Answers
The values of the terms, a₁, a₂, and a₁₀ of the sequence are;
(a) a₁ = 1, a₂ = 1/2, a₁₀ = 1/10
(b) [tex]\lim\limits_{n\to {\infty}}[/tex] aₙ = 0
What is a mathematical sequence?A sequence is an ordered collection of objects such as numbers in which repetition of the object is possible, and the objects (or numbers) are referred to as the terms of the sequence.
The sequence, [tex]\{a_n\}^\infty_{n = 1}[/tex] with values, aₙ = [tex]\int\limits^{\infty}_1 {\frac{dx}{x^{n + 1}} }[/tex]
(a) a₁ = [tex]\int\limits^{\infty}_1 {\frac{dx}{x^{1 + 1}} } = \int\limits^{\infty}_1 {\frac{dx}{x^2} } = -[\frac{1}{x} ]^{\infty}_1[/tex]
[tex]-[\frac{1}{x} ]^{\infty}_1 = -\frac{1}{\infty} - (-\frac{1}{1} ) = 0 + 1 = 1[/tex]
Therefore; a₁ = 1
a₂ = [tex]\int\limits^{\infty}_1 {\frac{dx}{x^{2 + 1}} } = \int\limits^{\infty}_1 {\frac{dx}{x^3} } = -[\frac{1}{2\cdot x^2} ]^{\infty}_1[/tex]
[tex]-[\frac{1}{2\cdot x} ]^{\infty}_1 = -\frac{1}{\infty} - (-\frac{1}{2 \times 1} ) = 0 + \frac{1}{2} = \frac{1}{2}[/tex]
Therefore; a₂ = 1/2
a₁₀ = [tex]\int\limits^{\infty}_1 {\frac{dx}{x^{10 + 1}} } = \int\limits^{\infty}_1 {\frac{dx}{x^{11}} } = [-\frac{1}{10\cdot x^{10}} ]^{\infty}_1[/tex]
[tex][-\frac{1}{10\cdot x^{10}} ]^{\infty}_1 = -\frac{1}{\infty} - (-\frac{1}{10\times 1} ) = 0 + \frac{1}{10} = \frac{1}{10}[/tex]
Therefore; a₁₀ = 1/10
b) The limit, [tex]\lim \limits_{n\to \infty} a_n[/tex] can be evaluated as follows;
aₙ = [tex]\int\limits^{\infty}_1 {\frac{dx}{x^{n+1} } } \, dx[/tex]
[tex]\int\limits^{\infty}_1 {\frac{dx}{x^{n+1} } } \, dx = [-\frac{1}{n\cdot x^n} ]^{\infty}_1 = -\frac{1}{n\times \infty } - (-\frac{1}{n\times 1^n} )= \frac{1}{n}[/tex]
aₙ = 1/n
Therefore;
[tex]\lim \limits_{n\to \infty} a_n = \frac{1}{\infty}[/tex] = 0
[tex]\lim\limits_{n\to {\infty}} a_n[/tex] = 0
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Related Questions
Find 2 numbers whose product is 57, and one of the numbers is 5 less than eight times the other number
Answers
One of the numbers is 21, and the other number is 2.5
What is an algebraic expression?
An algebraic expression is when we combine numbers and words to solve a mathematical problem.
Let's call one of the numbers "x". We know that x is 8 times greater than the other number, which we'll call "y". So we have:
y = x / 8
The product of x and y is 57, so:
x * y = 57
Substituting y = x / 8 into the equation above:
x * (x / 8) = 57
Expanding the right side of the equation:
[tex]x^2 / 8 = 57[/tex]
Multiplying both sides by 8:
[tex]x^2 = 8 * 57[/tex]
Solving for [tex]x^2[/tex]:
[tex]x^2 = 456[/tex]
Taking the square root of both sides:
[tex]x =\sqrt{456}[/tex]
x = 21
So, One of the numbers is 21, and the other number is 21 / 8 = 2.5.
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A cone has a volume of 392π cubic centimeters and a radius of 7 centimeters. What is the height of the cone? Be sure to show all work again (HINT: instead of solving for the radius, you will be solving for the height since you know what the radius measurement already is)!
Answers
The height of the cone is 24 centimetres.
