An equation of a line in a slope-intercept format that is parallel to
y = (1/2)x - 3 line and goes through the point (4, 2) is y = (1/2)x
What are lines and their slopes?We know lines have various types of equations, the general type is
Ax + By + c = 0, and the equation of a line in slope-intercept form is
y = mx + b.
Where slope = m and b = y-intercept.
the slope is the rate of change of the y-axis with respect to the x-axis and the y-intercept is the (0,b) where the line intersects the y-axis at x = 0.
We know parallel lines have equal slopes.
The equation of the line given is y = (1/2)x - 3 and our required equation is parallel to it and passes through (4, 2).
So, 2 = (1/2)(4) + b.
2 = 2 + b.
b = 0.
Hence, y = (1/2)x is the required equation of the line.
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resoudre [tex]30*\sqrt{5}² + 13*\sqrt{5}-6[/tex]
The simplified expression for this problem is given as follows:
[tex]43\sqrt{5} - 6[/tex]
How to simplify the expression?The expression for this problem is defined as follows:
[tex]30\sqrt{5} + 13\sqrt{5} - 6[/tex]
To simplify the expression, we combine the like terms, which are the terms that have the same rational, thus:
[tex]30\sqrt{5} + 13\sqrt{5} = 43\sqrt{5}[/tex]
Combining along with the subtraction by 6, the complete simplified expression is given as follows:
[tex]43\sqrt{5} - 6[/tex]
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The cost of a newspaper subscription
includes a discounted initial week. Beatriz
pays $55 for 7 weeks of the newspaper
subscription and $100 for 12 weeks.
As a result, the original discounted price is $1 while the weekly normal price is $9.
How does arithmetic work with discounts?The fundamental formula for calculating a reduction is to increase the initial price by the provided percentage rate in decimal form. We must deduct the reduction from the initial price to determine the item's sale price.
From the given information, we have:
d + 6x = 55 (discounted price for 7 weeks)
d + 11x = 100 (discounted price for 12 weeks)
We can solve this system of equations by eliminating d. Subtracting the first equation from the second, we get:
5x = 45
Dividing both sides by 5, we get:
x = 9
Substituting x = 9 into one of the equations, we can solve for d. Using the first equation, we get:
d + 6(9) = 55
d + 54 = 55
d = 1
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prove that a square matrix and its transpose have the same characteristic polynomial, and therefore the same set of eigenvalues. g
The eigenvalues of a matrix are roots of its characteristic polynomial using this we prove that a square matrix and its transpose have the same characteristic polynomial.
What is a matrix?A collection of numbers lined up in rows and columns to form a rectangular array is called a matrix. The elements, or entries, of the matrix are the integers. In addition to numerous mathematical disciplines, matrices find extensive use in the fields of engineering, physics, economics, and statistics. In computer graphics, where they have been used to describe picture rotations and other transformations, matrices have vital applications as well.
Recall that the eigenvalues of a matrix are roots of its characteristic polynomial.
Hence if the matrices A and A Transpose have the same characteristic polynomial, then they have the same eigenvalues.
So we show that the characteristic polynomial pA(t)=det(A−tI) of A is the same as the characteristic polynomial p[tex]A^T[/tex] (t)=det(A ^T−tI)of the transpose [tex]A^T[/tex].
We have,
p[tex]A^T[/tex](t)=det([tex]A^T[/tex] −tI) =det([tex]A^T[/tex] −tI T )
=det((A−tI) [tex]A^T[/tex] )
=det(A−tI)
=pA(t)
Since I ^T =I
Since det([tex]B^T[/tex] )=det(B)for any square matrix B
Therefore, we obtain p[tex]A^T[/tex] (t)=pA(t), we conclude that the eigenvalues of A and [tex]A^T[/tex] are the same.
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Write an equation in slope-intercept form of the line that satisfies the given conditions.
Through (6, 3); parallel to 7x-y=3
The equation of the line is
Step-by-step explanation:
The equation of a line in slope-intercept form is y = mx + b, where m is the slope and b is the y-intercept. If a line is parallel to another line, it means they have the same slope. To find the slope of the line 7x - y = 3, we can rearrange it into slope-intercept form: y = 7x - 3. So, the slope of the line 7x - y = 3 is 7.
