prove that if limnan = a and a + 0, then there exists a positve number k and a positve integer m such that Jan> k, whenever n > m.
Answers
After considering the given data we conclude that it is proven that \(limnan = a (and) a + 0\), and there exists a positive number k and a positive integer m such that Jan> k, whenever n > m.
To prove that if \(limnan = a (and) a + 0\), then there exists a positive number k and a positive integer m such that Jan> k, whenever n > m, we can apply the definition of a limit.
Definition of a limit: Let assume (an) be a sequence of real numbers. We interpret that the limit of (an) as n approaches infinity is a,
denoted limnan = a, if for every ε > 0, there exists a positive integer N such that \(\{|an - a| < \epsilon\} whenever}\{ n > N\}\).
Then lets proceed with the proof
Consider that \(limnan = a (and) a + 0.\)
Let \(\epsilon = a/2\). Since a + 0, we know that a > 0, so \(\epsilon\) > 0.
Applying the definition of a limit, there exists a positive integer \(N_1\) such that \(|an - a| < \epsilon (whenever) n > N_1.\)
Then \(k = a/\epsilon = 2\). Since \(\epsilon = a/2, (we have) k = 2.\)
Then \(m = max\{N1, k\}\) Hence, for n > m, we have:
\(n > N_1, (since) m \geq N_1\).
\(n > k,( since) m \geq k.\)
Therefore, we have:
\(|an - a| < \epsilon , (by the description) of N_1\).
\(\epsilon = a/2 < a/k, (since) k = 2.\)
\(|an - a| < a/k,\) by applying substitution.
\(an - a < a/k\), since |an - a| is positive.
\(an < a(1 + 1/k)\), by adding a to both sides.
\(an < a(1 + 1/2) = 3a/2.\)
Hence , we have shown that there exists a positive number k = 2 and a positive integer \(m = max\{N1, k\}\) such that Jan> k, whenever n > m.
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Related Questions
A foreign student club lists as its members 3 canadians, 2 japanese, 4 italians, and 3 germans. If a committee of 4 is selected at random, find the probability that all nationalities are represented;.
Answers
The probability that all nationalities are represented is 0.5%.
What does the math term "probability" mean?
Simply put, probability measures how probable something is to occur. We can discuss the probabilities of various outcomes, or how likely they are, whenever we are unsure of how an event will turn out.
Statistics is the study of events subject to probability.
Given that a foreign student club lists as its members 3 Canadians, 2 Japanese, 4 Italians, and 3 Germans, if a committee of 4 is selected at random, to fi nd the probability that (A) all nationalities are represented; (B) all nationalities except Italian are represented; the following calculations must be performed:
3 + 2 + 4 + 3 = 12
3/12 x 2/11 x 4/10 x 3/9 = X
0.0050 = X
0.0050 x 100 = 0.5
Therefore, the probability that all nationalities are represented is 0.5%.
7/12 x 6/11 x 5/10 x 4/9 = X
0.070 = X
0.070 x 100 = 7
In turn, the probability that all nationalities except Italian are represented is 7%.
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A gambler who on each bet either wins 1 with probability 18/38 or loses 1 with probability 20/38 The gambler will quit when he or she is winning a total of 5. What is the probability he or she plays exactly 15 times
Answers
The gambler wins 1 with probability 18/38 or loses 1 with probability 20/38 on each bet. Let X be the amount of the gambler's profit and losses after playing the game 15 times. For the gambler to win a total of 5 before he or she quits, the gambler must win 10 times and lose 5 times.
Thus the probability that the gambler wins exactly 10 times out of 15 is given by the binomial distribution as follows: $$P(X = 10) = \binom \({15}{10}(18/38)^{10}(20/38)^{5}$$\) .
Therefore, the probability that the gambler plays exactly 15 times before winning a total of 5 is equal to the probability that the gambler wins exactly 10 times,
i.e.,$$P(\text {play 15 times}) = P(X = 10)$$$$P(\text \(= P(X = 10)$$$$P(\text\) {play 15 times}) \(= \ binom{15}{10}(18/38)^{10}(20/38)^{5}$$$$P(\text{play 15 times}) \approx 0.174$$\)Thus, the probability that the gambler plays exactly 15 times is approximately 0.174.
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Simplify: 15.4 + 3.02
Answers
Answer:
18.42
Explanation:
15+3=18
0.4+0.02=0.42
18+0.42=18.42
help will give brainlist asap!
find m
and explain your reasoning
Answers
Answer:
90°
Step-by-step explanation:
its a square which means it has right angles. right angles always = 90°
a class of 20 students is divided into two groups of 10 and then each group is lined up in a particular order. how many ways are there to divide the class and line up the groups?
