The closed form sum of $$12 \left[ 1^2 \cdot 2 + 2^2 \cdot 3 + \ldots + n^2 (n+1) \right]$$ for $n \geq 1$ is $n(n+1)(n+2)(an+b).$ find $an +
b.$
Final answer:
The closed form sum of the given expression is obtained by expanding and simplifying it. Comparing this with the given expression, we can find the values of a and b.
Explanation:
The closed form sum of $$12 \left[ 1² \cdot 2 + 2²\cdot 3 + \ldots + n² (n+1) \right]$$ for $n \geq 1$ is $n(n+1)(n+2)(an+b).$ To find $an + b$, we can simplify the expression by expanding it:
Expanding the sum, we have:
$$\begin{align*} 12 &\left[ 1² \cdot 2 + 2² \cdot 3 + \ldots + n² (n+1) \right] \\ &= 2[1 + (n-1) + 3 + \ldots + (2n-3) + (2n-1)-(n-1)] \\ &= 2[n + 3 + \ldots + (2n-3) + n] \\ &= 2n² \end{align*}$$
Comparing this with the given expression $n(n+1)(n+2)(an+b)$, we can see that $2n² = n(n+1)(n+2)(an+b)$. So, $an+b = \boxed{2}$.
Samara is adjusting a satellite because she finds it is not focusing the incoming radio waves perfectly. The shape of her satellite can be modeled by y2=2(x+6), where x and y are modeled in inches. She realizes that the static is a result of the feed antenna shifting slightly off the focus point. What is the focus point of the satellite?
The focus point of the satellite is (-5.5,0).
This is a parabola by the looks of the equation and the fact that it's a satellite dish. The vertex of the equation is (-6,0) and the focus is 2/4, 1/2. 1/2 plus -6 is -5.5. the focus is (-5.5,0).
What is a Parabola?This refers to the plane curve that is U-shaped and is formed by the intersection of a cone.
Hence, we can see that based on the equation, y^2=2(x+6), we can see that the adjustments made by Samara where the x and y are in inches, the focal point based on the vertex of the equation which is (-6,0) and the focus is 2/4, 1/2.
Therefore, the focal point is 1/2 + -6 which is -5.5.
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Evaluate the function for an input of 10
Let n be the middle number of three consecutive even integers. Write an expression for the sum of these integers.
PLEASE HELPP
The expression for the sum of these integers is 3n
Let the first number be n-2
Let the middle number be n
Let the third number be n+2
The expression for the sum of these integers will be:
=(n - 2) + n + (n + 2)
= n - 2 + n + n + 2
Collect like terms
= 3n
The expression for the sum of these integers is 3n
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Points J and K lie in plane H.
How many lines can be drawn through points J and K?
0
1
2
3
1 is the answer bruvy
Answer: 1
Step-by-step explanation:
From the given picture, it can be seen that there is a plane H on which two pints J and K are located.
One of the Axiom in Euclid's geometry says that "Through any given two points X and Y, only one and only one line can be drawn "
Therefore by Axiom in Euclid's geometry , for the given points J and K in plane H , only one line can be drawn through points J and K.
Given ƒ(x) = 2x2 − 3x + 7, find ƒ(2.5)
Answer:
i need helppp
Step-by-step explanation:
this same problems but where is everyone getting the 6.25 from.
compare & contrast the graphs of y = 5x+9 and y - 5x=3
Final answer:
The equations y = 5x + 9 and y - 5x = 3 have the same slope of 5, indicating that their graphs are parallel lines. The difference lies in their y-intercepts; the first line crosses the y-axis at 9 and the second at 3, meaning they are parallel but at different vertical positions.
Explanation:
To compare and contrast the graphs of y = 5x + 9 and y - 5x = 3, we first need to rewrite the equations in slope-intercept form, which is y = mx + b, where m is the slope and b is the y-intercept. The first equation is already in this form, with a slope of 5 and a y-intercept of 9. We can rearrange the second equation by adding 5x to both sides to get y = 5x + 3, which has a slope of 5 and a y-intercept of 3.
Both lines have the same slope of 5, which means they are parallel and will never intersect. The difference between them is their y-intercepts; the first line intersects the y-axis at 9, whereas the second line intersects the y-axis at 3. This difference in y-intercepts means that while they have the same steepness and direction, they are located at different vertical positions on the coordinate plane.
Write each fraction as a repeating or circulating decimal. a. 2⁄3 b. 9⁄11 c. 2⁄9 d. 11⁄3 e. 5⁄6 f. 14⁄3
Answer:
Step-by-step explanation:
repeating or circulating decimal is decimal representation of a number whose digits are periodic or infinitely repeated.
