The Fibonacci sequence is 0, 1, 1, 2, 3, 5, 8, 13, . . . (where F0 0).
(a) Give the recursive definition for the sequence.
(b) Write out the first few terms of the sequence of partial sums: 0, 0 + 1, 0 + 1 + 1,. . .
(c) Give a closed formula for the sequence of partial sums in terms of Fn (for example, you might say F0 + F1 + · · · + Fn 3F 2 n−1 + n, although that is definitely not correct).
(Discrete Mathematics)

The number π/t is an irrational and positive number.

The rational numbers are composed by the set of all numbers that can be represented by fractions, such as whole numbers, terminating decimals, or repeating decimals.

The irrational numbers, conversely, are composed by the set of numbers that cannot be represented by fractions, such as non-exact roots and non-terminating decimals.

The number π is defined as follows:

π = 3.1415...

Which is a non-terminating decimal, hence it is an irrational number.

When an irrational number is divided by a rational number, as is the case in this problem, the result is irrational.

π is a positive number, just like t, hence the quotient is also a positive number.

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This steady-state number of fish is ...  1200 - 1058 = 142 . . . . below the original carrying capacity

The differential equation is modified by adding a -100 fish per year constant term:

 (dN)/(dt) = (0.4)/(1200)N(1200-N) -50

The steady-state value of the fish population will be when N reaches the value that makes dN/dt = 0.

 (0.4/1200)(N)(1200-N) -50 = 0

 N(N-1200) = -(50)(1200)/0.4) . . . . rewrite so N^2 has a positive coefficient

 N^2 -1200N + 600^2 = -150,000 +600^2 . . . . complete the square

 (N -600)^2 = 210,000 . . . . . simplify

 N = 600 + √210,000 ≈ 1058

This steady-state number of fish is ...

 1200 - 1058 = 142 . . . . below the original carrying capacity

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The equation of the graphed on the coordinate grid parallel to line M y = x and passing through the point P(4, -7) is equal to y = x - 11.

As given in the question,

Equation of the line M is given by :

y = x

Slope of the line M by comparing it with standard equation of the line with slope m is given by :

y = mx + c

Equation of the line M is y = x

Here m = 1

Slope of the required line is m₁ = 1 as it is parallel to line M

Equation of the required line which passes through ( x₁ , y₁) = ( 4, -7) with slope m₁ = 1 is given by:

y - y₁ = m₁( x - x₁)

⇒ y + 7 = 1( x - 4)

⇒ y= x - 4 -7

⇒ y= x - 11

Therefore, the equation of the required line which is parallel to line M and passing through point P is equal to y = x - 11.

Line M is graphed on the coordinate grid. y = x. Which equation can be used to represent a line parallel to M passing through the point P(4, -7)?

A.  x = 4          B. x = -7             C. y = -7              D. y = 4

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If the parent function f(x) = x² is horizontally shrunk by factor of 1/3 ad vertically stretch by factor of 4 then g(c) will be g(x) = 4(1/3x)² , the correct option is (c)  .

In the question ,

it is given that ,

the parent function is ⇒ f(x) = x² ;

the graph is horizontally shrunk by a factor of 1/3 that means

the function will become (1/3x)² .

next the function is then vertically stretch by a factor of 4 ,that means

then the function becomes g(x) = 4(1/3x)².

Therefore , the correct option is g(x) = 4(1/3x)²  .

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a brokerage firm keeps track of the total money spent on tesla stocks by their clients each day of the week is C(T) = $730 + $25T

R(T) = $35T

T = 193 transact

C = cost ; R = revenue ; T = number of transactions

Amount paid per transaction = $25

Cost keeping office open = $730

Amount collected on each transaction = $35

(a) Find a formula that gives C as a function of T.

C(T) = Cost of keeping office open + (cost per transaction × number of transactions)

C(T) = $730 + $25T

(b) Find a formula that gives R as a function of T.