Answers
There are ²⁰С₁₀ × 10! × 10! ways to divide the class and line up the groups.
Number of students in the class = 20
Selection of group one = ²⁰С₁₀ ways
Selection of group two = 1 way
Number of ways to arrange 10 students in a group = 10!
So, Total number of ways = ²⁰С₁₀ × 10! × 10! ways.
Hence there are ²⁰С₁₀ × 10! × 10! ways to divide the class and line up the groups.
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Factor by grouping.
64p2 – 48p – 27
(8p – 9)(8p – 3)
(8p – 9)(8p + 3)
(8p + 9)(8p + 3)
(8p + 9)(8p – 3)
Answers
Answer:
B
Step-by-step explanation:
hope this helps with the work
12.5% of $100 is what number? percent proption
Answers
Answer: $12.5
Step-by-step explanation:
The $100 Is 100 percent meaning that one percent is $1. Take that and multiply it by 12.5 and get $12.5
Answer:
12.5
Step-by-step explanation:
12.5 % of 100$ = 12.5
(\(\frac{12.5}{100}\)) x (100) = 12.5
a two-day mathematics conference has n participants. some of the participants give a talk on saturday, some others give a talk on sunday. nobody gives more than one talk, and there may be some people who do not give a talk at all. at the end of the conference, a few talks are selected to be included in a book. in how many different ways is this all possible if we assume that there is at least one talk selected for inclusion in the book?
Answers
There are (n + 2ⁿ- 1) * 2ⁿ different ways to select talks for inclusion in the book, given that at least one talk is selected.
We know that at least one talk must be selected, so we have two cases to consider: either one or more than one talk is selected.
Case 1: If only one talk is selected, then there are n possible talks to choose from since any participant could have given the selected talk.
Case 2: If more than one talk is selected, then we need to consider the number of talks that could have been selected. There are a total of 2ⁿ - 1 possible non-empty subsets of talks that could have been selected. However, we need to subtract 1 from this number to exclude the case where no talks are selected.
So the total number of ways to select talks for the book is n + 2ⁿ - 2.
Now we need to consider the number of ways to assign talks on Saturday and Sunday. There are 2ⁿ possible ways to assign talks since each participant can give their talk on Saturday or Sunday.
(n + 2ⁿ - 2) * 2ⁿ
However, we need to subtract the case where no talks are selected and included in the book, which is simply 2^n since each talk can be assigned to either Saturday or Sunday. So the final answer is:
(n + 2ⁿ - 2) * 2ⁿ - 2ⁿ = (n + 2ⁿ - 1) * 2ⁿ.
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help I’m in 5th 14 question
Answers
Consider the following current information for Galaxy Inc::
Output = 200 units
ATC = $50
What is the total cost of producing 200 units of output?
a. $10,000
b. $8,000
c. $1,100
d. Non
Answers
The answer is (a) $10,000.
How the total cost of producing 200 units of output can be found?The total cost of producing 200 units of output can be found by multiplying the output (200 units) by the average total cost (ATC) per unit, which is given as $50. Therefore, the total cost is:
Total Cost = Output x ATC
Total Cost = 200 x $50
Total Cost = $10,000
Therefore, the answer is (a) $10,000.
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Circle O is shown. Line segments B O and O C are radii. Point A is on the circle and is not within angle B O C.
The measure of Arc B A C is 240. What is the ratio of the measure of the major arc to the measure of the minor arc?
3:1
1:3
2:1
1:2
Answers
The ratio of major to minor arc = 2 : 1
What is a Circle?
A circle is the locus of a point such that its distance from a fixed point (center) is always constant.
Calculation
The length of an arc (L) with center angle θ is given by:
L = (θ/360) * 2πr
where r is the radius.
Putting the values in the above formula.
For the major arc:
L = (240/360) * 2πr
For the major arc:
l = ((360-240)/360) * 2πr
l = (120/360) * 2πr
The ratio of major to minor arc = [(240/360) * 2πr] / [(120/360) *2πr] = 2 : 1
The ratio of major to minor arc = 2 : 1.
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Answer:c
Step-by-step explanation:
ratio of 12/7 and 36/21
Answers
Find the values of x and y in parallelogram PQRS.
PT =y. TR = 3x + 1, QT = 3y. TS = 5x + 11
x=
y=
PLEASE HELPPP!!!!