To convert a fraction into decimal form we have to divide numerator by denominator.
a. 2/3 = 0.666666
b. 9/11 = 0.8181818
c. 2/9 = 0.222222
d. 11/3 = 3.666666
e. 5/6 = 0.833333
f. 14/3 = 4.666666
The upper limit for the function f(x) = x3 - 3x 2 - 2x + 15 is
3
4
5
interior angles on the same side of a transversal cutting parallel lines are _____ congruent
a) always
b) sometimes
c) never
The interior angles on the same side of a transversal cutting parallel lines are always congruent.
Explanation:The interior angles on the same side of a transversal cutting parallel lines are always congruent.
When a transversal intersects two parallel lines, it creates several pairs of corresponding angles. The corresponding angles that are on the same side of the transversal are congruent or equal in measure. This is known as the Alternate Interior Angles Theorem.
For example, if line m is parallel to line n and a transversal line t cuts both lines at points A and B, then the angle formed by the intersection of line t and line m, and the angle formed by the intersection of line t and line n on the same side of the transversal are congruent.
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find the resultant of two forces, one of 72 pounds and the other of 56 pounds if they act at and angle of 37°
What common angle do ΔCDG and ΔFCE share?
a. ∠C
b. ∠F
c. ∠E
d. ∠D
solve 5x=5/3 using multiplication property of equality
H E L P PLEASE!! COME ON! Grace receives a net pay of $483.75 biweekly. She has $170 withheld from her pay each pay period. What is her annual gross salary?
A. $3,653
B. $3,372
C. $16,997.50
D. $10,572
All the results I got didn't match so I'm apparently doing something wrong... :/ So please explain your answer CLEARLY. Thanks! Edit: this is the third time I've had to put this up guys. someONE HELP PLEASE
Lindsey has 10 inches of ribbon. she buys another 3 length of ribbon, each 5 inches long. how much ribbon does she have now?
535 students of the senior class voted to take a field trip. if 3/4 of the class voted, how many students voted for the trip? round your answer to the nearest whole number. 402 401 134 133
A fraction is a way to describe a part of a whole. The number of students who voted is 401. The correct option is B.
What is a Fraction?A fraction is a way to describe a part of a whole. such as the fraction ¼ can be described as 0.25.
Given that the total number of students in the senior class is 535. Now, if 3/4 of the class voted. Then we can write,
The number of seniors who voted = 3/4 of 535
= (3/4) × 535
= 401.25 ≈ 401
Hence, the number of students who voted is 401.
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To fence in 2 cows, josie used 64 feet of fence. for 3 cows, she used 96 feet of fence, and for 4 cows, she used 128 feet of fence. how many feet of fence will she use for 6 cows?
Answer:
192 feet fence would be used for 6 cows.
Step-by-step explanation:
Given,
64 feet of fence is used for 2 cows,
Ratio in fence and cows = [tex]\frac{64}{2}[/tex] = 32,
96 feet of fence is used for 3 cows,
Ratio in fence and cows = [tex]\frac{96}{3}[/tex] = 32,
128 feet of fence is used for 4 cows,
Ratio in fence and cows = [tex]\frac{128}{4}[/tex] = 32,
Thus, by the above explanation it is clear that in every case the ratio of the length of fence and number of cows is 32,
Let x be the length of cows that is used for x cows,
[tex]\implies \frac{x}{6}=32[/tex]
[tex]\implies x = 192[/tex]
Hence, 192 feet fence would be used for 6 cows.
Point R is between points S and T and SR = 29 and ST = 47
What is RT?
a.) 18
b.) 22
c.) 66
d.) 76
is 37% bigger than 0.37
In △ABC , the coordinates of vertices A and B are A(1,2), and B(−3,−1)?
For each of the given coordinates of vertex C, is △ABC a right triangle?
The correct answers are:
C(-3,-2) Not a Right Triangle C(0,-5) Right Triangle C(1,-2) Not a Right TriangleAnswer:
ABC would be right angle triangle if C(1,-1) or C(-3,2)
Step-by-step explanation:
In △ABC , the coordinates of vertices A and B are A(1,2), and B(−3,−1)
We need to find the third vertex of triangle ABC.
ABC is a right angle triangle.
A(1,2) and B(-3,-1)
Using A and B, Two possible right angle triangle.
Case 1: If we draw a vertical line at A and horizontal line at B then intersection point of vertical and horizontal would be third vertex of triangle ABC.
Vertical line at A, x=1
Horizontal line at B, y=-1
Vertex C: (1,-1)
Case 2: If we draw a horizontal line at A and vertical line at B then intersection point of vertical and horizontal would be third vertex of triangle ABC.
Vertical line at B, x=-3
Horizontal line at C, y=2
Vertex C: (-3,2)
Thus, ABC would be right angle triangle if C(1,-1) or C(-3,2)
Please see the attachment to see the process graphically.
Each calendar will sell for $5.00 each. Write an equation to model the total income, y, for selling x calendars
The equation to model the total income, y, for selling x calendars will be y=5x.
What is the equation?An equation is a statement that two expressions, which include variables and/or numbers, are equal. In essence, equations are questions, and efforts to systematically find solutions to these questions have been the driving forces behind the creation of mathematics.