How to find the height of the cone?A cone has a volume of 392π cubic centimetres and a radius of 7 centimetres. Therefore, the height of the cone can be found as follows;
Hence,
volume of a cone = 1 / 3 πr²h
where
h -= height of the coner = radius of the coneTherefore,
volume = 392π
r = 7 cm
392π = 1 / 3 × π × 7² × h
392π = 49 / 3 πh
cross multiply
1176π = 49πh
divide both sides by 49π
h = 1176π / 49π
h = 24 cm
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Which expressions are equivalent to 7^{-2} times 7^{6}
Answers
Answer:
(7^2)^2
Step-by-step explanation:
(7^-2)*(7^6)=7^-2+6
......since the base 7 is the same, when u multiply them, you should add the exponents and keep 7 as it is. That will be 7^4, which in equivalent to ans D(7^2)^2
does anybody know the answer to this?
Answers
Step-by-step explanation:
x=97+40 degree (being sum of two interior angle is equal to one exterior angle)
Choose whether the representation is or is not a function for the problem and select the
reasoning for the answer.
30
8
18
28
38
Y
77
77
77
- 77
Answers
To verify if the representation is a function for this problem, we should verify if each input is mapped to a single output. If there is an input that is mapped to more than one output, then the relation is not a function.
When does a relation represents a function?A relation represents a function when each input value is mapped to a single output value.
On a graph, a function is represented if the graph contains no vertical aligned points, that is, if there are no values of x at which we could trace a vertical line that would cross the graph of the function more than once.
With arrows, only one arrow can depart from each value of x, or more than one arrow departing from the same value of x but necessarily arriving at the same value of y.
Missing InformationThe problem is incomplete, hence the general procedure to verify if a relation represents a function was presented.
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If Tristan's pay was $1600 for the week, he sold ? vacuums
Answers
Answer: $3,000
(There are 7 days in a week) I don't know if i'm correct or not but it should be the answer.
What is equation of the line graphed below. Y= - 5/2 x-2 -this question was accidentally cut out
Answers
Answer: y= -2x+5-1/2
Answer:
y = - [tex]\frac{5}{2}[/tex] x - 2
Step-by-step explanation:
the equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
calculate m using the slope formula
m = [tex]\frac{y_{2}-y_{1} }{x_{2}-x_{1} }[/tex]
with (x₁, y₁ ) = (- 2, 3) and (x₂, y₂ ) = (0, - 2) ← 2 points on the line
m = [tex]\frac{-2-3}{0-(-2)}[/tex] = [tex]\frac{-5}{0+2}[/tex] = - [tex]\frac{5}{2}[/tex]
the line crosses the y- axis at (0, - 2 ) ⇒ c = - 2
y = - [tex]\frac{5}{2}[/tex] x - 2 ← equation of line
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A cone has a volume of 192π cubic inches. Its height is 9 inches, what it's radius? Show all your work.
Answers
Answer:
4.5 inches
Step-by-step explanation:
Geraldine is 4 years younger than Tom. If Tom is t years old, how old is
Geraldine? Also, if Steven is twice as old as Geraldine, how old is he?
Answers
Using equations, we know that the age of Geraldine and Steven is t - 4 and 2t - 8 respectively.
What are equations?It basically consists of a variable, perhaps with an additional numerical constant.
To easily understand this concept, consider the example below. 3x – 4 = 5. It is a straightforward class 7 equation.
The equals sign is a symbol used in mathematical formulas to denote the equality of two expressions.
So, we have the following algebraic expression after converting the word problem:
To Geraldine:
g = t - 4
To Steven:
s = 2g
Inputting Tom's age into equation 1 results in:
g = t - 4
To determine Steven's age:
s = 2(t-4)
s = 2t - 8
Therefore, using equations, we know that the age of Geraldine and Steven is t - 4 and 2t - 8 respectively.
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The area of the triangle is less than 18 square inches. Find the possible
values of a by writing and solving an inequality.