R(T) = (Amount collected per transaction * number of transactions)

R(T) = $35T

(c) Find the number of daily transactions that are needed to make the revenue $1200 more than the cost.

R = C + 1200

Substitute the value of R and C into the equation:

35T = 730 + 25T + 1200

35T - 25T = 730 + 1200

10T = 1930

T = 1930 / 10

T = 193 transactions

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Let X and Y be random variables that are independent, and their common pdf is f(x) = e x, (x > 0). Find the joint pdf of U = X/(X + Y), V = X + Y. Then, at that point, |J| = v(1 − u) + uv = v(> 0).

Keep in mind that 0 u  1, 0 v. The joint probability density function (joint pdf) of the continuous random variables X and Y is a piecewise continuous function that satisfies the following if they are defined on the same sample space S. P(Xa and Yb)=baf(x,y)dxdy = F(a,b).

A continuous random variable's probability density function, or PDF, indicates the relative likelihood of any continuum outcome occurring. For a continuous random variable, every outcome has a probability of zero, in contrast to discrete random variables.

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18. not a function

19. not a function

20. the left side graph is a function

18. Any relation to be a function, any x value should have only one y value

this is not a function because when x=7 , y= 9 and 12 (two y values for one x value)

19. x=-1 has two y values

and x=1 also has two y values

20. here, you can use the vertical line test.

the left graph passes the vertical line test which is a function because any vertical line drawn intersects the curve in only 1 point, but the right graph, the vertical line intersects the curve in 2 points, which is not a function.

To determine that an item is not in an unordered array of 100 items, total numbers of values which have to linear search examine on average is 100. The correct answer is C.

Linear Search is described as a sequential search algorithm which starts at one end and goes through each element of a list until the needed element is found, otherwise the search continues till the end of the data set. This is the simplest searching algorithm.

Starting at the beginning of the data set, each item of data is examined until a match is made. Once the item is found, the search ends.

Hence, in this case, since there are 100 items, then there will be 100 numbers of values.

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Based on probability, three of the four selected calculators will contain no defective.

Probability refers to the likelihood or chance that an expected outcome occurs in the midst of many possible outcomes or events.

Probability is zero when uncertain and one when 100% certain.  Otherwise, probability lies between zero and one, which yields a fractional value denoted by decimals, percentages, or fractions.

The number of shipped graphing calculators = 30

The number of defective calculators = 7

The probability of defective calculators = 0.233 (7/30)

The probability of no defective calculators = 0.767 (1- 0.233)

The number of calculators to be sent to a high school = 4

The number of these that will contain no defective calculators = 3.068 (4 x 0.767)

= 3

Thus, the local high school will probably receive 3 non-defective units.

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The commission earn by Mrs. Parker earn last month is $ 4,582.35

What does math's % mean?

In essence, percentages are fractions with a 100 as the denominator. We place the percent symbol (%) next to the number to indicate that the number is a percentage. For instance, you would have received a 75% grade if you answered 75 out of 100 questions correctly on a test (75/100).

What is the formula's percentage?

By dividing the value by the entire value and multiplying the result by 100, one may determine the percentage. The percentage calculation formula is (value/total value)100%.

Given that:

Total sales = $53,910

Commission = 8.5%

Commission in cash = 53910 * 0.085 = 4,582.35

Hence commission earn by Mrs. Parker earn last month is $ 4,582.35

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The length of the segment of the MN is 10 units, by using the midpoint formula and distance formula.

The algebraic phrase is known as the distance formula, which provides the distances between two places in terms of their coordinates (see coordinate system). The distance formulas for points in rectangular coordinates in two- and three-dimensional Euclidean space are based on the Pythagorean theorem.

Since M is the midpoint of side AB and N is the midpoint of side AC, we can use the midpoint formula to find the coordinates of points M and N.

