1. a. Write out the first five terms of the sequence, determine whether the sequence converges, and if so find its limit. (5 pts) n2 1 b. Use &n+1/to show that the given sequence {a} is strictly increasing or strictly decreasing (5 pts) {ne)=1 7100 Sn=1
Answers
a. The first five terms of the sequence are:
a1 = 2
a2 = 5
a3 = 10
a4 = 17
a5 = 26
To determine if the sequence converges, we can take the limit as n approaches infinity. The formula for the nth term of the sequence is an = n^2 + 1. So, we have:
lim n->∞ an = lim n->∞ (n^2 + 1) = ∞
Since the limit does not exist, the sequence does not converge.
b. To use &n+1/ to show whether the sequence is strictly increasing or strictly decreasing, we need to show that an+1 > an or an+1 < an for all n.
We have an = n^2 + 1, so an+1 = (n+1)^2 + 1 = n^2 + 2n + 2.
Using &n+1/, we have:
an+1 - an = (n^2 + 2n + 2) - (n^2 + 1) = 2n + 1
Since 2n + 1 is always positive for n > 0, we have an+1 > an for all n. Therefore, the sequence is strictly increasing.
Finally, we are given the sum of the first n terms of the sequence, which is Sn = 1 + 5 + 14 + ... + (n^2 + 1). We can find a formula for Sn by using the formula for the sum of an arithmetic series:
Sn = (n/2)(a1 + an)
In this case, a1 = 2 and an = n^2 + 1, so we have:
Sn = (n/2)(2 + n^2 + 1) = (n/2)(n^2 + 3)
Therefore, Sn = (n/2)(n^2 + 3).
1a. You provided the sequence term n^2 + 1. The first five terms of the sequence are:
n=1: 1^2 + 1 = 2
n=2: 2^2 + 1 = 5
n=3: 3^2 + 1 = 10
n=4: 4^2 + 1 = 17
n=5: 5^2 + 1 = 26
So, the first five terms of the sequence are 2, 5, 10, 17, and 26. As n increases, the terms get larger and larger without bound, so the sequence does not converge.
1b. To show whether the given sequence {a_n} is strictly increasing or decreasing, we'll examine the difference between consecutive terms a_(n+1) and a_n.
Difference = a_(n+1) - a_n = ((n+1)^2 + 1) - (n^2 + 1) = (n^2 + 2n + 1 + 1) - (n^2 + 1) = 2n + 1
Since 2n + 1 is always positive for all positive integer values of n, the sequence is strictly increasing.
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Related Questions
(Related to Checkpoint 5.6) (Solving for i) Kirk Van Houten, who has been married for 21 years, would like to buy his wife an expensive diamond ring with a platinum setting on their 30-year wedding anniversary. Assume that the cost of the ring will be $13,000 in 9 years. Kirk currently has $4,532 to invest. What annual rate of return must Kirk earn on his investment to accumulate enough money to pay for the ring? The annual rate of return Kirk must earn on his investment to accumulate enough money to pay for the ring is %. (Round to two decimal places.)
Answers
Kirk must earn an annual rate of return of approximately 6.07% on his investment to accumulate enough money to pay for the ring.
To calculate the annual rate of return Kirk must earn on his investment, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
where:
A = final amount (the cost of the ring, which is $13,000 in 9 years)
P = initial amount (the amount Kirk currently has, which is $4,532)
r = annual interest rate (what we need to solve for)
n = number of times interest is compounded per year (assuming it's compounded annually, n = 1)
t = number of years (9 years)
Plugging in the given values:
$13,000 = $4,532(1 + r/1)^(1*9)
Next, simplify the equation:
(1 + r)^(9) = 13000/4532
Take the ninth root of both sides:
1 + r = (13000/4532)^(1/9)
Solve for r:
r = (13000/4532)^(1/9) - 1
Now, plug the value into a calculator to find r:
r = 0.0607
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PLSS HELP ASAP!!
I WILL GIVE YOU BRAINLYEST!!
*LOOK AT THE PICTURE*
Answers
Answer:
A
Step-by-step explanation:
Every single color is listed, and each pants color is connected to every shirt color.
Answer:
A
Step-by-step explanation:
You need the tree diagram that starts with options for pants.
Then you need for each pant color to have the option of the colors for the 3 shirts
A is the only one that has the colors for the pants and then all the colors for the shirts.
