Use the ratio test to determine whether ∑n=14∞n+2n! converges or diverges. (a) Find the ratio of successive terms. Write your answer as a fully simplified fraction. For n≥14, limn→∞∣∣∣an+1an∣∣∣=limn→∞.

Answer with explanation:

The Inequality should be:

   [tex](1+x)^n\geq 1+n x[/tex]

Where, n and x are any integers.

⇒For, x= -1

L HS

 [tex]=[1+(-1)]^n\\\\=(0)^n\\\\=0[/tex]

R HS

→1+n × (-1)

=1-n

If, n is any Integer, then for, n=1

1-1=0

For, n=2

1-2= -1

....

So,   [tex](1+x)^n\geq 1+n x[/tex]

for, x=-1.

⇒For, x=1

L HS

 [tex]=[1+(1)]^n\\\\=(2)^n[/tex]

For, n=1

L H S=1

For, =2

L H S=4

R HS

→1+n × (1)

=1+n

If, n is any Integer, then for, n=1

1+1=2

For, n=2

1+2= 3

....

So,    [tex](1+x)^n\geq 1+n x[/tex]

for, x=1.

Answer:

Step-by-step explanation:

I'm making the assumption you are looking for a graph of y=log_3(x)

So 3^0=1          which means log_3(1)=0              graph (1,0)

     3^1=3          which means log_3(3)=1              graph (3,1)

     3^2=9         which means log_3(9)=2             graph (9,2)

     3^3=27       which means log_3(27)=3           graph (27,3)

Can you find a graph that fits these points?

Answer: 0.5237

Step-by-step explanation:

Mean : [tex]\mu=192\text{ days}[/tex]

Standard deviation : [tex]\sigma = 12\text{ days}[/tex]

The formula to calculate the z-score is given by :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x = 188 ​days ,

[tex]z=\dfrac{188-192}{12}\approx-0.33[/tex]

For x = 107 miles per day ,

[tex]z=\dfrac{107-92}{12}=1.25[/tex]

The P-value =[tex]P(-0.33<z<1.25)=P(z<1.25)-P(z<-0.33)[/tex]

[tex]0.8943502-0.3707=0.5236502\approx0.5237[/tex]

Hence, The probability that a randomly selected pregnancy lasts less than 188 days is approximately 0.5237.

Answer:

The percentage of the distribution is less than 75 is 97.72%.

Step-by-step explanation:

Given,

Mean of the distribution,

[tex]\mu=67[/tex]

Standard deviation,

[tex]\sigma = 4[/tex]

Thus, the z-score of the score 75,

[tex]z=\frac{x-\mu}{\sigma}[/tex]

[tex]=\frac{75-67}{4}[/tex]

[tex]=\frac{8}{4}[/tex]

[tex]=2[/tex]

With the help of z-score table,

[tex]P(x<75)=0.9772=97.72\%[/tex]

Hence, the percentage of the distribution is less than 75 is 97.72%.

Answer: The value of n is  374. The value of p is 0.21. The value of q is 0.79.

Step-by-step explanation:

Given : In a clinical trial of a cholesterol drug, 374 subjects were given a placebo, and 21% of them developed headaches.

∴ Sample size : [tex]n=374[/tex]

The probability that cholesterol drug developed headaches :[tex]p=0.21[/tex]

Then , the probability that cholesterol drug did not develop headaches :[tex]q=1-p=1-0.21=0.79[/tex]

Hence, The value of n is  374.

The value of p is 0.21.

The value of q is 0.79.

Answer: a) 0.7814

b) False

Step-by-step explanation:

To find : The p-value of z , where

[tex]-1.23<z< +1.23[/tex]

[tex]P(-1.23<z< +1.23)=1-P(z<-1.23)\\\\=1-0.1093=0.7814[/tex]

Hence, the standard normal probability [tex]P(-1.23<z< +1.23)=0.7814[/tex]

In the formula used to convert "real-world" data values into z numbers, the standard deviation of the data is considered.

The formula to calculate the z score is [tex]z=\dfrac{x-\mu}{\sigma}[/tex].

If sample size (n) is given then , the [tex]z=\dfrac{x-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

Hence, the statement is false.

Answer:

-∞ < x < ∞

Explanation:

x³ is the inverse of ∛x and x³ has range of all real numbers and is one to one function, so its inverse will have domain of all real numbers.