Area of a triangle = (base height)
a
6 in
⚠️HELP IM FAILING MATH⚠️
Answers
Answer:
0 < a < 6
or
(0, 6) in interval notation
Step-by-step explanation:
The area of the triangle is
[tex]Area \;A = \dfrac{1}{2}(base \cdot height)\\[/tex]
We are given base = 6 in, height = a and area A < 18 in²
Using the formula for area we get
[tex]A = (1/2)(6 \cdot a) = 3a[/tex]
We know this area is < 18
So 3a < 18
Dividing both sides by 3 gives us
3a/3 < 18/3
a < 6
So height a must be less than 6 inches.
Height cannot be 0 so the lower bound on height is
a > 0
Putting these together we get the rang of a as
0 < a < 6
In Interval notation this is written as (0, 6)
Hope that helps you. Have confidence in yourself and seek help for your doubts here or elsewhere on the net. You will succeed
HELP!!!!!!!ASAP!!! I NEED TO SUBMIT! WILL GIVE 100 POINTS AND BRAINLIEST!!!!!!!!!!
A group of students was asked about the number of flights they have taken. The data is shown in the line plot. Which of the following best describes the spread of the data? Explain its meaning in this situation. The data has a range of 5, which is a wide spread. This means the greatest number of flights was 5. The data has a range of 15, which is a narrow spread. This means that there is not a large difference in the number of flights taken by the students. The data has a range of 5, which is a narrow spread. This means that most of the students have taken a similar number of flights. The data has a range of 15, which is a wide spread. This means that there is a difference of 15 flights between each of the students.
Answers
Answer:
In the case presented, the range is the difference between the maximum value and the minimum value of the data. The range describes the breadth of the data and how much they vary from the minimum value to the maximum value. Therefore, the range indicates the dispersion of the data.
The statement indicates that the range is 5, which means that the most flights made by students is 5. Since the data has a narrow dispersion, we can conclude that most students have taken a similar number of flights, implying that most students have flown an amount close to the maximum number observed.
Therefore, the option that best describes the dispersion of the data is: "The data has a range of 5, which is a narrow dispersion. This means that most students have taken a similar number of flights."
Step-by-step explanation:
Find a solution to this system of equations:
5x-3y=20
15x+6y=60
Write your solution in (x,y) form.
Please help!!!!!
Answers
Answer:
(x, y) = (4, 0)
Step-by-step explanation:
You want the solution to the system of equations ...
5x -3y = 2015x +6y = 60Standard formIt often works well to start the solution process by putting both equations into standard form. We do that by removing any common factors from the coefficients. Here, we see that the second equation's coefficients each have a factor of 3, so we can remove that and rewrite the system as ...
5x -3y = 205x +2y = 20ObservationChanging the coefficient of the y-term has no effect, telling us that y=0, and the equation reduces to ...
5x = 20
x = 4
More formally, if we subtract the first equation from the second, we get ...
(5x +2y) -(5x -3y) = (20) -(20)
5y = 0 . . . . . . simplify
y = 0 . . . . . . . divide by 5
The solution is (x, y) = (4, 0).
6. Find the measure of an angle if five times the measure of its complement is 24° less than
twice the measure of its supplement.
Equation:
Measure:
Answers
Answer: Let's call the measure of the angle x.
The complement of the angle is (90 - x) degrees, and the supplement is (180 - x) degrees.
So, using the information given in the problem, we can set up an equation:
5 * (90 - x) = 2 * (180 - x) - 24
Expanding both sides and simplifying, we get:
450 - 5x = 360 - 2x - 24
Solving for x, we get:
6x = 294
x = 49
Therefore, the measure of the angle is 49 degrees.
Step-by-step explanation:
solve the system of equations below
Answers
The solution of the system of equations is:
x = 3
y = 2
How to solve the system of equations?Here we want to solve the system of equations below:
2x + y = 8
x - 3y = -3
First, let's isolate one variable in one of the equations, I will isolate y on the first one.
y = 8 - 2x
Now we can replace that in the other one to get:
x - 3*(8 - 2x) = - 3
x - 24 + 6x = -3
7x = -3 + 24
7x = 21
x = 21/7 = 3
And the value of y is:
y = 8 - 2x = 8 - 2*3 = 2
The solution is x = 3 and y = 2.
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On a recent trip, Jeremy and Frank drove
790 miles on
33}, gallons of gas. How
many miles per gallon did their
car get
on this trip?
Answers
Answer: Their car got an average of 23.6 miles per gallon.
BTW I just searched it up so I'm sorry if it is wrong.