Midpoint Formula: (x₁ + x₂)/2, (y₁ + y₂)/2

For M:

(A x + B x)/2, (A y + B y)/2

= (18 + 0)/2, (0 + 0)/2

= 9, 0

For N:

(A x + C x)/2, (A y + C y)/2

= (18 + 20)/2, (0 + 0)/2

= 19, 0

Step 4: Now that we have the coordinates of M and N, we can use the distance formula to find the length of the segment MN.

Distance Formula: √((x₂ - x₁)² + (y₂ - y₁)²)

For MN:

√((19 - 9)² + (0 - 0)²)

= √(100)

= 10

Therefore, the length of the segment MN is 10 units.

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The value of x that will make the two given solids similar is: d. x = 31.

Similar solids are solids that are scale copy of each other, that is, either of the solids can produce the other when enlarged by a scale factor to get the dimension of the other solid.

Therefore, if two solids are similar, their corresponding linear dimensions would have an equal ratio, or they would be proportional.

Applying the definition of similar solids, the equation below would be created in order to determine the value of x:

x/15.5 = 29/14.5

Cross multiply:

14.5x = 15.5 × 29

14.5x = 449.5

14.5x/14.5 = 449.5/14.5

x = 31

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The two possible variables design are; "COHORT STUDY" and "CROSS-SECTIONAL STUDY".

Cohort study design is an observational study in which people who receives a particular treatment or are under a particular condition, are been observed and studied. This type of research model also allows a researcher to compare people affected by a condition to people unaffected by the same conditions.

The investigator is likely to use the cohort design because it will help in studying the children that are always exposed to smoke, and to observe the affected and unaffected children.

The CROSS-SECTIONAL design is an observational study in which a specific population are studied at a particular time. This type of research allows the researcher to study the difference in that population at a time interval.

This study design is likely to be used by the investigator, because it will help the investigator to determine the amount of secondhand smoke each child inhale at a particular time. It will also help the investigator to know the outcome of each child after 5years

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Answer:

8584946569585889594934u

Based on the null hypotheses, the type of error is Decrease the significance level to decrease the probability of Type I error.

What does null hypotheses means?

In math, null hypotheses refers a type of statistical hypothesis that proposes that no statistical significance exists in a set of given observations.

Here we have give that researchers are testing a new diagnostic tool designed to identify a certain condition. the null hypothesis of the significance test is that the diagnostic tool is not effective in detecting the condition. for the researchers, the more consequential error would be that the diagnostic tool is not effective, but the significance test indicated that it is effective.

And we need to find in which of the following should the researchers do to avoid the more consequential error.

According to the given question, the null hypothesis is that the new diagnostic tool is not effective in detecting the condition.

Here the more consequential error described is that the diagnostic tool is not effective, but the significance test indicated that it is effective. This is a type I error.

And in Hypothesis testing, a type I error involves rejecting the null hypothesis and accepting the alternative hypothesis when in reality, the null hypothesis is true.

Therefore, it involves saying there is significant evidence to show that the new diagnostic tool is effective in detecting the condition when in reality, it isn't.

When the level of significance (α) of a hypothesis test directly gives the probability of a type I error. So, by setting it lower, we reduce the chances of a type I error.

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Hamiltonian group is  an example of a nonabelian group g with the property that every subgroup h of g is a normal subgroup.

To prove that  h be a subgroup of g. and suppose h is the only subgroup of g of order(h) h is normal in g is that it is proven by homomorphism .

This is a small contribution, however in reaction to a remark above, you may additionally clear up this with out the use of any homomorphism properties. Assume the subgroup H has order n and select out g∈g.

Since H has order n. The above holds for all h∈H, so the subgroup gHg−1 has order n, so it's miles identical to H through assumption. Then each h∈H has a few h∈H such that ghg−1=h′, implying gh=h′g, so gH=H

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Answer:

[tex]\frac{dy}{dx}=\frac{4}{3}x^{1/3}+e^x +\cos x[/tex]

Step-by-step explanation:

[tex]\frac{d}{dx} x^{4/3}=\frac{4}{3}x^{1/3} \\ \\ \frac{d}{dx} e^x=e^x \\ \\ \frac{d}{dx} \sin x=\cos x \\ \\ \therefore \frac{dy}{dx}=\frac{4}{3}x^{1/3}+e^x +\cos x[/tex]

89 is the money you will need to take to the sari-sari store and x=-1±√13/4 is the value of polynomial 4x² + 2x - 3.

