Answers
The sequence converges to 5 as n goes to ∞. We can prove this rigorously:
The claim is that, for [tex]n\in\Bbb N[/tex],
[tex]\displaystyle \lim_{n\to\infty} \frac{5n-1}{n+1} = 5[/tex]
By definition of the limit, this means for any ε > 0, there exists some N such that for any n larger than N, we have
[tex]\left| \dfrac{5n-1}{n+1} - 5 \right| < \varepsilon[/tex]
In order to ensure this inequality, we work backwards to find N in terms of ε :
[tex]\left| \dfrac{5n-1}{n+1} - 5 \right| = \left| \dfrac{5n-1 - 5(n+1)}{n+1} \right| = \dfrac4{|n+1|} < \varepsilon[/tex]
n is a natural number, so n + 1 > 0 and |n + 1| = n + 1. Then
[tex]\dfrac4{n+1} < \varepsilon \implies \dfrac{n+1}4 > \dfrac1\varepsilon \implies n+1 > \dfrac4\varepsilon \implies n > \dfrac4\varepsilon - 1[/tex]
Let N be the smallest natural number greater than or equal to 4/ε - 1, i.e.
[tex]N = \left\lceil \dfrac4\varepsilon - 1 \right\rceil[/tex]
Then the proof follows: suppose
[tex]n \ge N = \left\lceil \dfrac4\varepsilon - 1 \right\rceil[/tex]
Then
[tex]n \ge \left\lceil\dfrac4\varepsilon - 1\right\rceil \implies n > \dfrac4\varepsilon - 1 \\\\ \implies n+1 > \dfrac4\varepsilon \\\\ \implies \dfrac4{n+1} < \varepsilon \\\\\implies \left|\dfrac{5n-1}{n+1}-5\right| < \varepsilon[/tex]
as required.
Or we can avoid the rigor and use some facts we know about limits:
[tex]\displaystyle \lim_{n\to\infty} \frac{5n-1}{n+1} = \lim_{n\to\infty} \frac{5-\frac1n}{1+\frac1n} = 5[/tex]
since 1/n converges to 0 as n goes to ∞.
Related Questions
A new coffee shop can hold no more than 50 seats. The owner wants at least 20 of the seats to be stools and the remaining seats to be recliners. If x is the number of stools and y is the number of recliners, which graph represents the solution to the system of inequalities?
x + y ≤ 50
x ≥ 20
On a coordinate plane, 2 solid straight lines are shown. The first line is horizontal to the y-axis at y = 20. Everything below the line is shaded. The second line has a negative slope and goes through (0, 50) and (50, 0). Everything to the left of the line is shaded.
On a coordinate plane, 2 solid straight lines are shown. The first line is vertical to the x-axis at x = 20. Everything to the left of the line is shaded. The second line has a negative slope and goes through (0, 50) and (50, 0). Everything to the right of the line is shaded.
On a coordinate plane, 2 solid straight lines are shown. The first line is horizontal to the y-axis at y = 20. Everything above the line is shaded. The second line has a negative slope and goes through (0, 50) and (50, 0). Everything to the left of the line is shaded.
Answers
Answer: On a coordinate plane, 2 solid straight lines are shown. The first line is vertical to the x-axis at x = 20. Everything to the right of the line is shaded. The second line has a negative slope and goes through (0, 50) and (50, 0). Everything to the right of the line is shaded.
Step-by-step explanation:
x = # of stools = 20
y = # of recliners = 50-20 = 30
Answer:
On a coordinate plane, 2 solid straight lines are shown. The first line is vertical to the x-axis at x = 20. Everything to the right of the line is shaded. The second line has a negative slope and goes through (0, 50) and (50, 0). Everything to the left of the line is shaded.
this is from 7th grade
Answers
Answer:
3, 4
Step-by-step explanation:
The expanded form of the given product is found using the distributive property. An equivalent form can be found using the commutative property of multiplication.
__
7(2 -3n) = 7(2) +7(-3n) = 14 -21n . . . . . . . matches 3
Swapping the factors gives ...