Since the line we want to find is parallel to 7x - y = 3 and has a slope of 7, the equation of the line can be written as y = 7x + b, where b is the y-intercept. To find b, we use the point (6, 3) as a solution:
y = 7x + b
3 = 7 * 6 + b
3 = 42 + b
b = -39
So, the equation of the line that passes through (6, 3) and is parallel to 7x - y = 3 is y = 7x - 39.
a group of students and adults are going to the museum and have spent a total of $115 on tickets. Adult tickets cost $5 each and student tickets cost $3 each. There are 25 more students than adults in the group. How many students are in the group and how many adults are in the group?
A group of students and adults are going to the museum and have spent a total of $115 on tickets. Adult tickets cost $5 each and student tickets cost $3 each. There are 25 more students than adults in the group therefore the number of students are in the group are 35 and the number of adults in the group are 5.
What is Cost?This is referred to as the value of money that has been used up to produce something or deliver a service.
Since we were told that Adult tickets cost $5 each and student tickets cost $3 then $5 × 5 = $25 and $3 × 30 = $90. Therefore the total is $115.
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When planning a more strenuous hike, Erin figures that she will need at least 0.8 liters of water for each hour on the trail. She also plans to always have at least 1.50 liters of water as a general reserve. If x represents the duration of the hike (in hours) and y represents the amount of water needed (in liters) for a hike, the following inequality describes this relation: y greater or equal than 0.8 x plus 1.5
Answer:
the inequality y ≥ 0.8x + 1.5 accurately describes the relation between the duration of the hike (x) and the amount of water needed (y). This inequality means that for any duration of the hike (x), the amount of water needed (y) must be greater than or equal to 0.8x + 1.5.
Answer:
Step-by-step explanation:
3 liters of water for 4 hours of hiking.
A Triangular Prism with equilateral triangle faces has Volume of 84cm³ and a Length of 12cm. Find the base area of this prism.
Got this on my test, don't know how to do it lol.
Answer:
7cm²
Step-by-step explanation:
volume of a triangular prism=1/2×b×h×l
1/2×b×h=cross sectional area
84cm³=C.A×12
C.A=84/12
base area=7cm²
The May 1, 2009, issue of a certain publication reported the following home sale amounts for a sample of homes in Alameda, CA that were sold the previous month (1,000s of $).
592 817 577 608 353 1,281 412 535 550 679
(a) Calculate and interpret the sample mean and median.
The sample mean is x = thousand dollars and the sample median is x tilde= thousand dollars.
This means that the average sale price for a home in this sample was $ and that half the sales were for less than the _____mean median price, while half were more than the _____ mean median price.
(b )Suppose the 6th observation had been 985 rather than 1,281. How would the mean and median change?
(c) Calculate a 20% trimmed mean by first trimming the two smallest and two largest observations. (Round your answer to the nearest hundred dollars.)
(d) Calculate a 15% trimmed mean. (Round your answer to the nearest hundred dollars.)
This means that the average sale price for a home in this sample was "$773" and that half the sales were for less than the median price of "$577", while half were more than the median price of "$577".
(a) To calculate the sample mean, we add up all the values and divide by the number of values:
x = (592 + 817 + 577 + 608 + 353 + 1,281 + 412 + 535 + 550 + 679)/10 = 7,727/10 = $773
The sample median is the middle value when all the values are ordered from smallest to largest:
353, 412, 535, 550, 577, 608, 592, 679, 817, 1,281
The median is 577.
This means that the average sale price for a home in this sample was $773 and that half the sales were for less than the median price of $577, while half were more than the median price of $577.
(b) If the 6th observation had been 985 rather than 1,281, the new mean would be calculated as:
x = (592 + 817 + 577 + 608 + 353 + 985 + 412 + 535 + 550 + 679)/10 = 7,610/10 = $761
The new median would still be $577.
(c) To calculate a 20% trimmed mean, we first need to find the two smallest and two largest observations, then remove them and take the average of the remaining values:
Two smallest: 353, 412
Two largest: 1,281, 817
Remaining values: 535, 550, 577, 608, 592, 679
The trimmed mean is (535 + 550 + 577 + 608 + 592 + 679)/6 = $592
(d) To calculate a 15% trimmed mean, we first need to find the three smallest and three largest observations, then remove them and take the average of the remaining values:
Three smallest: 353, 412, 535
Three largest: 817, 1,281, 679
Remaining values: 550, 577, 608, 592
The trimmed mean is (550 + 577 + 608 + 592)/4 = $576.
