if n is a positive integer: 1²+ 2² + 3² + ... +n² = 1/6n(n+1)(2n+1) IS 1
What is meant by integer?An integer, which is pronounced "IN-tuh-jer," is a whole number that can be positive, negative, or zero and is not a fraction. Examples of integers include: -5, 1, 5, 8, 97, and 3,043. 1.43, 1 3/4, 3.14, and other numbers that are not integers are examples. The set of whole numbers and their antipodes is known as the integers. The set of integers excludes fractions and decimals. For instance, 2,5,0,12,244,15, and 8 are all numbers. Any positive or negative number that does not contain fractions or decimal places is known as an integer, often known as a "round number" or "whole number." For instance, the numbers 3, -10, and 1,025 are all integers, whereas the numbers 2.76 (decimal), 1.5 (decimal), and 3 12 (fraction) are not.P(n) :
1²+ 2² + 3² + … +n² = 1/6n(n+1)(2n+1)
P(1) :
[tex]1^2=\frac{1(1+1)(2(1)+1)}{6}[/tex]
[tex]1=\frac{6}{6}=1[/tex]
∴ LHS = RHS
Assume P(k) is true
P(k):
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L)
express the given in terms os sin x or cos x
sin ( 4pi - x )
how
The value of sin(4π-x) is -sinx
What are trigonometric functions?The trigonometric functions are real functions which relate an angle of a right-angled triangle to ratios of two side lengths.
Given that, express the given in terms of sinx or cosx sin (4π-x)
sin(4π-x) = sin(2π+2π-x)
We know, that, sin(a+b) = sina·cosb+sinb·cosa
Here, a = 2π and b = 2π-x
Therefore,
sin(2π+2π-x) = sin2πcos(2π-x)+sin(2π-x)cos2π
Now, sin2π = 0 and cos(2π) = 1
Therefore,
sin2πcos(2π-x)+sin(2π-x)cos2π
= sin(2π-x)
As we know that, sin(2π-x) = -sinx
Hence, sin(4π-x) = -sinx
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difference between -0.4 and 44.4
Answer:
44.8
Step-by-step explanation:
44.4 - (-0.4) = 44.8
Plz help this is due today!!!
Answer: 636.28
Step-by-step explanation:
first calculate the area of the square and the semicircle after u find the area of semicircle, multiply by four and add it to the area of the square. formula for area of semicircle = [tex]\frac{\pi r^2 }2\\[/tex]
formula of area of square [tex]side^2[/tex]
Please Help?
The amount of rain 90 minutes after it started raining was the same as the amount 120 minutes after it started raining.
Answer:
ok, the correct answer should be R(1.5)=R(2)!
Step-by-step explanation:
I did this my class before!
A fifth grader takes a standardized achievement test (mean = 125, standard deviation = 15) and
scores a 148. What percent of children made higher than a 148?
the child's percentile rank is 93 th percentile. It means he performed better than [tex]$93 \%$[/tex]of the total number of students.
What is Percentile rank?When compared to other students in the same grade and topic, for instance, a student's percentile rank shows how well they performed. The percentile rank of a student indicates whether their score was on par with or higher than that of the norm group's percentage of pupils. In statistics, the percent of scores in a frequency distribution that are lower than a specific score is known as the percentile rank of that score. A percentile rank returns a numerical value, which really indicates the proportion of people who fall inside that range. Since it is a percentile, its range is always between 1 and 99, with 50 serving as its mean or average rank. Only normalized values or data are ever usable in this fashion.- Mean, $\boldsymbol{\mu}=125$.- Population standard deviation, $\sigma=15$.- Marks obtained by child $(X)=148$Need to find the percentile rank for child's score.
Let[tex]$\mathrm{X}$[/tex]denote marks obtained by the child in the achievement test.