7(2 -3n) = (2 -3n)·7 . . . . . . . . . . matches 4
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About the other choices
1. The value is not the same as 14 -21n. It appears to have been the result of (more or less) adding 7 to the coefficients inside parentheses.
2. It is a common mistake to eliminate parentheses after applying the outside factor to the first term only. That seems to be what happened here. The distributive property says you apply the factor to all the terms inside parentheses.
5. The sum of terms (2 -3n) has been incorrectly replaced by the product of terms (2)(-3n).
which statement is true about the result considering a significance level of 5%?
oA. the result is not statistically significant, which implies that this results could me due to random chance.
oB. The result is statistically significant which implies that spraying the corn plants with the new type of fertilizer does not increase the growth rate
oC. The result is not statistically significant which implies that spraying the corn plants with the new type of fertilizer does increase the growth rate
oD. The result is statistically certificate which implies that spraying the corn plants with the new type of fertiliter does increase the growth rate
Answers
If the significance level is about 5% then the result is not statistically significant; which implies that spraying the corn plants with the new type of fertilizer does increase the growth rate.
What is the meaning of significance level?The term significance level has to do with the probability of rejecting the null hypothesis even though it is true.
The significance level tells us if a result is statistically significant. Thus, if the significance level is about 5% then the result is not statistically significant; which implies that spraying the corn plants with the new type of fertilizer does increase the growth rate.'
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A)The result is statistically significant, which implies that spraying the corn plants with the new type of fertilizer does increase the growth rate.
Did the project
73-74 express the limit as a definite integral.
73. [tex]\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{i^{4}}{n^{5}}[/tex] [hint: consider [tex]$f\mleft(x\mright)=x^4$.\rbrack[/tex]
Answers
Pull out a factor of 1/n from the summand and the result follows.
[tex]\displaystyle \lim_{n\to\infty} \sum_{i=1}^n \frac{i^4}{n^5} = \lim_{n\to\infty} \frac1n \sum_{i=1}^n \left(\frac in\right)^4 = \boxed{\int_0^1 x^4 \, dx}[/tex]
find the perimeter and the area of the figure and round answers to the nearest tenth
Answers
Answer:
Perimeter = 55.1 ftArea = 236.6 ft²Step-by-step explanation:
Perimeter
2(4) + 2 x 3.14 x 7.58 + 15 x 3.148 + 47.155.1 ftArea
15 x 4 + 3.14 x (7.5)²60 + 3.14 x 56.2560 + 176.625236.6 ft²Using the digits 0-9, no more than once, complete the puzzle so that the sum of each side is equivalent.
Answers
Answer:
[tex]\begin{matrix}8&2&7\\ 4&&1\\ 5&3&9\end{matrix}[/tex]
Step-by-step explanation:
[tex]\begin{matrix}a&e&b\\ f&&g\\ c&h&d\end{matrix}[/tex]
From the table,we can extract these equations:
a + e + b = a + c + f = b + g + d = c + h + d
a + e + b = a + c + f ⇒ e + b = c + f
b + g + d = c + h + d ⇒ b + g = c + h
[tex]\begin{Bmatrix}e+b=c+f\\ b+g=c+h\end{Bmatrix} \Longrightarrow g-e=h-f\Longrightarrow g+f=h+e[/tex]
2 + 3 = 4 + 1 then let’s consider :
e = 2 ; h = 3 ; f = 4 ; g = 1 ,which satisfies g + f = h + e
The table becomes:
[tex]\begin{matrix}a&2&b\\ 4&&1\\ c&3&d\end{matrix}[/tex]
From the equation b + e = c + f we get b + 2 = c + 4 then b = c + 2
If we consider c = 5 ⇒ b = c + 2 = 5 + 2 = 7
Then the table becomes
[tex]\begin{matrix}a&2&7\\ 4&&1\\ 5&3&d\end{matrix}[/tex]
Then a + 9 = d + 8 ⇒ d = a + 1
If we consider a = 8 ⇒ d = 9
[tex]\begin{matrix}8&2&7\\ 4&&1\\ 5&3&9\end{matrix}[/tex]
I know it’s not perfect reasoning ,but it may help.