Answer:

Option 1 negative (-) infinity < X < infinity

Step-by-step explanation:

Answer:

gragh

Step-by-step explanation:

its one of the few ways to analyze data and compare it to find the best choice

Final answer:

To calculate the probability of a shopper spending less than 20 minutes in the store, the Z-score is found using the formula Z = (X - μ) / σ, resulting in a Z-score of -0.25, which corresponds to a probability of approximately 0.40.

Explanation:

The question asks for the probability that a randomly selected shopper will spend less than 20 minutes in a store, given that the average time spent is 22 minutes with a standard deviation of 8 minutes, and that these times are normally distributed. To find this probability, we use the Z-score formula:

Z = (X - μ) / σ

Where X is the value we are checking (20 minutes), μ is the mean (22 minutes), and σ is the standard deviation (8 minutes). Plugging in the numbers, we get:

Z = (20 - 22) / 8 = -0.25

Next, we look up the Z-score in a standard normal distribution table, or use a calculator with normal distribution functions, to find the probability that a Z-score is less than -0.25. This probability is approximately 0.40.

Answer:

D. 1.25 grams

Step-by-step explanation:

Half-life is: 0.004 sec

Time spent : 0.016 sec

Quantity = 20 gram

In order to find the material after 0.016 sec, we have to calculate how many number of half-lives have been passed

No. of half-lives passed = 0.016/0.004

= 4

The number of lives passed will be raised to the power of 0.5.

0.5 ^ 4 = 0.0625

The answer will be multiplied with the quantity we started with.

Remaining material is:

20*0.0625 = 1.25 grams

Hence, Option D is correct ..

Answer:

Part A ⇒ n>-140/p and p≠0  

Part B. ⇒ d=-30-2a/5

Step-by-step explanation:

Part A. -np-80<60

First, add by 80 both sides of equation.

-np-80+80<60+80

Simplify.

60+80=140

-np<140

Then, multiply by -1 both sides of equation.

(-np)(-1)>140(-1)

Simplify.

np>-140

Divide by p both sides of equation.

np/p>-140/p; p≠0

Simplify to find the answer.

n>-140/p; p≠0 is the correct answer from part a.

___________________________________

Part B. 2a-5d=30

First add by 2a from both sides of equation.

2a-5d+2a=30+2a

Then, simplify.

-5d=30-2a

Divide by -5 from both sides of equation.

-5d/-5=30/-5-2a/5

Simplify, to find the answer.

d=-30-2a/5 is the correct answer from part b.

Part A:

For this case we have the following inequality:

[tex]-np-80 <60[/tex]

We add 80 to both sides of the inequality:

[tex]-np <60+80\\-np <140[/tex]

Dividing between -p on both sides, having to change the inequality sign:

[tex]n> - \frac {140} {p}[/tex]

Part B:

For this case we have the following equation:

[tex]2a-5d = 30[/tex]

Subtracting 2a on both sides:

[tex]-5d = 30-2a[/tex]

Dividing between -5 on both sides:

[tex]d = \frac {30-2a} {- 5}\\d = \frac {-30+2a} {5}\\d = - \frac {30} {5} +\frac {2a} {5}\\d = -6+ \frac {2a} {5}[/tex]

Answer:

[tex]n> - \frac {140} {p}\\d = -6+\frac {2a} {5}[/tex]

Answer:  B) 67%

Step-by-step explanation:

Find the Area of the Bullseye and Middle ring

A = π r²

A (inside) = π(8)² = 64π

Find the Area of the entire Target

A (target) = π (14)² = 196π

Find the Area of the Outer ring

A (outer ring) = A (target) - A(inside)

                      =   196 π    -    64π

                      =    132 π

The last step is to find the probability of landing on the outer ring:

[tex]P=\dfrac{success (area\ of\ outer\ ring)}{total\ possible\ outcomes(area\ of\ target)}=\dfrac{132\pi}{196\pi}=0.673=\large\boxed{67\%}[/tex]

Answer:

67%

Step-by-step explanation:

1 pencil and 4 pens = 5 total

Picking the pencil would be 1/5 ( 1 pencil out of 5 total items)

5 color pencils  + 5 crayons = 10 total items.