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The figure below is a parallelogram. Based on what we know about consecutive angles of a parallelogram, write and equation and solve for x.
The value of x in the given paralellogram is 12.
What is parallelogram?
"It is a quadrilateral with parallel opposite sides."
"The opposite angles of parallelogram are equal. "
For given question,
Two of the angles in a parallelogram measure 8x° and 7x°
The measure of an angle opposite to 8x° is also 8x°
In a similar manner, an angle opposite to 7x° is measured as 7x°.
So, we have the four angles of parallelogram as 8x°, 7x°, 8x°, 7x°
We are know that the total angle of a quadrilateral is 360°.
⇒ 8x° + 7x° + 8x° + 7x° = 360°
⇒ (5 + 7 + 5 + 7) x = 360
⇒ 30x = 360
⇒ x = 12
Therefore, the value of x is 15.
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Taye pays $0.06 sales tax on each dollar he spends. If he buys a new bike for $237.00, how much sales tax will Taye pay?
Answer: $14.22
Step-by-step explanation: 237*0.06
Graphing and substitution, please help!
covert to slope intercept form:
5x-3y=15
Answer:y=5/3 x-5
Step-by-step explanation: first find the y-intercept, when the x value is 0 that is the y intercept, then the equation becomes -3x=15, there you get
x=-5, then do the same to the other to get the x value so you get the points (0,-5), and (3,0) there you find the slope and plug in
Can someone answer this pls
-3y + 4 – 2 + 6y
Answer: 3y + 2
Step-by-step explanation: First, you can organize the equation into -3y + 6y + 4 - 2. Then, you get 3y + 2. :)))))
in a positive number of two digits, the sum of the digits is 15, if the digits are interchanged, the number is increased by 9. Find the number.
Answer:
the number is 78
Step-by-step explanation:
call x is the tens digit (0<x<9)
y is the unit digit (0[tex]\leq[/tex]y[tex]\leq[/tex]9)
we have:
10y+x=[tex]\oveline{yx}[/tex] (1)
10x+y= [tex]\overline{xy}[/tex] (2)
acording (1),(2) and the topic we have: (10y+x)-(10x+y)=9 ⇔9y-9x=9
⇔y-x=1 (3)
and x+y=15 (4)
from (3), (4) we have a system of equation
[tex]\left \{ {{y-x=1} \atop {x+y=15}} \right.[/tex]
⇔[tex]\left \{ {2y=16} \atop {x+y=15}} \right.[/tex]
⇔[tex]\left \{ {{y=8} \atop {x+8=15}} \right.[/tex]
⇔[tex]\left \{ {{y=8} \atop {x=7}} \right.[/tex] (conditions are satified)
the number is 78
What is the area of a rectangle with length of 7 and width of 4x? Simplify
]Answer:
The area of a rectangle is calculated by multiplying its length by its width. If the length of the rectangle is 7 and its width is 4x, the area can be found by performing 7 * 4x. The answer would be 28x square units, where x represents the unit of measurement.
At a local elementary school, 35 percent of all students have brown eyes, 45 percent have brown hair, and 60 percent have brown hair or brown eyes. A student will be selected at random from the school. Let E represent the event that the selected person has brown eyes, and let H represent the event that the selected person has brown hair.Are E and H mutually exclusive events?A- Yes, because P(E∩H)=0P(E∩H)=0.
B- Yes, because P(E∩H)=0.2P(E∩H)=0.2.
C- Yes, because P(E∩H)=0.6P(E∩H)=0.6.
D- No, because P(E∩H)=0.2P(E∩H)=0.2.
E- No, because P(E∩H)=0.6
The required events E and H are not mutually exclusive, because they can both occur at the same time.So D - No, because P(E∩H) = 0.20 is correct.
What are mutually exclusive events?The events E and H are not mutually exclusive, because they can both occur at the same time. In other words, a student can have both brown eyes and brown hair.
According to question:The probability that a student has both brown eyes and brown hair, P(E∩H), can be found using the formula:
P(E∩H) = P(E) + P(H) - P(E or H)
where P(E) is the probability of having brown eyes, P(H) is the probability of having brown hair, and P(E or H) is the probability of having either brown eyes or brown hair.