A student collecting money for a charity is donating $12 of his own money. He asks friends to give $5 each. Write a rule in function notation to find how much he will donate in all. Let p represent then number of friends who give money.
a. f(p) = 12p + 5
b. f(p) = 12 + 5p
c. f(p) = 12 + 5 + p
d. (p) = 17p
Answers
(8 x 3) x 6 = 8 x (3 x 6) is a true statement. Which multiplication property is this an example of?
Answers
Associative property
Step-by-step explanation:Multiplication properties are different statements that are always true. These statements help prove that different operations can be performed on equations and still remain equal.
Associative Property
The associative property states that (a x b) x c = a x (b x c). This means that in any equation where only multiplication is present, the parentheses can be moved and the equation remains equal.
Other Examples
The associative property is always true; this means that we can create other examples.
Let's take the expression 4 x (2 x 9). As the property states, we can move the parentheses and create the new expression (4 x 2) x 9. Now we can set these expressions equal to each other.
4 x (2 x 9) = (4 x 2) x 9Due to the property we know that this equation must be true.
We can prove this property by working out each side of the equation. If we multiple both sides out we are left with 72 = 72. Since this is true the equation above must also be true.
Answer:
Step-by-step explanation:
It's called The Associative Property of Multiplication
(a * b) * c = a * (b * c)
8 There are a combined 33 students who play a spring sport of track or
baseball. There are 3 more students on the track team than the baseball
team. How many students are on the track team? How many are on the
baseball team?
30
Answers
The number of students that are on the track team are 18.
The number of students that are on the baseball team are 15.
What are the linear equations that represent the question?a + b = 33 equation 1
a - b = 3 equation 2
Where:
a = number of students that are on the track teamb = number of students that are on the baseball teamHow many students that are on the baseball team?
Subtract equation 2 from equation 1
2b = 30
Divide both sides by 2
b = 30/2 = 15
How many students that are on the track team?
Subtract 15 from 33: 33 - 15 = 18
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Multiply and simplify: (1 + 5i)(3 - 8i)
Answers
Answer:
43 + 7i
Step-by-step explanation:
Given: (1 + 5i)(3 - 8i)
To find: Multiplication and simplification
Solution:
i=[tex]\sqrt{-1}[/tex],
Now, we simply open the brackets as follows:
(1 + 5i)(3 - 8i)
1(3 - 8i) + 5i (3 - 8i)
3 - 8i + 15i - 40[tex]i^{2}[/tex]
3 + 7i - 40[tex]i^{2}[/tex] - (i)
[tex]i^{2}[/tex] = i * i
[tex]i^{2}[/tex] = [tex]\sqrt{-1}[/tex] * [tex]\sqrt{-1}[/tex]
[tex]i^{2}[/tex] = -1
Now, put [tex]i^{2}[/tex] = -1 into the equation (i)
we get
⇒ 3 + 7i -40(-1)
⇒ 3 + 7i + 40
⇒ 43 + 7i
∴ Final Answer : 43 + 7i
[tex]\qquad\qquad\huge\underline{{\sf Answer}}[/tex]
Here we go ~
what we need to know here is that :
[tex]\qquad \sf \dashrightarrow \:i \cdot i = - 1[/tex]
Now, let's proceed accordingly ~
[tex]\qquad \sf \dashrightarrow \:(1 + 5i) \cdot(3 - 8i)[/tex]
[tex]\qquad \sf \dashrightarrow \:(1 \cdot3) - (1 \sdot8i) + (5i \cdot3) - (5i \cdot8i)[/tex]
[tex]\qquad \sf \dashrightarrow \:3 - 8i + 15i - ( - 1)(40)[/tex]
[tex]\qquad \sf \dashrightarrow \:3 + 7i - ( - 40)[/tex]
[tex]\qquad \sf \dashrightarrow \:3 + 40 + 7i[/tex]
[tex]\qquad \sf \dashrightarrow \:43 + 7i[/tex]
Dalia has a jar with 6 red pencils and 4 green pencils. She will randomly pick two pencils without replacement. What is the probability that Dalia will pick two red pencils?