The [tex]$\mathrm{Z}$[/tex]value is given as,
[tex]$$Z=\frac{X-\mu}{\sigma}$$[/tex]
Using equation (1), the required percentile rank for child's score is expressed as,
[tex]& Z=\frac{148-125}{15} \\[/tex]
[tex]& Z=1.533[/tex]
The probability value corresponding to the Z=1.533, obtained from the Z table is 0.93736.
Thus, the percentile is 0.93736.
Therefore, the child's percentile rank is 93 th percentile. It means he performed better than [tex]$93 \%$[/tex]of the total number of students.
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Find the area of the largest rectangle that can be inscribed in the ellipse x2 / a2 + y2 / b2 = 1.
The maximum area of the rectangle inside the ellipse is 2ab.
Given that,
Ellipse = x2 / a2 + y2 / b2 = 1
we know the general coordinates of any point in an ellipse of the equation
x2 / a2 + y2 / b2 = 1 is (acosθ,bsinθ)
So, we take general vertices of rectangles as
(acosθ,bsinθ), (acosθ,-bsinθ), (-acosθ,bsinθ), and (-acosθ,-bsinθ)
So, we can see that length and breadth of the rectangle is 2acosθ and 2bsinθ
So, its area will be ab4sinθcosθ (Length x Breadth)
Applying trigonometric formula 2sinθcosθ=sin2θ, let area equals to A
Now A=2absin2θ
Now as the area is the maximum given in the question, we have to differentiate it with respect to θ
dA/dθ=2abcos2θ×2 (using dsin2θ / dθ=2cos2θ)
Equating dA / dθ=2abcos2θ×2=0, it means cos2θ=0 , which means 2θ = π / 2 and θ=π / 4
So now we to put θ=π / 4 in the equation of area which is A=2absin2θ
We get as A=2absin2π / 4=2absinπ / 2=2ab
Hence the max area of the rectangle inside the ellipse is 2ab.
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Which of the following increases at the fastest rate for larger values of x?
a) f(x)=6^x+1
b) f(x)=6x^2+1
c) f(x)=6x+1
d) f(x)=6x^3+1
Answer:
Option d
d) f(x)=6x^3+1
Step-by-step explanation:
Given data
a) f(x)=6^x+1
b) f(x)=6x^2+1
c) f(x)=6x+1
d) f(x)=6x^3+1
In all option d is the function that increases at the fastest rate, this is because the power of x is the highest
Answer:
Option d
d) f(x)=6x^3+1
Step-by-step explanation:
Given data
a) f(x)=6^x+1
b) f(x)=6x^2+1
c) f(x)=6x+1
d) f(x)=6x^3+1
In all option d is the function that increases at the fastest rate, this is because the power of x is the highest
Step-by-step explanation:
help please give me the answer and no links please
Answer:
x= 69°
Step-by-step explanation:
First find the inner angle of the 159°
so you would do 180-159=21
next you add 21 ° to the 90° from the right angle
that gives you 111°
last you do 180-111 and you get 69
Answer:
x = 69
Step-by-step explanation:
180 - 159 = 21
The sum of the three angles of any triangle is equal to 180 degrees, so:
21 + 90 + x = 180
111 + x = 180
180 - 111 = x
x = 69
Have a nice day!
Points p and q are negative integers plotted on a number line. If |p| > |q|, which statement is true?
A. Point p is closer to 0 than point q, and to the right of point q.
B. Point p is closer to 0 than point q, and to the left of point q.
C. Point p is further from 0 than point q, and to the right of point q.
D. Point p is further from 0 than point q, and to the left of point q.
Point p is further from 0 than point q, and to the left of point q.
The correct option is D.
What is an absolute value?The absolute value of a number is the distance value from zero without considering the sign of a number. It is also represented as |x|, where x is any number.
Given:
Points p and q are negative integers plotted on a number line.
And |p| > |q|.
That means,
The absolute value of p is greater than the absolute value of q.
|p| and |q| are the positive integers.
So, p < q.
So, the q is closer to zero.
And p is left to point q.