Picking a color pencil would be 5/10 which reduces to 1/2

To find the probability of both happening, multiply them together:

1/5 x 1/2 = 1/10

The probability is 1/10

Answer:

  11

Step-by-step explanation:

Suitable software can generate these collections. They are ...

{13,1,1,1,1,1}, {11,3,1,1,1,1}, {9,5,1,1,1,1}, {9,3,3,1,1,1},

{7,7,1,1,1,1}, {7,5,3,1,1,1}, {7,3,3,3,1,1}, {5,5,5,1,1,1},

{5,5,3,3,1,1}, {5,3,3,3,3,1}, {3,3,3,3,3,3}}

The number of distinct collections of six positive odd integers that sum to 18 is indeed:[tex]\[\boxed{11}\][/tex]

[tex]\[2(k_1 + k_2 + k_3 + k_4 + k_5 + k_6) + 6 = 18\]\[k_1 + k_2 + k_3 + k_4 + k_5 + k_6 = 6\][/tex]

Here are the possible combinations of non-negative integers (up to permutations) that sum to 6:

[tex]1. \( (6, 0, 0, 0, 0, 0) \)2. \( (5, 1, 0, 0, 0, 0) \)3. \( (4, 2, 0, 0, 0, 0) \)4. \( (4, 1, 1, 0, 0, 0) \)5. \( (3, 3, 0, 0, 0, 0) \)6. \( (3, 2, 1, 0, 0, 0) \)7. \( (3, 1, 1, 1, 0, 0) \)8. \( (2, 2, 2, 0, 0, 0) \)9. \( (2, 2, 1, 1, 0, 0) \)10. \( (2, 1, 1, 1, 1, 0) \)11. \( (1, 1, 1, 1, 1, 1) \)[/tex]

Now let's map these back to the odd integers using [tex]\(a_i = 2k_i + 1\):[/tex]

[tex]1. \( (13, 1, 1, 1, 1, 1) \)2. \( (11, 3, 1, 1, 1, 1) \)3. \( (9, 5, 1, 1, 1, 1) \)4. \( (9, 3, 3, 1, 1, 1) \)5. \( (7, 7, 1, 1, 1, 1) \)6. \( (7, 5, 3, 1, 1, 1) \)7. \( (7, 3, 3, 3, 1, 1) \)8. \( (5, 5, 5, 1, 1, 1) \)9. \( (5, 5, 3, 3, 1, 1) \)10. \( (5, 3, 3, 3, 3, 1) \)11. \( (3, 3, 3, 3, 3, 3) \)[/tex]

These are all distinct collections (up to permutations) of positive odd integers that sum to 18.

Thus, the number of distinct collections of six positive odd integers that sum to 18 is indeed:[tex]\[\boxed{11}\][/tex]

(a) The probability of answering all 5 problems correctly is 0.08.

(b) The probability of answering at least 4 problems correctly is 0.5.

There are a total of 10 problems, and the student has figured out how to solve 7 of them which means there are 3 problems that the student hasn't figured out how to solve.

(a) To find the probability that the student answers correctly to all 5 problems, we need to consider that the student must select 5 out of the 7 problems they know how to solve and 0 out of the 3 problems they don't know how to solve.

The probability of selecting 5 specific problems out of 7 is given by:

(7 choose 5) / (10 choose 5)  [tex]=\frac{^7C_5}{^{10}C_5}[/tex]

=21/252

=0.083

(b) To find the probability that the student answers at least 4 problems correctly, we need to consider two cases: when the student answers 4 problems correctly and when the student answers all 5 problems correctly.

Case 1: Student answers 4 problems correctly and 1 problem incorrectly:

P(4 correct and 1 incorrect) = (7 choose 4) × (3 choose 1) / (10 choose 5)

Case 2: Student answers all 5 problems correctly:

P(5 correct out of 5) = (7 choose 5) / (10 choose 5)

Now, add the probabilities of these two cases to get the probability of answering at least 4 problems correctly:

P(at least 4 correct) = P(4 correct and 1 incorrect) + P(5 correct out of 5)

Calculate each part using combinations:

P(4 correct and 1 incorrect) = (35 × 3) / 252

= 0.4167

P(5 correct out of 5) = 0.08 (as calculated in part a)

Now, add these probabilities:

P(at least 4 correct) = 0.4167 + 0.08

= 0.5

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The probability that the student will correctly answer all 5 questions is 0.083 (8.3%) and the probability of correctly answering at least 4 questions is 0.5 (50%).