6/25
1/3
9/25
2/5
Answers
The probability that Dalia will pick two red pencils without replacement is 1/3.
What is the probability?Probability determines the chances that an event would happen. The probability the event occurs is 1 and the probability that the event does not occur is 0.
Probability of picking two red pencils = 6/10 x 5/9 = 1/3
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(8x) ³ in simplest radical form
Answers
Answer:
[tex]512x^3[/tex]
Step-by-step explanation:
[tex](8x)^3 = 8^3x^3 = 512x^3[/tex]
A 6-ounce jar of modeling clay costs $4.20. What is the unit price?
Answers
Answer:
$0.70 per ounce
Step-by-step explanation:
Concepts + Application
The unit price of an object, in this case, is the total amount of money spent divided by how many ounces there are. We're given the total amount of money as $4.20 and the number of ounces as 6. Thus, we just divide 4.20 by 6.
Solution
4.20 ÷ 60.70$0.70 per ounceHelp please I don’t understand.
Answers
How do you find the quadratics
Answers
Answer:
Put all terms to the side of the equals side, and make sure there are none on the other side. That would be unbalanced if there were terms on the opposite side.
Factor out the problem.
Set each factor of the problem equal to zero.
Solve each new equation.
Check to see if it makes sense by putting your new equation into the original equation.
Find the volume of the cylinder.
Either enter an exact answer in terms of 7 or use 3.14 for T.
6
units 3
Answers
Answer:
3.14×5(2)×6=471.24
Step-by-step explanation:
you have to take square of 5
Help im failing my math class
Answers
Answer:
88 m^2
Step-by-step explanation:
We can split the parallelogram into three shapes. 1 square and 2 triangles.
To find the area of a square, multiply one side by another side that does not face it. To find the area of a triangle, divide the base by half OR multiply the base by 0.5, and then multiply by the height.
Since these shapes are all pieces of the parallelogram, add together their total areas to get the area of the parallelogram.
Square: 7 x 8
Triangle: 0.5 x 4 x 8 or 4/2 x 8
(7 x 8) + 2 x (0.5 x 4 x 8)
56 + 2 x (2 x 8)
56 + 2 x 16
56 + 32
56 + 32 = 88 m^2
The midpoint of AB is (3,2). If point is is located at(8,4) what are the coordinates of point B
Answers
Step-by-step explanation:
8-3 = 5,
3-5 = -2 = the x coordinate of B
-2-4 =-6,
-2-6 =-8 = the y coordinate of B
(-2,-8) is the point B
A is 5 to the right of the midpoint, so B is 5 to the left of the midpoint
A is 6 up from the midpoint so B is 6 down from the midpoint
A standard deck of 52 cards is shuffled. One card is drawn at random. The probability that the card is red or an ace is
Answers
Answer:
prob of being a red=1/2
prob of being an ace=1/13
In the following question solve for X please help!
Answers
Answer:
x = 6
Step-by-step explanation:
Taking the sides in proportion :
EB/DC = AB/AC8/24 = -3 + 2x/271/3 = -3 + 2x/273(-3 + 2x) = 27-9 + 6x = 276x = 36x = 6Step-by-step explanation:
If you have any question about the sollution, ask me
Translate the phrase into an algebraic expression.The product of 4 and b
Answers
Answer:
the answer is 4×b=4b
Step-by-step explanation:
product=×
4×b=4b
I need to know this asap
Answers
Answer:
Output: 3
Step-by-step explanation:
(x, y)
(Input, Output)
(2, 3)
what would be 3/4 of 960.
Answers
Answer:
720
Step-by-step explanation:
Three-fourths of 960 are simply three fourths times 960, which can be written as follows:
Three/fourths x 960
Furthermore, you can convert "three" to "3" and "fourths" to "4" and then the equation and answer are:
3/4 x 960 = 720
Find the size of the final unknown exterior angle in a polygon whose other exterior angles are: 51°, 75°, 38°, 47°, 65° and 52º
Answers
PLS HELP! What is the linear function rule for this table?