A soda company would like to replace the label on a can of soda with a radius of 3 and one fourth cm. If the label touches end to end with no overlap, what is the length of the label? Use 22 over 7 for π.
143 over 56 cm
143 over 28 cm
143 over 14 cm
143 over 7 cm
Answers
The circumference of a circle is equal to the total length of its boundary. The length of label is 143 / 7 cm. the correct answer is option D.
What is a circle ?A circle is a geometrical shape having all of its points equidistant from a fixed point known as the center.
The angle subtended by a circle is 360°.
Given that,
the radius of the circle r = 3 + 1 / 4
= 13 / 4 cm
The length of label is equivalent to the circumference of circle having r = 13 / 4 cm.
The circumference is given as,
C = 2π × r
=> C = 2 × 22 / 7 × 13 / 4
=> C = ( 11 × 13 ) / 7
=> C = 143 / 7.
Hence, the length of the label is given as 143 / 7 cm.
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Answer:
d
Step-by-step explanation:
because i said so
What is the slope of a line that goes through the points (-3, 5) and (2, -2)?
Answers
Answer:
-7/5
Step-by-step Explanation:
What is the difference quotient for the function f (x) = negative startfraction 1 over 5 x minus 12 endfraction?
Answers
The difference quotient of f(x) is \(\frac{-\frac{1}{h}+ 5}{5x+ h -12}\) .
According to the given question.
We have a function
f(x) = -1/(5x -12)
As we know that, the difference quotient is a measure of the average rate of change of the function over and interval.
The difference quotient formula of the function y = f(x) is
[f(x + h) - f(x)]/h
Where,
f(x + h) is obtained by replacing x by x + h in f(x)
f(x) is a actual function.
Therefore, the difference quotient formual for the given function f(x)
= [f(x + h) - f(x)]/h
= \(\frac{\frac{-1}{5(x+h)-12} -\frac{-1}{5x-12} }{h}\)
= \(\frac{\frac{-1}{5x + 5h -12}+\frac{1}{5x-12} }{h}\)
= \(\frac{\frac{-1+5h}{5x + 5h-12} }{h}\)
= \(\frac{-1+5h}{(5x +h-12)(h)}\)
= \(\frac{-1+5h}{5xh + h^{2} -12h}\)
= \(\frac{h(-\frac{1}{h}+5) }{h(5x+h-12)}\)
= \(\frac{-\frac{1}{h}+ 5}{5x+ h -12}\)
Hence, the difference quotient of f(x) is \(\frac{-\frac{1}{h}+ 5}{5x+ h -12}\) .
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Write the equation of the line that passes through the point (3,-1) and has a slope of 2
Answers
you have 12 marbles and a scale. one marble is heavier than the others. you can only use the scale three times. how can you find the heavier marble? (you cannot feel them in your hands that is cheating)
Answers
You can weigh 4 marbles against each other first, then if none are heavier, weigh 3 of the remaining marbles against each other, and if none of those are heavier, weigh the last two marbles against each other to find the heavier one.
What is weight of an object?An object's power of attraction toward the Earth is measured by its weight. It is the result of multiplying the object's mass by the acceleration caused by gravity.
What instrument is used to weigh an objects?A scale or balance is a tool used to determine mass or weight. These are also referred to as weight scales, mass scales, weight balances, and mass balances.
To weigh the 12 marbles and find out the heavier one,
Take 4 marbles and weigh them against each other. If one of them is heavier, you have found the heavier marble. If they all weigh the same, go to step 2.Take 3 of the remaining marbles and weigh them against each other. If one of them is heavier, you have found the heavier marble. If they all weigh the same, go to step 3.Take the remaining 2 marbles and weigh them against each other. The heavier one is the heavier marble.This solution will always work because at each step, you are halving the number of marbles you need to check, so you will always be able to find the heavier marble in 3 weighings or less.