Two or more expressions with an Equal sign is called as Equation.

Given that need to buy 3 eggs cost 8.00

2 hotdogs cost 10.00

3 bread cost 5.00 each.

We need to find the money you will need to take to the sari-sari store.

3(8)+2(10)+3(15)

24+20+45

89

The given polynomial is 4x² + 2x - 3=0

a=4, b=2 and c=-3

x=-b±√b²-4ac/2a

x=-2±√4-4(4)(-3)/8

x=-1±√13/4

Hence, 89 is the money you will need to take to the sari-sari store and x=-1±√13/4 is the value of polynomial 4x² + 2x - 3.

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Tipton Tile expected to sell between, 5967 and 6902 square feet of tiles.

Multiplication is a mathematical arithmetic operation. It is also a process of adding the same types of expression for some number of times.

Example - 2 × 3 means 2 is added three times, or 3 is added 2 times.

Given,

Tipton Tile has seen an increase in tile sales over the last few weeks.

They sold 4,212 square feet of tile to 36 customers last week.

That means,

4212 / 36 = 117 square feet of tiles per person.

If they are expecting between 51 and 59 people to buy tile this week.

To find the number of square feet of tiles:

For 51 people,

117 x 51 = 5967 square feet of tiles.

And for 59 people,

117 x 59 = 6903 square feet of tiles.

Therefore, They expected to sell between, 5967 and 6902 square feet of tiles.

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If the base of the ladder is moving at 5 ft/s , then the speed of top of the ladder is 12ft/s .

Pythagoras Theorem states that the relation between the square of the longest side is equal to the squares of the other two sides .

the length of the ladder (L) = 13 ft ;

the rate of movement of the base (dx/dt) = 5 m/s ;

By Pythagoras Theorem ,

13 = x + y         ....eqn(1)

So , y = √(169 - x)

putting x = 12 , we get ;

y = √(169 - 144)  = 5 ft ;

Differentiating eqn(1) , w.r.t. "x" ,

we get ;

0 = 2x(dx/dt) + 2y(dy/dt)

So , dy/dt = (-x/y)(dx/dt)

putting the values ,

we get ;

dy/dt = -12 ft/s  .

Therefore , the ladder is sliding down the wall at 12 ft/s .

The given question is incomplete , the complete question is

A 13 foot ladder is leaning against the house when its base starts to slide away. By the time it is 12 feet from the house, the base is moving at the rate of 5 ft/sec.

How fast is the top of the ladder sliding down the wall at that moment ?

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Answer: The number is 3

Step-by-step explanation:

2(x+9)=24

x+9=12

x=3

As per the chi square test, the variable are dependent to each other.

What is meant by chi-square test?

In math, the chi square test refer a statistical test used to compare observed results with expected results.

From the table in the question; we can see the changes in employees adding, shedding, or not changing their staffing.

Converting the table to chi- squared using the relation.

Here the observed frequency is = 44 and the expected frequency is = 34.8

And the Number Change 0.1421 0.0079 0.15

=> Lay Off 4.8286 2.4083 7.2369

=> Total = 14.657

Therefore, the total chi-square = 14.657;

In order to find the value for p; we need to determine the degree of freedom

=> df = (2-1)(3-1)

Then the result to a degree of freedom of 2

Here from the chi square chart at the chi-square is 14.657 and degree of freedom is 2 ; the p value is between 0.1 and 0.005. Since this makes p-value less than 0.05.

Therefore, the variables are dependent.

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To test the CEO's claim about the mean wait time we need to conduct a hypothesis test.