Answers
Answer:
f(x) = 4x + 6
Step-by-step explanation:
Using f(x) = 4x + 6
lets find some answers
lets take x as (-2)
f(-2) = 4(-2) + 6
f(-2) = -8 + 6
= -2
Also, take x as (-1)
f(-1) = 4(-1) + 6
f(-1) = -4 + 6
= 2
Please help ASAP
Will mark as brainliest answer
Answers
Answer:
i hope this will help you
Answer:
a = 1b = √15Step-by-step explanation:
An equation is solved for one of the variables by undoing the operations done on that variable. Here, we can solve for 'a' and solve for 'b' to find the desired values.
__
value of aSolve for 'a', then substitute the value of 'b'.
[tex]\sqrt{9a}=\dfrac{b^2-3}{2}\\\\9a=\left(\dfrac{b^2-3}{2}\right)^2=\dfrac{(b^2-3)^2}{4}\\\\a=\dfrac{(b^2-3)^2}{36}=\dfrac{((-3)^2-3)^2}{36}=\dfrac{6^2}{36}\\\\\boxed{a=1}[/tex]
__
value of bSolve for 'b', then substitute the value of 'a'.
[tex]\sqrt{9a}=\dfrac{b^2-3}{2}\\\\2\cdot3\sqrt{a}=b^2-3\\\\6\sqrt{a}+3=b^2\\\\b=\sqrt{6\sqrt{a}+3}=\sqrt{6\sqrt{4}+3}=\sqrt{12+3}\\\\\boxed{b=\sqrt{15}}[/tex]
Please help me find the answer to this problem.
Answers
Answer:
numerator - is 2, 15, 18, 6
denominator - is 12, 14, 33, 8, 64
Step-by-step explanation:
2/10, 4/12, 8/14, 9/33, 1/8, 12/64, 15/55, 18/66, 6/32
2 1/6 - (-8/3) + (-4 7/9) what does it eqaul i cant figure it out
Answers
Answer:
- 95/18
Step-by-step explanation:
1st you have to convert the mixed number to an improper fraction so 2 ×6 = 12 +1 = 13 and that together equals 13/6 same goes for the next take away the parentheses 13/6 - 8/3 - 4 7/9
then you have to do the same thing with the 1st fraction convert it 4 × 9 = 36 + 7 = 43 that equals together 43/9
lastly you have to calculate the difference 13/6 - 8/3 - 43/9 and that equals negative 95/18
by the way the 95 isnt a negative the fraction itself is
A paint roller is 12 inches long and has a radius of 3/4 inch. what is its lateral area?
Answers
Answer: The answer is [tex]16[/tex], You need to reduce the expression, if possible, by canceling the common factors. Then its lateral area of 12 inches and the radius of 3/4 inches would be 16.
Therefore your answer would be [tex]16[/tex].
Based on the concrete of the lateral area and the content of the question, the right answer is 18π
What is the lateral area?The lateral area is the term that is used to describe the surface of an object as all of the sides of the object, excluding its base and top. Generally, the term lateral area is different from the surface area, which is the lateral surface area together with the areas of the base and top.
Hence, in this case, to calculate the lateral area of the paint roller, find the circumference of the roller and multiply it by its length.
The circumference of a circle can be found using the formula: C = 2πr where r is the radius.
Given that the roller has a radius of 3/4 inch let's substitute this value into the formula to find the circumference:
C = 2π(3/4)
C = 6π/4
C = 3π/2
Now we have the circumference of the roller.
To find the lateral area we multiply the circumference by the length of the roller which is 12 inches:
Lateral Area = Circumference x Length
Lateral Area = 3π/2 x 12
Lateral Area = 36π/2
Lateral Area = 18π
Therefore, in this case, it is concluded that the lateral area of the paint roller is 18π square inches.
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Mr. Brown took his class to see a play. Each student ticket was $3.00. Each adult ticket was $5.00. The class spent a total of $190 on tickets. Write an equation in standard form to represent this situation.