Therefore, the q is closer to zero and p is left to point q.
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Claire keeps track of the miles per gallon her car gets per week. She has accumulated the following data: (1, 22.42), (2, 22.84), (3, 23.26), (4, 23.68) What is the common difference or ratio
The common difference is 0.42. The comparison of each ratio's successive values over time for a single firm.
Now, According to the question:
What is meant by ratio?
When expressed in numbers or amounts, a ratio is a relationship between two things. For example, if a room contains ten boys and thirty girls, the boy-to-girl ratio is 1:3, or one to three.A ratio, or getting rationed, on the occurs when replies to a tweet vastly outnumber likes or retweets. This means that people are reacting negatively to the tweet's content.It usually means that your comment has received more replies than likes. This implies that there are more people who disagree with your comment than agree with it.Therefore, simplifying the equation then we get,
if you take the ranges (y) and subtract one from the other you get 0.42.
(22.84 - 22.42)/(2 - 1) = 0.42/1 = 0.42
(23.26 - 22.84)/(3 - 2) = 0.42
(23.68 - 23.26)/(4 - 3) = 0.42
Hence, The common difference is 0.42
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WILL GIVE BRAINLIEST!!!!
How are you doing today?
(This is for a survey for a class) We are learning percentages.
Answer: Hope This Helps!
Step-by-step explanation:
Im doing good today. And I hope you have a good rest of the day!
What is the percent of change if the price changes from $150 to $100? Round to the nearest whole percent if necessary.
Answer:
Percentual change= 33.33% decrease
Step-by-step explanation:
Giving the following information:
Present value (PV)= $150
Future value (FV)= $100
To calculate the percentual change, we need to use the following formula:
Percentual change= [(PV - FV) / PV]*100
Percentual change= [(150 - 100) / 150]*1''
Percentual change= 0.3333*100
Percentual change= 33.33% decrease
The half-life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801kg. Write an exponential function that models the
decay of the material.
Answer:
m(t) = 801 * (1/2)^(t/85)
Step-by-step explanation:
m(t) = 801 * (1/2)^(t/85)
m(t) is the mass after t days
Find the VOLUME of the figure below
Answer: 12 yd^3
Step-by-step explanation:
(base area x height)/3
1/2 x base x height = base
1/2 x 4 x 3 = 6
36/3 = 12yd^3
For Drew’s week 1 classes identify the domain and range is the relation a function? Explain
From the table for Drew's week 1 classes:
Domain = {English, Math, History, Biology, Biology Lab}
Range = {60. 90, 45, 0}
What is domain and range?A function's domain is the set of values that can be plugged into it. This set contains the x values in a function such as f. (x). A function's range is the set of values that it can take. This is the set of values that the function returns when we enter an x value. They are the y values.
A function can be thought of as a machine on an assembly line. On one end of the assembly line, there are a few screws and bolts, and on the other end, there is a car. The function is represented by the machine in the middle. The domain consists of the screws and bolts that we insert into the machine (our function).
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helppppppppppppp pleaseeeee
Answer:
e = 54
f = 20
d = 106
Step-by-step explanation:
Opposite angles are the same. Also, straight lines always equal 180 degrees.
Hope that this helps!
expand this expression using the distributive property. 4(-2g+4-3t)
Answer:
-8g+16-12t is the answer
Step-by-step explanation:
multiply everything by 4
Mrs. Knight is saving money for a trip. She already has some money saved in 1 point
her savings account and will add the same amount to her account each
week. At the end of 2 weeks, she has $130. At the end of 8 weeks, she has
$280. Write a linear function to represent the amount of money (y)
Mrs. Knight has saved after x weeks
The equation of line is given by y = 25x + 80.