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Graph the line that represents this equation: y = -5x +2
Answers
Answer:
Step-by-step explanation:
The +2 is the y-intercept, what value is y when x is zero. The -5/1 is your slope. Slope is always rise over run, y over x. Over 1, down 5
help me please this is due at 2:20
Answers
Answer:
94.2
Step-by-step explanation:
Answer:
C = pie x diameter
= 3.14 x 30
= 94.2 units
What is the completely factored form of f(x)=x^3+5x^2+4x-6
Answers
Answer:
(x+3)(x^2+2x-2)
Step-by-step explanation:
rewrote the expression
then factor out common numbers, variables, and exponents
Can anyone please help me will mark brainliest
Answers
Answer: the price is $19.60
Step-by-step explanation:
Consider a standard deck of 52 playing cards with 4 suits. If A is the event of drawing a 6 from the deck, and B is the event of drawing a black playing card from the deck, what is the intersection of A and B? (Remember that the black cards are spades and clubs.)
Answers
Answer: Intersection of A and B = 2
Step-by-step explanation:
Total cards = 52
Let A is the event of drawing a 6 from the deck, and B is the event of drawing a black playing card from the deck.
Total cards having 6 on them = 4
[There are 4 suits of two different colors red and black.]
Total black playing card =26
Intersection of A and B = Black cards having 6 = 2
hence, Intersection of A and B = 2
Which one of the following procedures would improve the reliability and validity of grading shortessay tests, thus refuting the complaint of sensitivity to bias and variability in grading?A) Administering more pretestsB) Grading on the curveC) Implementing a contract systemD) Using a scoring rubric
Answers
The most appropriate procedure from the given options that would improve the reliability and validity of grading short essay tests, thus addressing the complaint of sensitivity to bias and variability in grading, is: D) Using a scoring rubric.
A scoring rubric is a predefined set of criteria or guidelines that outlines the specific expectations and standards for evaluating student responses. By using a scoring rubric, the grading process becomes more objective, consistent, and transparent, thereby reducing bias and variability in grading.
A scoring rubric provides a structured framework that allows graders to assess and assign scores based on predetermined criteria, such as content knowledge, organization, clarity, coherence, and grammar. It ensures that all essays are evaluated using the same standards and eliminates subjective judgment as much as possible.
Administering more pretests (option A) may provide additional practice or feedback to students but does not directly address the issue of bias and variability in grading.
Grading on the curve (option B) adjusts grades based on the performance of other students in the class, which does not directly address bias or variability in grading. It can introduce its own set of issues and may not reflect the true quality of each individual student's work.
Implementing a contract system (option C) refers to a negotiated agreement between students and instructors regarding grading criteria, which may help clarify expectations but does not inherently address bias or variability.
Therefore, using a scoring rubric (option D) is the most appropriate procedure to improve the reliability and validity of grading short essay tests while minimizing sensitivity to bias and variability in grading.
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Miguel babysits for 3 hours and earn 15 which represents the unit rate?
Answers
Answer:
15
Step-by-step explanation:
Answer:
Miguel babysits for 3 hours and earns $15. Which represents the unit rate?
Option D
(D) $5 per hour
Step-by-step explanation:
$5.00 Per Hour Represents unit rate, 15 / 3 = 5
A group of friends were working on a student film that had a budget of $1200. They used 62% of their budget on props. How much money did they spend on props?
Answers
Answer: $804
Step-by-step explanation: 1. Set the equation to x/1200=67/100 2. Cross-multiply so you get 100x=80,400 3. Now divide both sides by 100 to get x=804 which is your final answer 4. Put the dollar sign in front to denote units.
which expressions are equivualent
Answers
Answer:
A (1\(3^4))
Step-by-step explanation:
3^-8\3^-4 = 3^4\3^8 =1\3^4
In public opinion surveys, a sample of the total population
a. must be at least 50 percent of the targeted population.
b. need be only 25 percent of the targeted population.
c. need be only 10 percent of the targeted population.
d. must be representative in that the views of those in the sample must accurately and proportionately reflect the views of the whole population.
e. must be based on all the registered voters in the targeted population.
Answers
In order for a sample of the entire population to be considered representative in public opinion polls, it must correctly and fairly reflect the views of the entire population.
Public opinion surveys are used to collect data and information on the opinions and views of people in a given population. For the results of a survey to be valid and reliable, the sample of people chosen for the survey must be representative of the entire population. This means that the views of those in the sample must accurately and proportionately reflect the views of the whole population.
Public opinion surveys are designed to measure the opinions and views of people in a given population. To ensure the results of the survey are accurate and reliable, the sample of people chosen for the survey must be representative of the entire population. This means that the views of those in the sample must accurately and proportionately reflect the views of the whole population. It is not necessary for the sample to exceed 50% of the targeted population, nor does it need to be based on all registered voters in the population. Therefore, option d. is the correct answer, which states that the sample must be representative in that the views of those in the sample must accurately and proportionately reflect the views of the whole population.
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Saad purchases 75 books and 75 dictionary .If a book costs Rs 476 and a dictionary costs Rs 128. Find the total money spent on books and dictionaries. *
Answers
Answer:
75.9
Step-by-step explanation:
Solve for X
Thank you
Answers
Answer:
X is equal to 5
Step-by-step explanation:
The tangents of a circle is always equal, which means 2x + 7 = 5x - 8. After you solve this equation, you will get that X = 5.
The data in this graph represent how long it took a child to complete a puzzle based on its number of pieces.
How many pieces were in the puzzle that took the child 30 seconds to complete?
Enter your answer in the box.
pieces
Coordinate plane titled Puzzle Completions with five points. The x-axis extends from 0 to 13 by ones and is labeled Number of pieces. The y-axis extends from 0 to 50 by fives and is labeled Time (seconds). Points are above the 4 on the x-axis and to the right of the 15 on the y-axis, above the 5 on the x-axis and to the right of the 15 on the y-axis, above the 7 on the x-axis and to the right of the 30 on the y-axis, above the 8 on the x-axis and to the right of the 25 on the y-axis, and above the 12 on the x-axis and to the right of the 40 on the y-axis.
Answers
There were 7 number of pieces that were in the puzzle that took the child 30 seconds to complete,that can be found using the data given & the coordinates on the graph (7,30) which indicates the number of pieces as 7 and time as 30 sec.
What is data?
Data is the collected information that can be arranged and used for analysing or interpretation. The collected data is also called as raw data, which further needs to be arranged. The data can be represented visually, in tabular form , grouped or ungrouped form. Pictographs, bargraphs, box-whisker plot, line plot, histogram, pie chart are the visual representations.
Here given that the data is arranged in the form of graph containing x-axis & y-axis. Data collected/arranged represents the number of pieces contained in the puzzle & the time taken to solve it.
X-axis represents the number of pieces in the puzzle & Y-axis represents the time taken to complete that.
The points plotted on graph are:
(4,15): Number of pieces=4
Time taken to complete=15 seconds
(5,15):Number of pieces=5
Time taken to complete=15 seconds
(7,30): Number of pieces=7
Time taken to complete=30 seconds
(8,25): Number of pieces=8
Time taken to complete=25 seconds
(12,40): Number of pieces=12
Time taken to complete=40 seconds
∴The puzzle that was completed by child in 30 seconds contained 7 pieces.
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Refer to the attachment for the graph.
Find the remainder term Rn in the nth order Taylor polynomial centered at a for the given function. Express the result for a general value of n. f(x)=e-2x, a-2 Choose the correct answer below. (-2)" e -2c (x- 2)" for some c between x and 2. (-2)1+1 e -2c (n+ 1)! O B. Rn(x)- -(x-2)"+1 for some c between x and 2. (-2)1+1e 2c Rn(x)=?(n+1)!-(x-2)n + 1 for some c between x and 2. n -2c OD. (x-2)"+1 for some c between x and 2.
Answers
Here is the correct answer in LaTeX code:
The correct answer is \($B\). \(R_n(x) = (-2)^{n+1} e^{-2c} (n+1)!$.\) The remainder term, \($R_n(x)$\) , in the \($n$th\) order Taylor polynomial for the function \($f(x) = e^{-2x}$\) centered at \($a = -2$\) is given by the formula:
\(\[R_n(x) = \frac{f^{(n+1)}(c) \cdot (x-a)^{n+1}}{(n+1)!}\]\)
where \($c$\) is a value between \($x$\) and \($a$\). In this case, \($a = -2$.\)
Taking the derivative of \($f(x) = e^{-2x}$\) , we have
\($f'(x) = -2e^{-2x}$, $f''(x) = 4e^{-2x}$, $f'''(x) = -8e^{-2x}$\) , and so on.