The null hypothesis is that the mean wait time is no more than 7 minutes, which we can represent as H0: μ ≤ 7. The alternative hypothesis is that the mean wait time is greater than 7 minutes, which we can represent as Ha: μ > 7. We can use a one-tailed t-test for this situation, with a significance level of alpha = 0.05.

To perform the t-test, we need to calculate the sample mean, sample standard deviation, and the t-statistic. The sample mean is the sum of the wait times divided by the sample size, which is 36 in this case. The sample standard deviation is a measure of the dispersion of the data. The t-statistic is calculated using the sample mean, sample standard deviation, and the hypothesized mean wait time of 7 minutes.

Once we have calculated the t-statistic, we can compare it to the critical value from the t-distribution table to determine the p-value. If the p-value is less than alpha, we can reject the null hypothesis and conclude that the mean wait time is indeed greater than 7 minutes. If the p-value is greater than alpha, we cannot reject the null hypothesis and must conclude that the mean wait time is no more than 7 minutes.

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The expressions that represents Carissa's distance to the right of the center of the skitrail, i.e, h in terms of the number of times Carissa has traveled since she started skiing,d is h = 1.3 cos( d/1.3). So, correct answer is option (c).

We have, Carissa skis along a circular ski trail. Radius of circle , r = 1.3 km

Carissa starts at the 3-o'clock position of the trail travels in the counter clockwise direction. let "d" km be total distance that Carissa has traveled since she started skiing and "h" km be Carissa's distance to the right of the center of the skitrail in km.

We have to wright an expressions for "h" in terms of d . Here, we see an sector is formed with one side length is r and arc's distance is d km . Let θ be angle formed by this sector . Using the sector formula,

θ = arc length /radius of circle

=> θ = d/r = d/1.3 --(1)

Also, ∆ formed here , h = r cos θ

=> h = 1.3 cos( d/1.3)

Hence, the required expression is h=1.3 cos( d/1.3).

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Complete question:

Carissa skis along a circular ski trail with a radius 1.3 km long. Carissa starts at the 3-o'clock position of the trail (the East-side) and travels in the CCW direction. Which of the following expressions represents Carissa's distance to the right of the center of the skitrail in km, h, in terms of the number of km that Carissa has traveled since she started skiing.d?

a)h 1.3 cos(d+ 1.3)

b) h = 1.3 cos(1.3/d)

c)h = 1.3 cos (d /1.3 )

d)h = 1.3 cos(d)

e) h = COS( 1.3 )

Submit Question 16. Points possible:

For the given velocity function , the object will not return to the origin .

In the question ,

it is given that ,

the velocity function is v(t) = { t/35  , 0 ≤ t ≤ 70

                                               { 2    ,  70 ≤ t ≤ 175

                                               { 7 - t/35  ,   175 < t  .

So , For t ≤ 175 , velocity is positive ,

So , [tex]\int\limits^T_0 {v(t) } \, dt[/tex]  = [tex]\int\limits^{70}_0 {t/35 } \, dt[/tex]  +  [tex]\int\limits^{175}_{70} {2 } \, dt[/tex]  + [tex]\int\limits^{T}_{175} {(7 - t/35) } \, dt[/tex]

On integrating ,

we get ,

x(T) = [t²/70]⁷⁰₀  + [2t]¹⁷⁵₇₀  + [7t - t/70[tex]]^{T} _{175}[/tex]

x(T) = (70 - 0) + (350 - 0) + (7T - T²/70) - (1225 - 875/2)

x(T) = 420 + 7T - T²/20 - 1575/2

x(T) = -T²/20 + 7T - 735/2

x(T) = ( -T² + 140T - 7350)/20

For the object to return to the origin , x(T) = 0

So , (-T² + 140T - 7350)/20 = 0

T² - 140T + 7350 = 0

Therefore , Discriminant (D) = b² - 4ac

= (-140)² - 4(1)(7350)

= 19600 - 29400

= -9800 < 0 .

So , the root is imaginary , hence at no value of T , the object will not return to the origin for the given information .