Answers
In the parallelogram PQRS, x = 2 and y=7
Parallelogram:
A parallelogram is a quadrilateral with two pairs of parallel sides. A parallelogram has opposite sides of equal length and opposite angles of equal length. Also, the interior angles on the same side of the horizontal line are supplementary angles. The sum of all interior angles is 360 degrees. A three-dimensional shape whose faces are parallelograms is called a parallelepiped. The area of a parallelogram depends on its base (one of its parallel sides) and its height (height drawn from top to bottom). The perimeter of a parallelogram depends on the length of its four sides.
Given that
PT=y, TR=3x+1, QT=3y, TS=5x+11
The diagonals of a parallelogram bisect each other
PT = TR and QT = TS
y = 3x+1 and 3y = 5x+11
Now,
We have a system of equations
Multiply the first equation by 3
3y = 9x+3 and 3y = 5x+11
Set them equal
9x + 3 = 5x + 11
⇒ 9x - 5x = 11 -3
⇒ 4x = 8
⇒ x =2
Now,
Putting the value of x =2 in equation :
y = 3x+1
=3(2)+1
=7
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A residual is the difference between _______.
Answers
A residual is the difference between "the measured and predicted values of the quantity of interest".
What is residuals?In regression analysis, a residual would be the difference between an observable and anticipated value.
The formula is;
Residual = Observed value – Predicted value
Some key features regarding the residuals are-
The purpose of linear regression would be to quantify the relationship among one or so more predictor variables one and or more response variables. Linear regression selects the line that best "fits" the data, defined as least squares regression line, to do this. This line generates a prediction for every observation in the dataset, however it is improbable that the regression line's prediction would perfectly match its observed value.The residual is the discrepancy between the predicted and the observed value. The residuals for every observation would represent the vertical distance between both the observation as well as the regression line if we plotted the observed values then overlaid the fitted regression line.To know more about the residual, here
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Angle a and b are vertical angles. If m angle a = (5x- 4) degree and m angle b = (2x - 1) degree. Find value of x. WILL MARK BRAINLIEST !!!!!!!!!!!!!!!
Answers
Answer:
Since the 2 angles are vertical angles:
5x-4=2x-1
3x=3
x=1
expand and simplify 3(x-2) + 2(4x-1)
Answers
Answer:
11x-8
Step-by-step explanation:
\(3(x - 2) + 2(4x - 1) \\ multiply \: inside \: the \: bracket \\ 3x - 6 + 8x - 2 \\ 11x - 8 \)
convert 336g to m³
\(336g \: to \: m3\)
Answers
The conversion 336g to m³ is impossible because they measure different units
How to convert 336g to m³From the question, we have the following parameters that can be used in our computation:
Convert 336g to m³
To convert a unit to another, the units must measure the same quantity
Take for instance:
Grams can be converted to kilograms because they both measure mass
Using the above as a guide, we have the following:
336g to m³ cannot be done because of the different quantities they represent
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como resuelvo esto y=1+2(4/5)
Answers
Answer:
Es 2.6
Step-by-step explanation:
Answer: translate
Step-by-step explanation:
Write a proportion for each set of similar polygons and solve for the unknown side. Type the FULL answer in the box, using the variable, an equal sign, and the number. Do not add any spaces or words.
Answers
Answer:
Please help me now Solve
The area of Australia/Oceania is approximately 7.69 × 106 square kilometers. Its population is approximately 3.11 × 107 people. What is the approximate population density (people per square kilometer) of Australia/Oceania? Write your answer in standard form. If necessary, round your answer to the nearest hundredth.
PLEASSSSEEEE HELPPPPPPPPPPP
Answers
Answer: 3.11*10^7/7.69*10^6
3110/769
4.04
Step-by-step explanation:
Given isosceles HIJ where HI ≈ IJ, m/H = (3x + 10)°, and m/I = (6x − 20)°, find the measures of angles H, I, and J.
Answers
The missing angles of the isosceles triangle are as follows:
m∠H = 55 degreesm∠I = 70 degreesm∠J = 55 degrees.How to find the angles of an isosceles triangle?An isosceles triangle is a triangle that has two sides equal to each other and two angles congruent to each other.
The base angles of an isosceles triangles are congruent and the two legs are also equal.
Therefore, ΔHIJ
where
HI ≅ IJ
m∠H = 3x + 10
m∠I = 6x - 20
Therefore,
m∠H ≅ m∠J
m∠H + m∠I+ m∠J = 180
3x + 10 + 6x - 20 + 3x + 10 = 180
12x = 180
x = 180 / 12
x = 15
Hence,
m∠H = 3(15) + 10 = 55 degrees
m∠I = 6(15) - 20 = 90 - 20 = 70 degrees
m∠J = 55 degrees.
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I WILL MARK BRAINEST!!
Could someone help me with this please, thanks!