Given that:
P(E) = 0.35
P(H) = 0.45
P(E or H) = 0.60
So,
P(E∩H) = 0.35 + 0.45 - 0.60 = 0.20
So, the answer is D - No, because P(E∩H) = 0.20.
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If ST = p + 31, RU = 6p + 14, and VW = 4p + 53, what is the value of p?
Answer:
If we subtract 14 from both sides of the equation for RU, we get 6p = RU - 14, or 6p = RU - 14. We can then substitute this expression for 6p into the equation for ST to get:
ST = p + 31
ST = (RU - 14) + 31
Expanding RU, we get:
ST = RU - 14 + 31
ST = RU + 17
Since RU = 6p + 14, we can substitute this expression into the above equation to solve for p:
ST = 6p + 14 + 17
ST = 6p + 31
Finally, we can subtract 31 from both sides of the equation to find p:
ST - 31 = 6p + 31 - 31
ST - 31 = 6p
ST - 31 = 6p
p = (ST - 31) / 6
So, to find the value of p, we need to know the value of ST.
Solve and explain the method you chose to use (distributive property, FOIL, multiplying special cases): (c+3)^2
After resolving [tex](c+3)^{2}[/tex] , c = -3, and the approach is an example of the distributive property of equations.
What is equations?Its simplest form has been thought to be the figure is probably smallest equivalent fraction. How or when to determine the basic form. Look for shared factors throughout the denominator and numerator. A fractional number can sometimes be checked to discover if it is a power of 2. A statement having two equal sides and an equal sign in the middle is referred to as a mathematical equation. Every variable's degree courses are listed in descending order as in general form of any equation. The formula for a linear equation is x + b = 0. The general form of a three separate linear equation is an x + b y + c = 0. (or something similar). Either conditioned equations or identities are categories for mathematics. An identity holds true for whatever value of the variables.
Given,
[tex](c+3)^{2}[/tex]
= by using formulas of [tex](a+b)^{2}[/tex]
by distributive property
[tex]c^{2}[/tex] + [tex]3^{2}[/tex] + 6c
= [tex]c^{2}[/tex] + 9 + 6c
= [tex]c^{2}[/tex] + 6c + 9 = 0
[tex]c^{2}[/tex] + 3c + 3c + 9 = 0
= c ( c + 3 ) + 3 ( c + 3)
= (c + 3 )(c+ 3)
c = - 3
After resolving [tex](c+3)^{2}[/tex], c = -3, and the approach is an example of the distributive property of equations.
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Please help I will give brainliest
l=7m
Step-by-step explanation:
here us the answer of the solution
if Δabc≅Δpqr, then ac≅
ac is congruent to pr because those sides correspond
Describe how you would obtain an equation in one variable to solve the system by substitution. 3x+5y=25 x-2y=-6
Solve for x in one of the equations and substitute it into the other equation.
To solve the system of equations 3x + 5y = 25 and x - 2y = -6 using substitution, we first solve for x in one of the equations. Let's solve for x in the second equation: x = 2y + 6. Next, we substitute this expression for x into the first equation: 3(2y + 6) + 5y = 25. This gives us 6y + 18 + 5y = 25, which simplifies to 11y + 18 = 25. Finally, we subtract 18 from both sides to obtain 11y = 7, and divide both sides by 11 to find y = 7/11. Now that we have y, we can substitute it back into either equation to find x.
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Intelligence Quotient (IQ) scores are often reported to be normally distributed with μ=100.0 and σ=15.0 . A random sample of 59 people is taken. Step 2 of 2 : What is the probability that the mean IQ score of people in the sample is less than 99 ? Round your answer to 4 decimal places, if necessary.
The likelihood that a stranger on the street has an IQ below 98 is 0.4470, or 44.70%.
what is probability ?The probability that an event will happen or a claim will be correct is measured by statistical mechanics, a branch of mathematics. The possibility of an occurrence is a positive integer and 1, where about 0 represents how likely the event is to occur and value of 1 indicates certainty. A probability is a numerical illustration of the possibility that a specific activity will take place. Probabilities can also be expressed using percentages extending from 0% to 100% or from 0 to 1. the ratio of the number of outcomes to the proportion of occurrences in a comprehensive set of equally likely choices that lead to a specific occurrence.
given
This is Z's pvalue for X = 98. So
Z = 98 -100 / 15
Z = -0.133
Z = -0.133 pvalue of 0.4470
The likelihood that a stranger on the street has an IQ below 98 is 0.4470, or 44.70%.