Therefore , the object will not return to the origin .

The given question is incomplete , the complete question is

An object at the origin at time has velocity  

                                      v(t) = { t/35  , 0 ≤ t ≤ 70

                                               { 2    ,  70 ≤ t ≤ 175

                                               { 7 - t/35  ,   175 < t  .

measured in meters per second, when, if ever, does the object return to the origin ?

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It is based on a sample and provides information about the sample based on a survey of 35 samples, this four cell phone is a statistic, so the response is a statistic.

When looking at populations, the term "median" is a significant and frequently used statistic. Most of the time, it's used to represent a set of numbers.

We sort the population numbers to find the one in the middle to find the median. It differs from the mean and can sometimes be more useful than the mean, such as when there are outliers that could have an excessive impact on the mean. We should take the average of the two values in the middle if the number of values is even.

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The sample mean is = 6.98, the standard deviation is = 4.53 (2) The pint estimate difference is = 20.2 (3) The confidence interval limits are $ 15943.6 and $24456.4

a)

Private Universities

Sample median: 42,500 dollars

Mean = (Σx)/N

where N = Sample size

Σx = sum of all variables

Private colleges

Σx = (52.8+43.2+45.0+33.3+44.0+30.6+45.8+37.8+50.5+42.0) = 425

N = 10

Mean = 425/10 = 42.5 thousand dollars

Standard deviation = S1 = √[Σ(x - xbar)²/N]

Σ(x - xbar)² = (52.8-42.5)² + (43.2-42.5)²

+ (45.0-42.5)² + (33.3-42.5)²

+ (44.0-42.5)² + (30.6-42.5)² + (45.8-42.5)² + (37.8-42.5)² + (50.5-42.5)² + (42.0-42.5)² = 438.56

N = 10

Standard deviation = √[Σ(x - xbar)²/N]

Standard deviation = √(438.56/10) = 6.62 thousand dollars

S1 or the standard deviation equals $6.62k.

b)

The sample means the difference in both circumstances is 42.5 minus 22.3, or 20.2 thousand dollars.

Going to a private college often costs 20.2 thousand dollars more than going to a public institution on average.

c)

A sample distribution's 95% confidence interval for the price difference between private and public colleges is provided as

(15.0, 25.4) dollars in thousands.

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The slope m = -5/4 , y - intercept is 5  and the equation in slope - intercept form is y = (-5/4)x + 5 .

Given :

The graph of a linear function is shown on the grid.

From the graph :

Take two points ( 0 , 5 ) and ( 4 , 0 )

slope m = 0 - 5 / 4 - 0

= -5 / 4

we know that ,

slope intercept form is y = mx + b

substitute m and ( 0 , 5 )

5 = -5 / 4 ( 0 ) + b

5 = 0 + b

b = 5 ( y - intercept )

substitute b and m value

y = (-5/4)x + 5

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To confirm that the solution is correct, we can use the method of substitution. Substitute the solution back into the original differential equation, and if the equation is satisfied, then the solution is correct.

To confirm that the proposed solution is indeed a valid solution to the original equation, we can use the method of substitution. This involves substituting the proposed solution into the original equation and verifying that it satisfies the equation. If the equation holds, then the solution is correct. This method is useful for verifying the correctness of a proposed solution. It can also be used to determine if a proposed solution is the only solution, or if there are other solutions to the equation.

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Answer:

see explanation

Step-by-step explanation:

1

[tex]\frac{bh}{2}[/tex] = A ( multiply both sides by 2 to clear the fraction )

bh = 2A ( divide both sides by h )

b = [tex]\frac{2A}{h}[/tex]

2

2l + 2w = P ( subtract 2l from both sides )

2w = P - 2l ( divide both sides by 2 )

w = [tex]\frac{P-2l}{2}[/tex]

3

C = 2πd ( isolate d by dividing both sides by 2π )

[tex]\frac{C}{2\pi }[/tex] = d