What is an equation of a line?The equation of a line is expressed as y = mx + b where m exists the slope and b exists the y-intercept
y - y₁ = m ( x - x₁ )
Where, y be the y-coordinate of second point
y₁ be y-coordinate of point one
m be the slope
x be the x-coordinate of second point
x₁ be the x-coordinate of point one
Let the equation of slope m be
m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Equation of line be represented as A
The two points are given as A ( 2 , 130 ) and B ( 8 , 280 )
The slope of the line between the two points exists given by
Slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Substituting the values in the equation , we get
Slope m = ( 280 - 130 ) / ( 8 - 2 )
simplifying the above equation, we get
m = 150 / 6
m = 25
Now , the equation of line is given by
y - y₁ = m ( x - x₁ )
Substituting the values in the equation , we get
y - 130 = 25 ( x - 2 )
On simplifying the equation , we get
y - 130 = 25x - 50
Adding 130 on both sides , we get
y = 25x + 80
The equation is of the form y = mx + b where b is the y intercept.
Therefore, the equation of line is given by y = 25x + 80
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The equation of line is given by y = 25x + 80, this represent the linear function between the amount of money and the saved amount.
What is an equation of a line?The equation of a line is expressed as y = mx + b where m is called the slope and b is called the y-intercept
y - y₁ = m ( x - x₁ )
Where, y be the y-coordinate of second point
y₁ be y-coordinate of point one
m be the slope
x be the x-coordinate of second point
x₁ be the x-coordinate of point one
Let the equation of slope m be
m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Equation of line be represented as A
The two points are given as A ( 2 , 130 ) and B ( 8 , 280 )
The slope of the line between the two points exists given by
Slope m = ( y₂ - y₁ ) / ( x₂ - x₁ )
Substituting the values in the equation , we get
Slope m = ( 280 - 130 ) / ( 8 - 2 )
simplifying the above equation, we get
m = 150 / 6
m = 25
Now , the equation of line is given by
y - y₁ = m ( x - x₁ )
Substituting the values in the equation , we get
y - 130 = 25 ( x - 2 )
On simplifying the equation , we get
y - 130 = 25x - 50
Adding 130 on both sides , we get
y = 25x + 80
The equation is of the form y = mx + b where b is the y intercept.
Therefore, the equation of line is given by y = 25x + 80
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Find the gradients of lines A and B.
Answer: The Gradient of line A and B are
2 and -1
Step-by-step explanation:
For the given two points A(x1, y1) and B(x2, y2)
The gradient of the line AB is y2-y1/x2-x1
So the gradient of line A is
5-1/2-0
2
the gradient of line B is
5-0/0-5
-1
A blood bank needs 13 people to help with a blood drive. 20 people have volunteered. Find how many different groups of 13 can be formed from the 20 volunteers.plsss help I'm stuck
Answer:
20 C 13 = 77520
Step-by-step explanation:
That's the answer that it told me on delta math.
Please help!
ponential functions are in the form y = a(b)'.
What values of b make it an exponential growth function?
What values of make b it an exponential decay function?
How can you solve inequalities explain and give an example?
To Solve the inequalities , we take the variables on a side and the numbers on the other side and then apply mathematical operations to solve it .
What is Inequality ?
The Inequality is defined as a relationship between two values that are not equal to each other .
The method of solving the inequalities is a little different than solving equations.
To solve the inequalities, some simple rules should be followed .
(i) The inequality does not change if any number or variable is added or subtracted from both sides.
Similarly, the value of inequality is not affected if the same number is multiplied on both the sides.
(ii) If the inequality is divided by a negative variable or number on both sides, then the inequality sign gets reversed.
(iii) On dividing inequality by a positive number, there is no change.
For Example : Solve the inequality [tex]4x+6 > 3x+7[/tex] ;
subtracting [tex]3x[/tex] from both the sides ,
we get ;
[tex]x+6 > 7[/tex],
Now subtracting [tex]6[/tex] from both sides of the inequality;
we get ;
[tex]x > 1[/tex] .
Therefore , the solution of the inequality [tex]4x+6 > 3x+7[/tex] is [tex]x > 1[/tex] .