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Factor 1 - 2.25x8.
(1 - 1.5x 4) 2
(1.5 + x 4)(1.5 - x 4)
(1 + 1.5x 4)(1 - 1.5x 4)
The factor of 1 - 2.25x⁸ is (1 + 1.5x⁴) (1 - 1.5x⁴).
The correct option is C.
What is factorization?Factorization in mathematics is the process of dividing a large number into smaller ones that, when multiplied together, give you the original number. Factorization is the process of dividing a number into its components or divisors.
Given:
An expression,
1 - 2.25x⁸.
Simplifying to the factored form.
1² - (1.5x⁴)²
= (1 + 1.5x⁴) (1 - 1.5x⁴)
Therefore, (1 + 1.5x⁴) (1 - 1.5x⁴) is the factored form.
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The ratio of hydrogen atoms to nitrogen atoms in a container is 2:3 what is the value of 2:3
The value of 2:3 can be written as a fraction: 2/3.
What is the ratio?
In mathematics, a ratio is a comparison of two quantities, often expressed as a fraction or using the colon (:) symbol. Ratios are used to indicate the relative size or amount of two or more things.
For example, the ratio of the length of a rectangle to its width can be expressed as a fraction, where the numerator represents the length and the denominator represents the width.
The ratio of hydrogen atoms to nitrogen atoms is 2:3.
This means that for every 2 hydrogen atoms, there are 3 nitrogen atoms in the container. The value of the ratio 2:3 is a way of expressing the relative quantity of hydrogen atoms to nitrogen atoms in a simplified form. It can also be written as a fraction: 2/3.
Hence, the value of 2:3 can be written as a fraction: 2/3.
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Write 2^-x × 3^-x × 6^2x × 3^2x × 2^2x as a power of 6
Answer:
[tex]6 {}^{3x} [/tex]
Step-by-step explanation:
Given expression to us is ,
2^{-x} . 3^{-x} . 6^{2x} . 3^{2x} . 2^{2x}
We can rewrite it as ,
[2^{-x} . 2^{2x} ] [ 3^{-x} . 3^{2x}] . 6^{2x}
we know a^m . a^n = a^{m+n} , so ;
2^{-x +2x} . 3^{-x+2x} . 6^2x
simplify,
2^x . 3^x . 6^{2x}
we know a^m . b^m = (ab)^m , so ;
(2*3)^x . 6^{2x}
6^x . 6^2x
6^{x+2x}
6^{3x}
And we are done!
Answer:
[tex]6^{3x}[/tex]
Step-by-step explanation:
Given exponential expression:
[tex]2^{-x} \cdot 3^{-x} \cdot 6^{2x} \cdot 3^{2x} \cdot 2^{2x}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^n \cdot c^n=(a \cdot c)^n:[/tex]
[tex](2\cdot 3)^{-x} \cdot 6^{2x} \cdot (3 \cdot 2)^{2x}[/tex]
Multiply the numbers:
[tex]6^{-x} \cdot 6^{2x} \cdot 6^{2x}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}:[/tex]
[tex]6^{(-x+2x+2x)}[/tex]
Therefore:
[tex]6^{3x}[/tex]
Use the graphs to evaluate the expressions below
Answer:
5,4,2,0
Step-by-step explanation:
f(g(4))=5
g(f(3))=4
f(f(2))=2
g(g(5))=0
The value of composition function is:
1. f(g(4)) = 5
2. g(f(3)) = 4
3. f(f(2)) = 2
4. g(g(5)) = 0
To find the value of function focus on the graph of the function.
1. f(g(4))
From the graph of g(x), when x= 4 the value of y is 2.
So, g(4) = 2
Now, the value of f(g(4))
f(g(4))= f(2)
= 5.
2. g(f(3))
From the graph of f(x), when x= 3 the value of y is 0.
So, f(3) = 0
Now, the value of g(f(3))
g(f(3))= g(0)
= 4.
3. f(f(2))
From the graph of f(x), when x= 2 the value of y is 5.
so, f(2)= 5
Now, the value of f(f(2))
f(f(2)) = f(5)
= 2
4. g(g(5))
From the graph of g(x), when x= 5 the value of y is 3.
so, g(5)= 3
Now, the value of g(g(5))
g(3) = 0
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Joanna is currently watching 6.4 hours of T.V per day. She made a goal to reduce the amount to 1.6 hours per day . what is the amount now?