A fitness center has several rectangular rooms where exercise classes are held. The area of Room A is 27x−18 square units. What are possible dimensions of Room A?
Answers
The possible dimensions of rectangular room A are 9 and 3x - 2.
What is rectangular shape?A quadrilateral with parallel sides that are equal to one another and four equal vertices is known as a rectangle.
Given:
A fitness center has several rectangular rooms where exercise classes are held.
The area of Room A is 27x − 18 square units.
The area of the rectangle = length x width
27x − 18 = length x width
9 (3x - 2) = length x width.
Therefore, 9 and 3x - 2 are the dimensions.
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Given the point g' at (4,6) and the rule (x,y) > > > (x-7,y+3). Find the coordinates of g.
Answers
The new co-ordinates of the point after transformation is (-3,9)
what is the Cartesian coordinate system?A Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates,
Given here: The point g'(4,6)
The point transforms to (x-7,y+3)
Thus the new points are 4⇒4-7=-3
6⇒6+3=9
Thus, The new co-ordinates of the point after transformation is (-3,9)
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Find the roots of the function f(x)=x^2+15x-16
Answers
Answer:
x = -16 and x = 1
Step-by-step explanation:
"Finding the roots" means solving the equation. We'll set the equation equal to 0 (replace f(x) with zero) and factor.
0 = x^2 + 15x - 16
0 = (x + 16)(x - 1)
This is factored.
Using the Zero Product Property, we separate this into two small equations.
x+16 = 0 and x-1 = 0
Solve these one-step equations.
x = -16 and x = 1
These are the roots (and roots are solutions, are zeros, are x-intercepts) If the question ask for zeros, roots, solutions or x-intercepts, set it equal to 0, factor and solve.
In a survey of 500 tourists, 350 like Pokhara, 200 like Chitwan and number of tourist who like both is half of who like Chitwan and 20 like neither of two, using Venn-diagram find how many like only one place.
Answers
Answer:
160 tourists like only one place. This can be determined by using a Venn diagram. Of the 500 tourists surveyed, 350 like Pokhara, 200 like Chitwan, and 50 like both places. That leaves 20 who like neither of the two places. To get the number of tourists that like only one place, subtract the number of tourists who like both (50) and the number of tourists who like neither (20) from the total number of tourists surveyed (500), resulting in a total of 160 tourists who like only one place.
The bookstore had 56 copies of magazine yesterday and sold 1/4 of them today sold 4/7 of what remain how many copies does the book store have 
Answers
Step-by-step explanation:
Find 1/4 of 56
56 / 4 = 14
Now
56-14=42
Now
42/7=6
6x4=24
42-24=18
18 copies left
Tito purchased 11 tickets to a baseball game for $374. About how much did one ticket cost?
Answers
Answer: It is 34$ per one ticket
Step-by-step explanation:
$374 divided by 11 is 34
Answer:
$34 per ticket
Step-by-step explanation:
We were told $374/11tickets.
We are looking for how much per 1 ticket.
Divide to find the per-ticket cost.
Use either 374÷11
374/11 .
374÷11 = 34
This means $34 per ticket.
Answer. please.........
Answers
The area of the given parallelogram is 6.6 square units.
What is the area of the parallelogram?The region or space occupied by a parallelogram in a two-dimensional plane is referred to as the area of a parallelogram. A unique category of a quadrilateral is a parallelogram.
A quadrilateral is referred to as a parallelogram if it has two sets of parallel opposite sides. A parallelogram can be anything from a rectangle to a square to a rhombus.
The area of the partogram is calculated as:-
Area = Base x Height
Area = 2.2 x 3
Area = 6.6 square units
Therefore, the parallelogram will have an area of 6.6 square units.
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ABCD is a quadrilateral.if AL is perpendicular to BD and CM is perpendicular to BD,prove that : ar ( quad.ABCD) = 1/2 ×BD×(AL +CM )
Answers
Using the Pythagorean theorem, if AL is perpendicular to BD and CM is perpendicular to BD, then quadrilateral ABCD is a parallelogram.
How can we prove this?In a parallelogram, the opposite sides are equal in length, i.e. AL = CD and BM = AD.