This is a probability question related to the field of combinatorics. To answer the student's question, we have to calculate the probability of correctly answering the questions out of the known ones.

The total ways the instructor can select 5 problems out of 10 is represented by the combination formula C(10,5). This equates to 252.

The student can answer 7 questions, so a) the number of ways of getting all 5 correctly is represented by C(7,5) which equals 21. Therefore, the probability of answering all 5 correctly is 21/252 = 0.083 or 8.3%.

For b), the student aims to answer at least 4 correctly. This means that we calculate the probability for getting 4 and 5 problems correct. For 4 problems it's C(7,4)*C(3,1) = 105. Adding the ways to get 5 problems correct, we get 105+21=126. So, the chance of answering at least 4 correct is 126/252 = 0.5 or 50%.

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The problem is to calculate the number of additional men needed to build an airstrip in less time given two men already took five hours. The additional man required is one after solving the equation, but this is not an option provided, indicating a potential error in the question.

The question involves calculating the number of additional men required to build an airstrip in less time. If it took 2 men 5 hours to build an airstrip, we can say they have a combined work rate of 1 airstrip per 5 hours, or ​​(1/5) airstrip per hour. To finish the job in 4 hours, which is 1 hour less than the original time, we would need a work rate of 1 airstrip per 4 hours.

So, if we let the number of additional men be X, we can set up the equation as follows:

Since we cannot hire half a person, we round up to the nearest whole number. Hence, one additional man would be sufficient to complete the work in 1 hour less time. However, none of the options given (A) Two, (B) Three, (C) Four, or (D) Six, are correct. Therefore, the answer is not provided in the given options and this represents a possible error in the question itself.

Final answer:

Stephanie cuts the rope into five 1-meter-long pieces by dividing the total length (5 meters) by the number of pieces (5).

Explanation:

If Stephanie cuts a five-meter-long piece of rope into five equal parts, we would divide the total length of the rope by the number of parts to find the length of each piece. In this case, the calculation would be:

So, the length of each part would be 5 meters ÷ 5, which equals 1 meter. Therefore, each piece of rope would be 1 meter long.

Answer: The cost of child's ticket = $10

The cost of adult ticket = $ 13

Step-by-step explanation:

Let x be the cost of a child ticket and y be the cost of an adult ticket.

Then According to the question, we have

[tex]6x+2y=86..........................(1)\\\\10x+3y=139.......................(2)[/tex]

Multiply equation (1) by 3 and equation (2) by 2, then we have

[tex]18x+6y=258.......................(1)\\\\20x+6y=278...........................(2)[/tex]

Subtract equation (1) from equation (2), we have

[tex]2x=20\\\\\Rightarrow\ x=10[/tex]

Substitute the value of x in equation (1), we get

[tex]60+2y=86\\\\\Rightarrow\ 2y=26\\\\\Rightarrow\ y=13[/tex]

Hence, the cost of child's ticket = $10

The cost of adult ticket = $ 13

Answer:

The second choice is correct.

Step-by-step explanation:

    These are triangles, so the interior angles have to add up to 180 degrees. Since 45 degrees and m<1 are vertical angles, they will have the same measure. So right off the back we know that m<1 =45. To find the measure<2 all you need to do is add 45 and 59, then subtract that answer from 180. M<2= 76. That leaves only one option. The second  one.

The measurements of the angles are Angle 1 = 45 degrees, Angle 2 = 76 degrees, Angle 3 = 80 degrees. The correct option is b) 1 = 45, 2 = 76, 3 = 80.

Let's solve this step by step to find the measurements of angles 1, 2, and 3.

We named the two triangles as OAB and OCD, with a common vertex O. angle O makes vertically opposite angle 45 degree in triangle OAB and angle 1 in OCD, angle A makes angle 2, angle B makes angle 59 degree, angle C makes 55 degree and angle D makes angle 3.Given the information:

In triangle OAB:

Angle OAB (angle 2) = 59 degrees

Angle O = 45 degrees (vertically opposite to angle 1)

In triangle OCD:

Angle O = 45 degrees (again, vertically opposite to angle 1)

Angle OCD (angle 3) = ?