Answer: 4.8
Step-by-step explanation:
6.4 - 1.6 = 4.8
The size of fish is very important to commercial fishing. A study conducted in 2012 found the length of Atlantic cod caught in nets in Karlskrona to have a mean of 49.9 cm and a standard deviation of 3.74 cm.
Round the probabilities to four decimal places.
It is possible with rounding for a probability to be 0.0000.
this is a homeeork problem
rv X = the length of a randomly selected Atlantic cod
g) What length do 57% of all Atlantic cod have more than?
Round your answer to two decimal places in the first box.
Put the correct units in the second box.
The z score is -0.18 and 57 % of all Atlantic cod have more than 49.23 cm in length and
What is a z-score?The relationship between a value and the mean of a set of values is expressed numerically by a Z-score. The Z-score is computed using the standard deviations from the mean. A Z-score of zero indicates that the data point's score and the mean score are identical.
The Z-score is calculated using the formula:
z = (x - μ)/σ
where z: standard score
x: observed value
μ: mean of the sample
σ: standard deviation of the sample
Given data ,
Let the probability that a randomly selected Atlantic cod has a length of 53.68 cm or less be represented as P
Now , the equation will be
The observed value of be x
The mean of the sample μ = 49.9 cm
The standard deviation of the sample σ = 3.74 cm
Now , z-score is calculated using the formula:
z = (x - μ)/σ
And , the length of 57% of all Atlantic cod have more than is calculated by
Now , the z score with a p value of 0.43 is
z = -0.18
Substituting the values in the equation , we get
-0.18 = ( x - 49.9 ) / 3.74
Multiply by 3.74 on both sides of the equation , we get
-0.6732 = x - 49.9
Adding 49.9 on both sides of the equation , we get
x = 49.2268 cm
Hence , 57 % of all Atlantic cod have more than 49.2268 cm in length and the z score is -0.18
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Patricia decides to buy a pair of boots for $90.00. She was willing to pay $120.00. When her friend Cecilia sees the boots, she loves them and thinks they are worth $170.00. So she offers Patricia $145.00 for the boots, and Patricia accepts. Patricia and Cecilia are both thrilled with the exchange.
The total surplus received by both Patricia and Cecilia is $
Suppose that Patricia purchased the boots from Marie’s Boutique. Marie and other boutique owners in town are upset that customers like Patricia buy boots at their stores, and then resell them for a higher price. Marie and the other owners convince the city government to pass a law preventing such resale. Assuming the law is successful, how much surplus is lost if Patricia cannot sell the boots to Cecilia?
The total surplus received by both Patricia and Cecilia is $55.
What is Addition?The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.
Given that;
Patricia decides to buy a pair of boots for $90.00. She was willing to pay $120.00.
And, Her friend Cecilia sees the boots, she loves them and thinks they are worth $170.00. So she offers Patricia $145.00 for the boots, and Patricia accepts.
Hence, Consumer surplus is the difference between the willingness to pay of a consumer and the price of the good.
Consumer surplus = willingness to pay – price of the good
Consumer surplus received by Patricia = $120 - $90 = $30
Consumer surplus received by Cecilia = $170 - $145 = $25
Thus, Sum of the surpluses = $30 + $25 = $55
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prove by mathematical induction that the sum of the cubes of the first n positive integers is equal to the square of the sum of these integers
The formula for the sum of cubes of the first n natural numbers is:
(1³+2³+3³+ ... + n³) = (1 + 2 + 3 + ... + n)²
Let us consider:
A(n)=1³+2³+⋯+n³--------(1)
B(n)=(1+2+⋯+n)²---------(2)
Now plug the numbers in the above equations(1)&(2) then,we get:
A(1)=B(1)=1
A(2)=B(2)=9
A(3)=B(3)=36
A(4)=B(4)=100
A(5)=B(5)=225
So both A and B are polynomials of degree at most 4 on n, and they coincide at 5 points.
Hence A(n) = B(n) for all n.
In general, p(n) is a polynomial of degree d on n,
∑kn=1 P(k)
is defined as a polynomial of the degree of d+1 on n. This trick will help solve every single issue involving the summation of polynomials, and then find the value on d+2 points, and we will determine the sum for all n.
To know more about the Sum of cubes of positive integers:
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