Answers
Answer:a
a=y
Step-by-step explanation:
Horizontal line (Coordinate Geometry) Definition: A straight line on the coordinate plane where all points on the line have the same y-coordinate. ... A horizontal line is one the goes left-to-right, parallel to the x-axis of the coordinate plane. All points on the line will have the same y-coordinate.
Answer:
a) y-coordinates
b) undefined
c) x=8
d) y= x-6
e) y= -2x+9
2. y=6x+17
Step-by-step explanation:
2. y=mx+b use point (-2,5) and slope to find b
5=6(-2)+b
5=-12+b
5+12=b
b=17
y=6x+17
Find the volume of a
cone with a radius of 3
cm and a height of
5 cm
Answers
Answer:
V≈47.12cm³
Step-by-step explanation:
i really hope this helps
Tree rings in dendrochronology are counted to yield precise dates in years, however, if the tree ring record for the specific region is not well established in order to utilize this dating technique samples from the tree have to be collected from a tree that is:
Answers
In dendrochronology, tree rings are counted to determine precise dates. However, if the tree ring record for a specific region is not well-established, samples need to be collected from a tree that is...
In order to utilize dendrochronology as a dating technique when the tree ring record for a particular region is not well-established, researchers must collect samples from a tree that serves as a reference or anchor. This tree, known as a master or living chronology, is typically an older, long-lived tree species that has a well-defined and uninterrupted growth pattern. By analyzing the tree rings of this master chronology, researchers can establish a timeline that extends back several centuries or even millennia. They can then compare the pattern of growth rings in the collected samples with the master chronology and determine the dates associated with each ring. This process relies on the principle that trees in the same region will respond similarly to environmental factors and exhibit similar growth patterns. Through careful analysis and cross-referencing, scientists can create a reliable tree ring record for the specific region, enabling accurate dating of samples collected from other trees in that area.
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If 5(x - y) =18, find 10x-10y
Answers
Answer:
36
Step-by-step explanation:
5(x-y)=18
5x-5y=18
To make it 10x-10y we multiply entire equation by 2
5x-5y=18
*2. *2. *2
10x+10y=36
Hopes this helps please mark brainliest
what is minus sixteen minus minus twenty
Answers
Answer: (-16)-(-20)= 4
Step-by-step explanation:
Our math team had m students last year, but this year it has already n students! By what percent did the number of students grow?
Answers
Answer:
(n-m)/m * 100%
Step-by-step explanation:
The original amount is m
The new amount is n
Take the new amount and subtract the original amount
(n-m)
Divide by the original amount
(n-m)/m
Multiply by 100 %
(n-m)/m * 100%
Answer:
(n-m)/m*100
Step-by-step explanation
trust me its the answer i checked with rsm math homework and it was correct
Find the 11th term in the geometric sequence: 4, 6, 9,....
Answers
Answer:
69
Step-by-step explanation:
let the first term be n
second term = 2 + n
third term = 3 + second term so we can conclude that the other term will be:
forth term = 4 + third term
fifth term = 5 + forth term etc.
eleventh term will be 11 + 10 term
based on observations the tenth term is 58 so
eleventh term= 11 + 58
= 69
here is concern that regular customers will find out about the special order, and in order to retain all of the x company's regular customers, the regular selling price would have to be reduced by $0.32. if the selling price were reduced and next year's unit sales turn out to be the same as this year's sales, firm profits would fall by
Answers
Profit fall in one year = 116.8$
What is profit?
In accounting, a profit is an income that is given to the owner throughout a successful market producing process. Profitability, which is the owner's primary interest in the income-formation process of market production, is measured by profit. There are several profit metrics that are frequently used.
Let cost price = C
And selling price before reducing = S
Profit for 1 day = S-C
Now, After reducing selling price to = S-0.32
Profit= S - 0.32 - C for 1 day
Profit fall by = (S-C)-(S - 0.32 - C)
= S - C - S + 0.32 + C
Profit fall by = 0.32 for 1 day
Profit fall by one year = 0.32×365
= 116.8$
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The circumference of a circle is 91.688 millimeters. what is the radius of the circle? use 3.14 for π. 29.2 mm 183.4 mm 14.6 mm 45.8 mm
Answers
Answer:
14.6mm
Step-by-step explanation:
The formula for finding the circumference of a circle is 2πr.
The circumference is 91.688 millimeters inorder to find the radius we have to do the opposite or inverse of multiplication which is division.
So we divide the circumference by 2 and pi which is 3.14 which will give us the radius
r = radius = 91.668 ÷ 3.14 ÷ 2 = 14.6
Answer: its c
Step-by-step explanation: