Let and .
a) Study the monotony of the sequence (un).
b) What is its limit?
Answers
We are given the sequence (un) defined by un = (n^3 + 2n^2 - 3) / (n^2 + 1), and we need to determine the monotonicity of the sequence and find its limit. The sequence (un) is strictly increasing, and its limit as n approaches infinity is infinity.
a) To study the monotonicity of the sequence (un), we examine the behavior of consecutive terms. We can calculate the difference between successive terms by subtracting un+1 from un. Let's denote this difference as Δun = un+1 - un. If Δun is always positive or always negative, the sequence is monotonic.
Calculating Δun:
Δun = (n+1)^3 + 2(n+1)^2 - 3 - (n^3 + 2n^2 - 3)
= (n^3 + 3n^2 + 3n + 1) + 2(n^2 + 2n + 1) - 3 - n^3 - 2n^2 + 3
= 6n + 3
From the expression of Δun, we observe that Δun is a linear function of n with a positive coefficient. Therefore, Δun is always positive, indicating that the sequence (un) is strictly increasing.
b) To find the limit of the sequence (un), we examine its behavior as n approaches infinity. Taking the limit of the expression for un as n approaches infinity, we have:
lim(n→∞) un = lim(n→∞) [(n^3 + 2n^2 - 3) / (n^2 + 1)]
By applying the rules of limits, we can simplify the expression:
lim(n→∞) un = lim(n→∞) (n^3/n^2) = lim(n→∞) n = ∞
Therefore, the limit of the sequence (un) as n approaches infinity is infinity.
In summary, the sequence (un) is strictly increasing, and its limit as n approaches infinity is infinity.
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Related Questions
how do you expand 2(p-j)
Answers
Answer:
Disturbute
Step-by-step explanation:
2p-2j
The distance between two cities is 450 miles. On a map, they are 9 inches apart. What is the scale of the map?
Answers
Answer:
1 inch=50 miles
Step-by-step explanation:
450÷9=50
the most important thing to notice about a table of q-sort correlates is the a. smallest correlation. b. exact value of the correlations. c. wording of specific items. d. general patterns that emerge.
Answers
The most important thing to notice about a table of q-sort correlates is d. general patterns that emerge.
Q-sorting is a method of ranking items in order of preference or importance, and Q-sort correlates are used to identify relationships between the ranked items. The correlations in the table represent how strongly the items are associated with each other based on their rankings.
While the exact value of the correlations is important for statistical analysis, the most valuable information in a table of q-sort correlates is the general patterns that emerge. These patterns can help to identify groups of items that are closely related to each other, as well as those that are more distantly related.
Understanding the wording of specific items may also be important, as it can impact how the items are ranked and how they relate to each other. However, the primary focus of a table of q-sort correlates is on the relationships between the items, rather than the items themselves.
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What is an equation of the line that passes through the point (3,-1) and has a slope of 2?
Answers
y=2x-1
or
y=2x+(-1)
hope this helps:)
X-3=2(y+1)
X-3=2y+2
X-3-2y-2=0
Hope this helps ;)
Antonio quiere el suelo de la cocina de losetas cuadradas del mayor tamaño posible. Si la cocina mide 4,4 m de largo por 3,2m de ancho ¿Cuántos centímetros debe medir el lado de la loseta?
Answers
The centimeters that the side of the tile measure when Antonio wants the square tile kitchen floor as large as possible is 15.2 cm
How to illustrate the information?The complete question is: Antonio wants the square tile kitchen floor as large as possible. If the kitchen is 4.4 m long by 3.2 m wide, how many centimeters should the side of the tile measure?
It should be noted that from the information, Antonio wants the square tile kitchen floor as large as possible and the kitchen is 4.4 m long by 3.2 m wide.
It should be noted that this question is simply about the calculation of the perimeter. It should be noted that the perimeter of a rectangle will be:
Perimeter = 2(length + width)
= 2(4.4 + 3.2)
= 2 × 7.6
= 15.2 cm
Therefore, the centimeters that the side of the tile measure when Antonio wants the square tile kitchen floor as large as possible is 15.2 cm
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Antonio wants the square tile kitchen floor as large as possible. If the kitchen is 4.4 m long by 3.2 m wide, how many centimeters should the side of the tile measure?
Sally has 3:4 as many beads as Kelly. Kelly has 18 more beads than Sally. Find the average number of beads the girl have
Answers
The average number of beads that the girls have is 63
Let's start by using algebra to represent the given information:
Let b be the number of beads that Sally has.
Then, Kelly has 3/4 times as many beads as Sally, which can be expressed as (3/4)b.
Also, we know that Kelly has 18 more beads than Sally, which can be expressed as (b + 18).
Putting these together, we can write the equation:
(3/4)b = b + 18
Solving for b, we get:
b = 72
So, Sally has 72 beads, and Kelly has (3/4) × 72 = 54 beads.
The average number of beads that the girls have is (72 + 54)/2 = 63 beads
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4. Tim has a wheelbarrow full of sand. He can fill a 12 litre bucket with the sand 15 times. (a) How many times could he fill a bucket with a capacity of 18 litres, with the sand from the wheelbarrow?
Answers
Tim could fill the 18-litre bucket in 22.5 times.
How many times could Tim fill an the bucket?To get the number of times he could fill an 18-litre bucket with the sand from the wheelbarrow, we wil set up proportion using the known information.
The amount of sand that can fill a 12-litre bucket is proportional to the number of times it can fill the bucket.
The proportion will be:
12 litres / 15 times = 18 litres / x times
12x = 15 * 18
12x = 270
x = 270 / 12
x = 22.5.
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Cochineal was a dye of the New World, used by the Aztecs long before the arrival of the Spanish conquistadors. Obtained from the crushed bodies of the female cochineal beetle, cochineal (also known as caine ) was expensive. It took 70,000 insects to make 1lb of the dye. a
Answers
Cochineal, a dye obtained from the crushed bodies of the female cochineal beetle, was used by the Aztecs and required 70,000 insects to produce 1lb of dye.
How did the Aztecs use cochineal dye and why was it expensive?The Aztecs utilized cochineal dye for various purposes before the Spanish conquistadors arrived. They employed it in the coloring of textiles, especially for producing vibrant reds and purples.
Cochineal dye was highly valued due to its exceptional color quality and permanence. Its vibrant hues made it a sought-after pigment for religious rituals, clothing of nobles, and even for painting murals.
The reason behind cochineal dye being expensive was primarily due to the labor-intensive process of obtaining it. To yield 1lb of dye, a staggering amount of 70,000 female cochineal beetles had to be crushed.
Additionally, the insects were native to the New World and were not found in Europe, making cochineal dye a rare and exotic commodity. The scarcity, combined with the high demand for its vivid and long-lasting color, contributed to its high cost in the market.
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A popular video game retailer develops apps. The total profit, in dollars per day, after x days can be modeled by the function R(x) = 3x3 – 21x2 + 21x + 45. The total cost, in dollars per day, after x days, can be modeled by the function A(x) = x2 – 2x – 3. After how many days does the retailer start making money?
Answers
The difference between the cost and revenue functions is the profit function.
The retailer will start making money after 3 days.
Given that:
\(R(x) = 3x^3\:-\:21x^2\:+\:21x\:+\:45\)
\(A(x) = x^2 - 2x - 3\)
The retailer will start making money when R(x) > 0 i.e. when the profit is greater than 0.
So, we have:
\(3x^3 - 21x^2 + 21x + 45 > 0\)
Factor out 3
\(3(x^3 - 7x^2 +7x + 15) > 0\)
Divide through by 3
\(x^3 - 7x^2 +7x + 15 > 0\)
Expand
\(x^3 - 8x^2 + 15x + x^2 - 8x + 15 > 0\)
Further expand
\(x^3 - 5x^2 - 3x^2 + 15x + x^2 - 5x - 3x + 15 > 0\)
Factorize
\(x^2(x - 5) - 3x(x - 5) + x(x - 5) - 3(x -5) > 0\)
Factor out the common factors
\((x^2 - 3x)(x - 5) + (x - 3)(x - 5) > 0\)
Factor out x
\(x(x - 3)(x - 5) + (x - 3)(x - 5) > 0\)
Factor out (x - 3)(x - 5)
\((x - 3)(x - 5)(x + 1) > 0\)
Split
\(x - 3> 0\) or \(x - 5> 0\) or \(x + 1 > 0\)
Solve for x
\(x > 3\) or \(x > 5\) or \(x > -1\)
x cannot be negative.So:
\(x > 3\) or \(x > 5\)
Because 3 is less than 5, then the reasonable solution to the set is:
\(x > 3\)
Hence, the retailer will start making money after 3 days.
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An investor has an account with stick from two different companies last year his stock yk company a was worth 6500 and his stock h was worth 7450 increased 4
Answers
The percentage increase in the value of the stock is 14.62%.
How to calculate the percentage?Thw.investor has an account with stock from two different companies last year his stock yk company a was worth 6500 and his stock h was worth 7450.
The percentage increase will be:
= Change in value / Initial value × 100
= (7450 - 6500) / 6500 × 100.
= 950/6500 × 100
= 14.62%
Therefore, percentage increase in the value of the stock is 14.62%.
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An investor has an account with stick from two different companies last year his stock yk company a was worth 6500 and his stock h was worth 7450. Calculate the percentage increase.
RIGHT ANSWER GETS BRAINLIEST AND 11 POINTS ‼️‼️‼️‼️‼️‼️‼️‼️
Answers
Answer:
Less
Step-by-step explanation:
Look at the hundreds number in set L the numbers are greater in Set L than Set K.
Q: S and T are relations on the real numbers
and are defined as follows:
S = {(x, y) ∣ x < y}
T = {(x, y) ∣ x > y}
What is T ∘ S?
A) R x R (all pairs of real numbers)
B)
C) S
D) T
Answers
B) ∅ (empty set); The composition T ∘ S is an empty set (∅) because there are no ordered pairs that satisfy both the conditions of the relations T and S.
To find the composition T ∘ S, we need to determine the set of ordered pairs that satisfy both relations S and T. Let's analyze the definitions of S and T:
S = {(x, y) ∣ x < y}
T = {(x, y) ∣ x > y}
To find T ∘ S, we need to check if there exists an element z such that (x, z) is in T and (z, y) is in S for any (x, y) in the given relations. However, if we observe the definitions of S and T, we can see that there is no common element that satisfies both relations.
For any (x, y) in S, we have x < y, but in T, the relation is defined as x > y. Therefore, there are no elements that satisfy both conditions simultaneously.
As a result, T ∘ S will be an empty set (∅) because there are no ordered pairs that satisfy the composition of the two relations.
The composition T ∘ S is an empty set (∅) because there are no ordered pairs that satisfy both the conditions of the relations T and S.
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A cell phone that was bought for 2500 was sold at 15% calculate the selling price
Answers
Answer:
The answer to your question is $375
The selling price of the cell phone is $375.
Step-by-step explanation:
15% x 2500
100 = 375
I hope this helps and have a wonderful day!
A pancake recipe asks for one and two thirds times as much milk as flour. If three and three quarters cups of
milk is used, what quantity of flour would then be needed, according to the recipe
Answers
Answer: 1 whole
Step-by-step explanation:
Brain list for correct answer!!!
Answers
Answer:
1.58 is the correct answer
\( \huge{\tt{Answer:}} \)
option 1 .86
\( \sf {sinA= {\dfrac{opposite~side}{hypotenuse}} = {\dfrac{9.5}{11}} = .86} \)
Plan B of a monthly payment of 336 for 6 years (72 months) at 4.5 %. What is the total amount paid to the principal in the first month's payment?
Select the correct answer below:
A. $256.63
B. $261.92
C. this value can’t be determined
D. $79.37
Answers
Answer:
The answer is "A" $256.63
Step-by-step explanation:
336 = (.045/12)(pv)/(1-(1+.045/12)^-72)
PV = $21,166.65
(.045*$21,166.65)/12 = $79.37
$336 - $79.37 = $256.63
Please help with my geometry hw
Answers
Using the given description < NOP is found to be 46 degrees
How to find angle NOPgiven data]
< OPQ = ( 9x - 19 ) degrees
< PNO = ( 2x + 5 ) degrees
< NOP = ( 3x + 16 ) degrees
From the figure described
let < NPO = y
< PNO + < NOP + y = 180 degrees ( sum of angles of triangle )
< OPQ + y = 180 degrees ( linear pair theorem )
Hence we equate both
< PNO + < NOP + y = < OPQ + y
< PNO + < NOP = < OPQ + y - y
< PNO + < NOP = < OPQ
( 2x + 5 ) + ( 3x + 16 ) = ( 9x - 19 )
2x + 3x + 5 + 16 = 9x - 19
5x + 21 = 9x - 19
21 + 19 = 9x - 5x
40 = 4x
x = 40 / 4
x = 10
< NOP = ( 3x + 16 ) degrees
< NOP = ( 3 * 10 + 16 ) degrees
< NOP = ( 30 + 16 ) degrees
< NOP = 46 degrees
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a researcher is interested in the relationship between happiness and gpa of high school students. after surveying 50 students, he determines that there is a correlation between these two variables of .90. this is considered a: group of answer choices strong negative linear correlation strong positive linear correlation weak negative linear correlation weak positive linear correlation
Answers
The correlation coefficient of 0.90 indicates a strong positive linear correlation between happiness and GPA of high school students.
A correlation coefficient measures the strength and direction of the relationship between two variables. In this case, the correlation coefficient of 0.90 indicates a strong positive linear correlation between happiness and GPA of high school students.
A positive correlation means that as one variable (in this case, happiness) increases, the other variable (GPA) also tends to increase. The magnitude of the correlation coefficient, which ranges from -1 to 1, represents the strength of the relationship. A value of 0.90 indicates a very strong positive linear correlation, suggesting that there is a consistent and significant relationship between happiness and GPA.
This means that as the level of happiness increases among high school students, their GPA tends to be higher as well. The correlation coefficient of 0.90 suggests a high degree of predictability in the relationship between these two variables.
It is important to note that correlation does not imply causation. While a strong positive correlation indicates a relationship between happiness and GPA, it does not necessarily mean that one variable causes the other. Other factors or variables may also influence the relationship between happiness and GPA.
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construct a quadrilateral BEST,in which ES= 7.5 cm, BS= 6cm, BT=6cm,ST=5cm and ET=10 cm
Answers
The resulting quadrilateral BEST will have the following properties:
Side ES = 7.5 cmSide BS = 6 cmSide BT = 6 cmSide ST = 5 cmSide ET = 10 cmTo construct the quadrilateral BEST with given side lengths, follow these steps:
Start by drawing a straight line segment of length 6 cm and label it BS.At point B, construct an angle of 90 degrees.From point B, measure 7.5 cm along the perpendicular bisector of BS and mark a point as E.Draw a line segment connecting points B and E, which will be the diagonal of the quadrilateral.From point B, measure 6 cm along the line segment BE and mark a point as T.From point T, draw a line segment perpendicular to BE, measuring 5 cm, and mark the point of intersection with line segment BE as S.Finally, draw a line segment connecting points E and S, which completes the quadrilateral BEST.Remember to use a ruler and a protractor to ensure accurate measurements and angles while constructing the quadrilateral.
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HELP FAST PLEASEEEEEE I NEED HELPPPPP
Answers
The median means that as many as friends have less than A. 1. 5 pets as those that have more than A. 1. 5 pets.
What does the median mean ?The median is a measure of central tendency in statistics that represents the middle value of a dataset when it is ordered from smallest to largest. The median is often used as a more robust measure of central tendency than the mean, because it is less affected by extreme values in the dataset.
From the box plot, the data set of friends with pets would be:
0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 2, ,2 ,2, 2, 2, 2, 3, 4, 4, 7.
The median here is:
= ( 10 th position + 11 th position ) / 2
= ( 1 + 2 ) / 3
= 1. 5
This therefore means that as many friends have more than 1. 5 pets as those with less than 1. 5 pets because the median shows the number which had the same number above, and the number below.
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Which statement best describes the intersection of three planes? The planes could form one or two lines. The planes could form one, two, or three lines. The planes will form two lines. The planes could form one, two, or three lines, or they could intersect at exactly one point.
Answers
Answer:
fhe planes coukd form ons or two limez
5x + 2y = 8
-4x + 10y = 40
Answers
Answer:
(0, 4 )
Step-by-step explanation:
5x + 2y = 8 → (1)
- 4x + 10y = 40 → (2)
multiplying (1) by - 5 and adding to (2) will eliminate y
- 25x - 10y = - 40 → (3)
add (2) and (3) term by term to eliminate y
- 29x = 0 , then
x = 0
substitute x = 0 into (1) and solve for y
5(0) + 2y = 8
0 + 2y = 8
2y = 8 ( divide both sides by 2 )
y = 4
solution is (0, 4 )
7/8 divided by 1/3 as a reduced fraction
Answers
Name and discuss a minimum of two (2) geophysical survey methods. • How can this be used in geometric road design?
Answers
Geophysical survey methods are used to gather information about the subsurface properties of an area. Two commonly used methods in geometric road design are seismic surveys and ground-penetrating radar (GPR) surveys.
Seismic surveys involve sending sound waves into the ground and measuring the time it takes for the waves to bounce back. This helps determine the depth and characteristics of different layers of soil and rock. Seismic surveys can be used to identify areas of soft soil or rock, which may require additional engineering measures during road construction.
GPR surveys use radar signals to image the subsurface. The radar waves are sent into the ground and reflected back by different layers and objects, such as buried utilities or geological features. GPR surveys can provide detailed information about the thickness and composition of subsurface layers, helping engineers determine the best route for a road and avoid potential hazards.
By using these geophysical survey methods in geometric road design, engineers can gather important information about the subsurface conditions and make informed decisions about road alignment and construction techniques. This can help ensure the road's stability, durability, and safety.
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Zara has forgotten her 4-digit PIN code.
She knows the first digit is a factor of 10 and the 4 digits make a number divisible by 5.
How many different sets of 4 digits could it be?
Answers
Since the first digit is a factor of 20, the factors of 20 are 1,2,4,5,10,20. We only need the single digit factors which are 1,2,4 and 5. These 4 numbers can be permuted in 1 way for the first digit, so we have ⁴P₁.
For the second digit, we have 10 digits permuted in 1 way, ¹⁰P₁ and also for the third digit, we have 10 digits permuted in 1 way, ¹⁰P₁ and for the last digit, which is divisible by 5, it is either a 0 or 5, so we have two digits permuted in 1 way, ²P₁.
So, the number of different 4 digit number that Zara'2 4-digit PIN code could be is ⁴P₁ × ¹⁰P₁ × ¹⁰P₁ × ²P₁ = 4 × 10 × 10 × 2 = 800 different sets of digits
Answer:
800 different 4 digit passwords can be the pin
Step-by-step explanation:
Since the first digit is a factor of 20, the factors of 20 are 1,2,4,5,10,20. We only need the single digit factors which are 1,2,4 and 5.
These 4 numbers can be permuted in 1 way for the first digit, so we have ⁴P₁.
For the second digit, we have 10 digits permuted in 1 way, ¹⁰P₁ and also for the third digit, we have
10 digits permuted in 1 way, and for the last digit, which is divisible by 5, it is either a 0 or 5, so we have two digits permuted in 1 way,
So, the number of different 4 digit number that Zara'2 4-digit PIN code could be is
4 × 10 × 10 × 2 = 800 different sets of digits
Each statement describes a transformation of the graph of y=x. which statement correctly describes the graph of y=x-9?
A. It is the graph of y+x translated down 9 units.
B. It is the graph of y+x, where the slope is decreased by 9
C. It is the graph of y+x translated to the left 9 units.
D. It is the graph of y+x translated up 9 units.
Answers
Simplify. Please show work.
Answers
Answer:
Step-by-step explanation:
2(√x+2/2 - 1)² + 4 (√x+2/2 - 1)
2((√x+2/2)² + 1² - 2(√x+2/2)(1)) + 4√x+2/2 - 4
2(x+2/2 + 1 - 2(√x+2/2)) + 4√x+2/2 - 4
x + 2 + 2 - 4√x+2/2 + 4√x+2/2 - 4
x + 4 - 4
x
Answer:
the answer is x
Step-by-step explanation:
Helppp!!!!!!!!!!!!!!!!!!!!!
Answers
The value of x is equal to 15°
How to determine the value of x?In Mathematics and Geometry, the sum of the exterior angles of both a regular and irregular polygon is always equal to 360 degrees.
Note: The given geometric figure (regular polygon) represents a pentagon and it has 5 sides.
By substituting the given parameters, we have the following:
3x + 4x + 8 + 5x + 5 + 6x - 1 + 5x + 3 = 360°.
3x + 4x + 5x + 6x + 5x + 8 + 5 - 1 + 3 = 360°.
23x + 15 = 360°.
23x = 360 - 15
23x = 345
x = 345/23
x = 15°.
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can someone help me solve D, a, or/and b
Answers
Answer:
\(slop = (175 - 125) \div (2 - 1) = 50 \\ y - 125 = 50 \times (x - 1) \\ y = 50x - 50 + 125 \\ y = 50x + 75\)
you can solve problems by this equation
I will only give points if you give me a long answer (not THAT long.). Now, what is 100 x 300. Make sure to give a long answer but not that long. So enjoy your day.
Answers
Answer:
100×300=3000
Step-by-step explanation:
hope the picture attached helps