First, let's find angle 2 in triangle OAB:

In triangle OAB, the sum of angles in a triangle is 180 degrees.

So, angle 2 + angle O + angle OAB = 180 degrees.

59 + 45 + angle OAB = 180 degrees.

Now, solve for angle OAB:

angle OAB = 180 degrees - 59 degrees - 45 degrees

angle OAB = 76 degrees

Now, let's find angle 3 in triangle OCD:

In triangle OCD, the sum of angles in a triangle is also 180 degrees.

So, angle O + angle OCD + angle C = 180 degrees.

45 + angle 3 + 55 = 180 degrees.

Now, solve for angle 3:

angle 3 = 180 degrees - 45 degrees - 55 degrees

angle 3 = 80 degrees

So, the measurements of angles 1, 2, and 3 are as follows:

Angle 1 = 45 degrees

Angle 2 = 76 degrees

Angle 3 = 80 degrees

The correct option is:

b) 1 = 45, 2 = 76, 3 = 80

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Answer: $5,224.74

Step-by-step explanation:

You need to calculate the 6% of  $4,929.00.

Convert the percentage to decimal form:

[tex]\frac{6}{100}=0.06[/tex]

Now multiply  $4,929.00 by 0.06:

[tex](\$4,929.00)(0.06)=\$295.74[/tex] (This is the 6% of $4,929.00)

Finally, you need to add $295.74 to $4,929.00. Then you get:

[tex]\$4,929.00+\$295.74=\$5,224.74[/tex]

Therefore, the 6% sales tax on $4,929.00 is: $5,224.74

Answer:

53.27

Step-by-step explanation:

To begin, we want to figure out what percent of products are defective and non defective, and of which type, so we can figure out probabilities. So, we start with that 5% of chairs are defective. We know that 35% of product sales are chairs, so 5% of 35% is 35%*0.05 (a percent can be divided by 100 to convert to decimals)= 1.75%. For tables, 12% of 55 is 55*0.12=6.6 percent. For beds, we first must figure out what percent of sales are beds. For this, we must take our total (100%) and subtract everything that is not beds, which is tables and chairs. This, 100-35-55=10, which is our percent of beds. Then, 8% of that is 0.8%. So, we know that the probability that an item is defective is 0.8+6.6+1.75=9.15%. The item pulled was not defective, so we want to figure out the probability of that, which would be the total-defective=100-9.15=90.85%. We then need to figure out which of that 90.85 is divided by tables, as we want to figure out what the probability of a table is. We know that 12% of tables are defective, so 100-12=88% are not. 88% of 55% is 55*0.88=48.4, so there is a 48.4% chance that if you picked out anything, it would be a non defective table. However, we are only picking things out from nondefective items, or the 90.85%. We know that 48.4 is 48.4% of 100, but we want to figure out what percent 48.4 is of 90.85. To find this, we do (48.4/90.85) * 100, which is 53.27 rounded. Feel free to ask further questions!

Answer: 11

Step-by-step explanation:

5% × 220 =

(5 ÷ 100) × 220 =

(5 × 220) ÷ 100 =

1,100 ÷ 100 =

11;

5% of 220 = 11

Answer:

44% is your answer.

Step-by-step explanation:

So, 220/5= 44/1= 44%

Because if you divide 220 by 5 you'll get 44 and 5 divided by 5 ofcourse is 1.

(I'm doing these also in my class, so hopefully i helped you.)

The gradient of function f is (∂f/∂x, ∂f/∂y) = (2cos(2x + 5y), 5cos(2x + 5y)). The gradient at the point P(-5, 2), is ∇f(-5, 2) = (2cos(-20), 5cos(-20)). Rate of change of f at P in the direction of the vector u is  (-1/2, 3). Duf(-5, 2) = ∇f(-5, 2) · (-1/2, 3).

(a) To find the gradient of the function f(x, y) = sin(2x + 5y), we need to compute the partial derivatives with respect to x and y:

Gradient of f(x, y) = (∂f/∂x, ∂f/∂y).

Taking the partial derivative with respect to x:

∂f/∂x = ∂/∂x(sin(2x + 5y)) = 2cos(2x + 5y).

Taking the partial derivative with respect to y:

∂f/∂y = ∂/∂y(sin(2x + 5y)) = 5cos(2x + 5y).

