45 / x^2+10x+9
Answer: x - 5/x + 1
Step-by-step explanation:
This algebraic fraction
The task to be performed here is factorisation and simplification. Now going by the question,
x² + 4x - 45/x² + 10x + 9, the factorisation of
x² + 4x - 45 = x² + 9x - 5x - 45
= x(x + 9 ) - 5(x + 9 )
= ( x + 9 )(x - 5 ), don't forget this is the algebraic fraction's Numerator
The second part
x² + 10x + 9 = x² + x + 9x + 9
= x(x + 1) + 9( x + 1 )
= ( x + 9 )( x + 1 ), this is the algebraic denominator.
Now place the second expression which is the denominator under the first expression which is the numerator.
( x + 9 )( x - 5 )/( x + 9 )( x + 1 ).
You can see that, ( x + 9 )/( x + 9 ) divide each other , therefore therr then cancelled and left with
x - 5/x + 1
A pitcher throws a baseball 90 miles per hour. Convert this speed into feet per second.
write each statement as a decimal to two decimal places. $6.00 to 95 cents 3 hours to 35 minutes 42 inches to 2 feet
Answer:
1. $6.00 to 95 cents = 6.32
2. 3 hours to 35 minutes = 5.14
3. 42 inches to 2 feet = 1.75
Step-by-step explanation:
Given
$6.00 to 95 cents
3 hours to 35 minutes
42 inches to 2 feet
Required
Express in 2 decimal places
1. $6.00 to 95 cents
Express as ratio
[tex]\$6.00 : 95\ cents[/tex]
Convert dollars to cents
[tex]6.00 * 100\ cents: 95\ cents[/tex]
[tex]600\ cents: 95\ cents[/tex]
Convert ratio to division
[tex]\frac{600\ cents}{95\ cents}[/tex]
[tex]6.31578947368[/tex]
Approximate
[tex]6.32[/tex]
2. 3 hours to 35 minutes
Express as ratio
[tex]3\ hours : 35\ minutes[/tex]
Convert hours to minutes
[tex]3 * 60\ minutes: 35\ minutes[/tex]
[tex]180\ minutes: 35\ minutes[/tex]
Convert ratio to division
[tex]\frac{180\ minutes}{35\ minutes}[/tex]
[tex]5.14285714286[/tex]
Approximate
[tex]5.14[/tex]
3. 42 inches to 2 feet
Express as ratio
[tex]42\ inches: 2\ feet[/tex]
Convert feet to inches
[tex]42\ inches:2 * 12\ inches[/tex]
[tex]42\ inches : 24\ inches[/tex]
Convert ratio to division
[tex]\frac{42\ inches}{24\ inches}[/tex]
[tex]1.75[/tex]
-5m-2 (1-7m) show work please
Answer:
9m - 2
Step-by-step explanation:
Order of Operations: BPEMDAS
Step 1: Write out expression
-5m - 2(1 - 7m)
Step 2: Distribute -2
-5m - 2 + 14m
Step 3: Combine like terms
9m - 2
I need help again....
help please I don't understand
Answer:
x = 8AM = 64 cmBM = 64 cmM is the midpointStep-by-step explanation:
Part A. The total length of a line segment is the sum of the lengths of its parts.
AM +MB = AB
Substituting the given information, this equation becomes ...
(7x +8) cm + (9x -8) cm = 128 cm
16x = 128 . . . . . . . . . collect terms, divide by cm
x = 128/16 . . . . . divide by the coefficient of x
x = 8
__
Part B. Then the length of AM is ...
AM = (7x +8) cm = (7(8) +8) cm = (56 +8) cm
AM = 64 cm
__
Part C. The length of BM is ...
BM = (9x -8) cm = (9(8) -8) cm = (72 -8) cm
BM = 64 cm
__
Part D. AM = BM = 64 cm, so point M is the midpoint of AB. Point M divides the segment into two equal parts.