Substituting these derivatives into the remainder term formula, we get:
\(\[R_n(x) = (-2)^{n+1} e^{-2c} (n+1)! \cdot (x-(-2))^{n+1} / (n+1)!\]\)
Simplifying, we have:
\(\[R_n(x) = (-2)^{n+1} e^{-2c} \cdot (x+2)^{n+1}\]\)
So, the correct answer is \($B\). \(R_n(x) = (-2)^{n+1} e^{-2c} (n+1)!$.\)
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Solve 6(1 − 3d) -3/16
d 3/16
Answers
Answer:
d > -3/16
Step-by-step explanation:
A cone has a volume 8pie cubic inches. And 6in height
What is the radius of the cone?
inches
Answers
Answer:
r=1.3in
Step-by-step
hope this helps:)
Victoria has part-time job. She wants to make $125 per week at her part-time job. Last month she was within $5.50 of her goal. What are the maximum and minimum amounts that Victoria might have made last month?
Answers
Answer:
homeopatía oscar papal odiarlo
Explain how the rules for dividing two negative fractions relate to the rules for multiplying two negative fractions.
Answers
The relation between multiplying and dividing two negative fractions is that when multiplying and dividing two negative fractions the answer will be always positive.
How to explain the fraction?The first set of rules addresses the issue of combining positive and negative signs.
If both signals are positive (either both positive or both negative), the response is yes.If the signs are opposite (one positive and one negative), the answer is negative.Division of fractions is equivalent to multiplication by a reciprocal fraction. When you take a fraction and swap the numerator and denominator, you get a reciprocal. The reciprocal of 2/3, for example, is 3/2. However, keep in mind that the divisor, or the number doing the actual work of division, is always flipped. The divisor in division problems is the second fraction.
In this case, an example will be:
= -2/5 × -1/5
= 2/25
In this case, the negative signs are now positive. It should be noted that this also applies to division if negative fractions too as it will become positive.
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SAmina walked nine times round a square Field. If she covered a total distance of 12 96 meters, what is the area of the field?
Answers
Explanation: You want to take 1,296 as the perimeter and divide it by 9 as she walked around it 9 times. From there plug the answer (144) into the equation to find a area and you get 1296
if the 10 musical acts for the winter show are randomly selected, what is the probability that 5 acoustic acts and 5 electric acts get to play in the show?
Answers
The probability that 5 acoustic acts and 5 electric acts get to play in the show is 63, 540.
What is combination?Combinations are mathematical operations that count the number of potential configurations for a set of elements when the order of the selection is irrelevant.
Given:
Total musician = 10
So, the probability that 5 acoustic acts and 5 electric acts select
= C(10, 5) x C(10, 5)
= 10!/ 5! 5! x 10!/ 5! 5!
= 10 x 9 x 8 x 7 x 6 x 10 x 9 x 8 x 7 x 6 / 5 x 4x 3 x 2 x5 x 4x 3 x 2
= 91,44,57,600/ 14400
= 63540
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Tom and Brian are hiking towards each other at the opposite ends of a trail that is 21 miles long. Tom hikes uphill at a speed of 2 miles per hour. Brian is hiking downhill at a speed of 4 miles per hour. How long will it take for the two to meet?
Answers
Answer:
3.5 hours
Step-by-step explanation:
Distance = 21 miles
Speed * time
Speed * time = distance
Tom speed = 2 mph
Brian = 4 mph
Let time = t
Brian distance + Tom's distance = total distance
4t + 2t = 21
6t = 21
t = 21 / 6
t = 3.5 hours
18 x 1 8/9 write in a simplest form.
Answers
Hope this helps !!!!!
Answer:
34
Key:
" \(*\) " is a multiplication sign :)
Step-by-step explanation:
\(\frac{18}{1} * 1\frac{8}{9} \\\\\)
1) Make mixed nimber \((1\frac{8}{9})\) an improper fraction
\(\frac{18}{1} * \frac{17}{9} \\\\\) ( (9 × 1) + 8) = \(\frac{17}{9}\)
2) Multiply across
\(\frac{18}{1} * \frac{17}{9} = \frac{18 * 17}{1 * 9} = \frac{306}{9}\)
3) Simplify
\(\frac{306}{9}\) ÷ \(9\) = \(\frac{34}{1} = 34\)
HOPE THIS HELPS :DThe circular lid of a jar has a diameter of 3 inches. Which of the following
expressions could be used to find the radius of the jar?
The circular lid of a jar has a diameter of 3 inches. Which of the following
expressions could be used to find the radius of the jar?
1. 2 • 3
2. 3π
3. 3 • 3
4. 3 ÷ 2
Answers
So the answer is:
4. 3 / 2