By the Pythagorean theorem, we know that the length of the diagonal BD can be found as the square root of the sum of the squares of AL and CM:
BD = √(AL^2 + CM^2)
Since AL = CD, the area of quadrilateral ABCD can be found as:
ar (quad.ABCD) = AL × BD
Substituting BD = √(AL^2 + CM^2) and combining the two equal length sides, we get:
ar (quad.ABCD) = (AL + CM) × √(AL^2 + CM^2) / 2
Finally, dividing both sides by √(AL^2 + CM^2) and using the fact that BD = √(AL^2 + CM^2), we get:
ar (quad.ABCD) = 1/2 × BD × (AL + CM)
Thus, we have proved that the area of quadrilateral ABCD is equal to 1/2 × BD × (AL + CM).
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Write the inequality for the graph below. (DO NOT put any spaces between the number, symbols, and letters.)
Answers
First you find the slope which is rise/run which in this problem is -1/1 or -1 then you find the y value in which y=8 and you plug into equation
Answer:
y = -x +8
Step-by-step explanation:
The line crosses the x axis at 8. So we know when y = 0 then x = 8
The line cross the y axis at 8. so we know when x = 0 then y = 8
the equation of a line we know to take the form y = mx + c
When x = 0 and y = 8, and y = mx+ c then then c =8
when y = 0 x = 8. y = mx + c, 0 = m(8) +8. for this to be true m must be equal to -1
so the equation of the line, the inequality is y = -x +8
forty two out of fifty six students voted Michael for president. What fraction of the workers did not vote for him?
Answers
Answer: [tex]\frac{1}{4}[/tex]
Step-by-step explanation:
1) Subtract Michael's voters from all the voters 56/56 - 42/56. When you have the same denominator, subtract the numerator and keep the denominator the same to get 14/56.
2) Simplify, both the numerator and denominator is divisible by 14, so divide 14/14 to get 1 as our new numerator and divide 56/14 to get 4 as our denominator leading us to our answer of 1/4
In a bag of 10 marbles, there are 4 blue, 3 red, 2 green, and 1 yellow. What is the probability that you draw one marble that is blue, replace it, and draw another marble that is green? Enter your answer as a fraction in lowest terms. Do not add spaces to your answer. (EX: 1/2)
Answers
Work Shown:
A = the 1st marble is blue
B = the 2nd marble is green
Replacement is applied. This makes A and B independent.
P(A) = (4 blue)/(10 total) = 4/10 = 2/5
P(B) = (2 green)/(10 total) = 2/10 = 1/5
P(A and B) = P(A)*P(B) ... see note below
P(A and B) = (2/5)*(1/5)
P(A and B) = 2/25
Note: The equation is valid because events A and B are independent.
once again im tired and i dont know how to do this
Answers
First option is correct
If these two triangles are similar, find the value of x
Answers
In exercise 11-14 find the value of x for each one thank you whoever gets all of it right gets brainliest photo is below please help I have one day to turn this in or I fail
Answers
Answer:
For exercise 11: x = 64
For exercise 12: x = 66
For exercise 13: x = 89
For exercise 14: x = 99
Step-by-step explanation:
The sum of interior angles of an n-sided polygon = [tex](n- 2 )[/tex] x [tex]180^{o}[/tex]
where n = number of sides
For all the four figures in this exercise, n = 4
Therefore, Sum of interior angles = [tex](4 - 2)[/tex] x [tex]180^{o}[/tex]
= 2 x [tex]180^{o}[/tex]
= [tex]360^{o}[/tex]
For Exercise 11:
[tex]x^{o} + 100^{o} + 130^{o} + 66^{o} = 360^{o}[/tex]
[tex]x^{o} + 296^{o} = 360^{o}[/tex]
[tex]x = 360 - 296[/tex]
x = [tex]64^{o}[/tex]
Apply the above stated concept in the other three exercises
Which number has a digit that is 10x greater
than 70?
a) 3,607
b) 7,391
c) 1,738
d) 7,246
Answers
Answer:
C
Step-by-step explanation:
Look at 70. What digit is it in? The tens.x10 moves our digit once to the left (70 => 700) AKA 7 hundreds!Look at the answers.3607 ( "7" ones)
7391 ( "7" thousands)
1738 ( "7" hundreds)
7246 ( "7" thousands)