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PLZ HELP ME QUICK!!
Does the data provide convincing statistical evidence that at women pay more than men for the same catalytic converter repair?
Answer:
Yes the data provide convincing statistical evidence that at women pay more than men for the same catalytic converter repair
Which function has zeros at x=23 and x=−4?
Answer:
g(x) = a(x - 23)(x + 4)
Step-by-step explanation:
Weren't there possible answer choices to share?
if this function has zeros at 23 and -4, then the function has the following form:
g(x) = a(x - 23)(x + 4)
A laboratory uses liquid nitrogen tanks of two different sizes. The combined volume of 3 large tanks and 2 small tanks is 24 liters. The combined volume of 2 large tanks and 3 small tanks is 21 liters. What is the volume of each size of tank? JUSTIFY YOUR ANSWER.
The size of the large tank is 6 liters and the size of the small tank is 3 liters.
What is the size of each type of tank?The first step is to form a system of equations using the information provided in the question:
3l + 2s = 24 equation 1
2l + 3s = 21 equation 2
Where:
l = volume of the large tank s = volume of the small tankThe elimination method would be used to solve the equations:
Multiply equation 1 by 2 and equation 2 by 3:
6l + 4s = 48 equation 3
6l + 9s = 63 equation 4
Subtract equation 3 from equation 4:
5s = 15
Divide both sides of the equation by 5
s = 15 / 5
s = 3 liters
Substitute for s in equation 1 :
3l + 2(3) = 24
3l + 6 = 24
3l = 24 - 6
3l = 18
l = 18 / 3
l = 6
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Which is equal to 100%?
A. 0.001
B. 1/100
c. 0.1
D 1/10
Need ASAP
Answer:
1/100
or we can say 100/1
What must be added to x³ 3x² 12x 19?
The expression x3 3x2 12x 19 is a polynomial expression in standard form.
To add 24 to this expression, we need to add 24 to each of the terms. This means that the answer is x3 + 3x2 + 12x + 43. To check our work, we can use theFOIL method to expand the expression and make sure that the coefficients of each of the terms are correct. The FOIL method stands for First, Outer, Inner, Last. The first and last terms are the x3 and 43, the outer terms are 3x2 and 12x, and the inner terms are 3x2 and 12x. When we multiply these terms, we get x3 + 3x2 + 12x + 43, which is our final answer.
Calculation
x3 + 3x2 + 12x + (19 + 24)
= x3 + 3x2 + 12x + 43
The expression x3 3x2 12x 19 is a polynomial expression in standard form.
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Line 1 has slope 2 and goes through (1,3). Which is one form of the equation for line ??
Step-by-step explanation:
y = 2x + b ---> 3 = 2(1) + b ---> b = 1
Therefore, y = 2x + 1
On Friday, 134 MP3s were downloaded from an online music store. Of the 73 country songs. 52 were downloaded by women. Of the 61 alternative
songs. 47 were downloaded by men.
a. Make a two-way frequency table for the data.
b. What is the joint relative frequency of men who downloaded a country song?
c. What is the marginal relative frequency of people who downloaded an alternative song?
Answer:
13
Step-by-step explanation:
2x³ +x² -25x +12 factorised
The correct solutions for "x" after factorizing the cubic equation are:
x1 = -4
x2 = 0.5
x3 = 3
Cubic Equation:
A cubic polynomial in mathematics is a polynomial whose highest degree is three. A cubic equation is one that contains a cubic polynomial.
Either one real root or three real roots are present in every cubic equation. The cubic equation is written as ax^3+bx^2+cx+d=0.
Therefore, standard form of the cubic equation is: ax^3 + bx^2 + cx + d = 0
Given: 2x³ +x² -25x +12
On comparing with the standard form of the cubic equation, we get;
a=2
b=1
c=-25
d=12
After factorizing the cubic equation, we have three roots.
x1 = -4
x2 = 0.5
x3 = 3
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