So, the gradient of f is (∂f/∂x, ∂f/∂y) = (2cos(2x + 5y), 5cos(2x + 5y)).

(b) To evaluate the gradient at point P(-5, 2), we substitute these values into the gradient expression:

∇f(-5, 2) = (2cos(2(-5) + 5(2)), 5cos(2(-5) + 5(2))).

Calculating these values gives the gradient at P.

The gradient at point P is ∇f(-5, 2) = (2cos(-20), 5cos(-20)).

(c) To find the rate of change of f at point P(-5, 2) in the direction of the vector u = (1/2, 3) - (1, 0) = (-1/2, 3), we use the dot product:

Duf(-5, 2) = ∇f(-5, 2) · (-1/2, 3).

Calculate this dot product to find the rate of change of f in the direction of u at point P.

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Answer:

honesty you cant realy use a calculator because you need to meadian mode and range them first

Step-by-step explanation:

Answer:

  the graph is shown below

Step-by-step explanation:

The solution region is that quadruply-shaded area above the line y=-2 and to the left of the line x=1. It is further bounded above by the line y = -1/2x +3.5.

Answer:

Step-by-step explanation:

You can't simplify the sum, but you can factor it.

t^2 + 2t - 3

(t  + 3)(t - 1 )

That's about all you can do.

Answer:

It is already in simplest form.

Step-by-step explanation:

It cannot be further simplified because it does not have any like terms.

Step-by-step explanation:

o - odd number

e - even number

n × e = e, n is either odd or even...rule 1

n - e = o, n must be odd...rule 2

n - e = e, n must be even...rule 3

n^2 = o, n must be odd...rule 4

n^2 = e, n must be even...rule 5

6n is even, no matter if n is odd or even following rule 1

if n^2 - 6n = o, n must be odd following rule 2

if n^2 = o, n must be odd following rule 4

To find the probability of the soft drink machine filling a cup between 30 and 31 ounces, calculate the z scores for 30 and 31 ounces, use them to find the cumulative probabilities from a standard normal distribution table, then subtract the two probabilities. The result is 0.0919 or 9.19%.

This is a question about probability in a normal distribution. In this case, we want to find the probability of the output being between 30 and 31 ounces, given a mean of 28 ounces and a standard deviation of 2 ounces.

First, we find the z-scores for 30 and 31 ounces. The z-score is calculated by subtracting the mean from the value and dividing the result by the standard deviation. For 30 ounces, the z-score is (30-28)/2 = 1. For 31 ounces, the z-score is (31-28)/2 = 1.5.

Next, we use these z-scores to find the cumulative probabilities from a standard normal distribution table. The cumulative probability for a z-score of 1 is 0.8413 and for 1.5, it's 0.9332.

The probability of filling a cup between 30 and 31 ounces is the difference between the cumulative probabilities of the two z-scores. So, the answer is 0.9332 - 0.8413 = 0.0919.

Therefore, the probability of the soft drink machine filling a cup between 30 to 31 ounces is 0.0919 or 9.19% when rounded to four decimal places.

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The probability of filling a cup between 30 and 31 ounces is approximately [tex]\(\boxed{0.0919}\)[/tex]

The probability of filling a cup between 30 and 31 ounces when the mean output is 28 ounces and the standard deviation is 2 ounces can be found using the Z-score formula for a normal distribution.

 First, we calculate the Z-score for 30 ounces:

[tex]\[ Z_{30} = \frac{X - \mu}{\sigma} = \frac{30 - 28}{2} = \frac{2}{2} = 1 \][/tex]

Next, we calculate the Z-score for 31 ounces:

[tex]\[ Z_{31} = \frac{X - \mu}{\sigma} = \frac{31 - 28}{2} = \frac{3}{2} = 1.5 \][/tex]

Now, we look up the probabilities corresponding to these Z-scores in the standard normal distribution table or use a calculator.

The probability of getting a value less than or equal to [tex]\( Z_{30} \)[/tex]is:

[tex]\[ P(Z \leq 1) \approx 0.8413 \][/tex]

 The probability of getting a value less than or equal to is:

[tex]\[ P(Z \leq 1.5) \approx 0.9332 \][/tex]

To find the probability of filling a cup between 30 and 31 ounces, we subtract the probability of filling up to 30 ounces from the probability of filling up to 31 ounces:

[tex]\[ P(30 < X < 31) = P(Z \leq 1.5) - P(Z \leq 1) \][/tex]

[tex]\[ P(30 < X < 31) \approx 0.0919 \][/tex]

 Rounded to four decimal places, the probability is 0.0919.