It is given that, each calendar will sell for $5.00. the total income is y and the calendars to be sold be x.
Since each calendar costs $5, the total revenue (y) is equal to the number of calendars sold (x).
y=5x
The above equation is in the one variable known as the linear equation in one variable. It had the form y = mx. x and y are the variables in the equation.
Thus, the equation to model the total income, y, for selling x calendars will be y=5x.
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What is 187.023 I word form
An artist is creating a large butterfly sculpture outside a museum. There is a circular dot on each wing made out of a metal ring. The distance around each dot is 24pi inches. The artist plans to fill the inside of each dot with blue colored glass.
What is the area of the blue glass will be needed to fill each butterfly dot?
Give your answer in terms of pi.
The area of the blue glass that will be needed to fill each butterfly dot would be 144π.
What is the circumference of the circle?The circumference of the circle that has a radius of r is defined as the product of diameter to the pie value.
The circumference of the circle = 2πr
We have given the distance around each dot is 24pi inches.
So, The circumference of the circle = 2πr
24π = 2πr
r = 12
The artist plans to fill the inside of each dot with blue-colored glass.
Now, we know that
The area of the circle = [tex]\pi r^{2}[/tex]
= π x 12 x 12
= 144π
Thus, the area of the blue glass that will be needed to fill each butterfly dot would be 144π.
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Can someone help me with this problem too?
The base of a triangle exceeds the height by 7 feet. if the area is 114 square feet, find the length of the base and the height of the triangle.
Final answer:
The base and height of the triangle are found using the area formula for triangles, with the base being 7 feet more than the height. Using this relationship and the fact that the area is 114 square feet, we solve a quadratic equation to find the height to be 12 feet and the base to be 19 feet.
Explanation:
The student is given that the base of a triangle exceeds the height by 7 feet and that the area of the triangle is 114 square feet. To find the base and height of the triangle, we can use the area of a triangle formula A = (1/2) × base × height. Let's denote the base as b and the height as h. From the given information, b = h + 7.
Substituting the values into the area formula, we get:
114 = (1/2) × (h + 7) × h. Simplifying this, we get 114 = (1/2) × h² + 7/2 × h. Multiplying both sides by 2 to clear the fraction, we obtain 228 = h² + 7h. This is a quadratic equation in the form of h² + 7h - 228 = 0, which we can solve by factoring or using the quadratic formula. Factoring gives us (h + 19)(h - 12) = 0, leading to two potential solutions for h: -19 or 12. Since a height cannot be negative, we discard -19, leaving us with h = 12 feet. Therefore, the base b = h + 7 = 19 feet
The base of the triangle is 19 feet, and the height is 12 feet.
Why do you multiply a percent by a number to find the percent of that number?
Sheila purchase a plot of land for $23,500. The land appreciates about 1.9% each year. What is the value of the land after seven years?
A snack stand sold hot dogs and chips at a recent game. The hot dogs cost $2.50 and the snacks cost $0.75. After the game, they tallied 175 transactions totaling $262.50.
Represent the linear system in an augmented matrix:
x = hot dogs
y = snacks
x +y = 175
x = 175-y
2.50x + 0.75y = 262.50
2.50(175-y) +0.75y = 262.50
437.50-2.50y + 0.75y = 262.50
-1.75y = -175
y = -175 / -1.75 = 100
100 snacks & 75 hot dogs
The number of hot dogs sold is 75 while the number of snacks sold is 100.
Let the number of hot dogs be represented by xLet the number of snacks sold be represented by y.Based on the information, the equation to solve the question will be:
x + y = 175 ....... i
2.50x + 0.75y = 262.50 ....... ii
From equation i, x = 175 - y.
We'll put the value of x into 2.50x + 0.75y = 262.50 and this will be:
2.50x + 0.75y = 262.50
2.50(175 - y) + 0.75y = 262.50
437.50 - 2.50y + 0.75y = 262.50
Collect like terms
-2.50y + 0.75y = 262.50 - 437.50
-1.75y = -175
y = 175/1.75
y = 100
The number of snacks sold is 100.
Since x + y = 175
x = 175 - 100
x = 75
The number of hot dogs sold is 75
Therefore, 75 hot dogs and 100 snacks were sold.
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Western boundary currents are deflected by the western coastlines of the continents.True False
Solve for a. 5a2=320
To solve for 'a' in the equation 5a^2 = 320, first divide both sides of the equation by 5 to get a^2 = 64. Then, take the square root of both sides which provides two possible solutions, a = 8, and a = -8.
Explanation:The student is asking to solve for 'a' in the algebraic equation 5a2 = 320. To find the solution, we first need to isolate 'a' on one side of the equation. Begin by dividing both sides of the equation by 5:
a2 = 320 / 5
This simplifies to:
a2 = 64
To solve for 'a', we take the square root of both sides:
a = √64
Therefore, the solutions are:
a = 8, a = -8
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