 Therefore, the probability of filling a cup between 30 and 31 ounces is approximately [tex]\(\boxed{0.0919}\)[/tex]

Answer:

we need to prove : for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.

1) check divisibility for n=1, [tex]f(1)=(1)^{5}-1=0[/tex]  (divisible)

2) Assume that [tex]f(k)[/tex] is divisible by 5, [tex]f(k)=(k)^{5}-k[/tex]

3) Induction,

[tex]f(k+1)=(k+1)^{5}-(k+1)[/tex]

[tex]=(k^{5}+5k^{4}+10k^{3}+10k^{2}+5k+1)-k-1[/tex]

[tex]=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k[/tex]

Now, [tex]f(k+1)-f(k)[/tex]

[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-(k^{5}-k)[/tex]

[tex]f(k+1)-f(k)=k^{5}+5k^{4}+10k^{3}+10k^{2}+4k-k^{5}+k[/tex]

[tex]f(k+1)-f(k)=5k^{4}+10k^{3}+10k^{2}+5k[/tex]

Take out the common factor,

[tex]f(k+1)-f(k)=5(k^{4}+2k^{3}+2k^{2}+k)[/tex]      (divisible by 5)

add both the sides by f(k)

[tex]f(k+1)=f(k)+5(k^{4}+2k^{3}+2k^{2}+k)[/tex]

We have proved that difference between [tex]f(k+1)[/tex] and [tex]f(k)[/tex] is divisible by 5.

so, our assumption in step 2 is correct.

Since [tex]f(k)[/tex] is divisible by 5, then [tex]f(k+1)[/tex] must be divisible by 5 since we are taking the sum of 2 terms that are divisible by 5.

Therefore, for every integer n>1, the number [tex]n^{5}-n[/tex] is a multiple of 5.

Final answer:

The statement is true for all integers greater than 1.

Explanation:

The n5 - n is a multiple of 5 for every integer n > 1, we will use proof by induction.

Base Case: For n = 2,
n5 - n = 25 - 2 = 32 - 2 = 30,
which is clearly a multiple of 5. Hence, our base case holds true.

Inductive Step: Assume that for some integer k > 1, the statement holds true, i.e.,
k5 - k is a multiple of 5. We need to show that k5 - k + 5(k4 + k3 + k2 + k + 1) is also a multiple of 5.

If k5 - k is a multiple of 5, then adding a number which is a multiple of 5 (5 times a sum of powers of k) to it will also result in a multiple of 5. This means that (k + 1)5 - (k + 1) will be a multiple of 5, hence the statement holds for k + 1. By the principle of mathematical induction, the statement holds true for all integers n > 1.

Answer:

x = -5 sin (2t)

Step-by-step explanation:

k is the spring stiffness.  The unstretched length of the spring is L.

When the mass is added, the spring stretches to an equilibrium position of L+s, where mg = ks.  When the mass is displaced a distance x (where x is positive if the displacement is down and negative if it's up), the spring is stretched a total distance s + x.

There are two forces on the mass: weight and force from the spring.  Sum of the forces in the downward direction:

∑F = ma

mg − k(s + x) = ma

mg − ks − kx = ma

Since mg = ks:

-kx = ma

Acceleration is second derivative of position, so:

-kx = m d²x/dt²

Let's find k:

F = kx

400 = 2k

k = 200

We know that m = 50.  Substituting:

-200x = 50 d²x/dt²

-4x = d²x/dt²

d²x/dt² + 4x = 0

This is a linear second order differential equation of the form:

x" + ω² x = 0

The solution to this is:

x = A cos (ωt) + B sin (ωt)

Here, ω² = 4, so ω = 2.

x = A cos (2t) + B sin (2t)

We're given initial conditions that x(0) = 0 and x'(0) = -10 (remember that down is positive and up is negative).

Finding x'(t):

x' = -2A sin (2t) + 2B cos (2t)

Plugging in the initial conditions:

0 = A

-10 = 2B

Therefore:

x = -5 sin (2t)