Use mathematical induction to prove that for each integer n ≥ 4, 5^n ≥ 2^2n+1 + 100. (it is 5 to power of n and 2 to the power of 2n+1)

Answer:

The statement is true

Step-by-step explanation:

We will prove by mathematical induction that, for every natural n,

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

We will prove our base case, when n=1, to be true.

base case:

[tex]\sum^{1}_{i=1}\frac{1}{(2-1)(2+1)} =\frac{1}{3}=\frac{n}{2n+1}[/tex]

Inductive hypothesis:

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

Now, we will assume the induction hypothesis and then uses this assumption, involving n, to prove the statement for n + 1.

Inductive step:

[tex]\sum^{n+1}_{i=1}\frac{1}{(2i-1)(2i+1)} =\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)}+\frac{1}{(2(n+1)-1)(2(n+1)+1)}=\frac{n}{2n+1}+\frac{1}{(2n+1)(2n+3)}=\frac{n(2n+3)+1}{(2n+1)(2n+3)}=\frac{2n^2+3n+1}{(2n+1)(2n+3)}=\frac{(2n+1)(n+1)}{(2n+1)(2n+3)}=\frac{n+1}{2n+3}=\frac{n+1}{2(n+1)+1}[/tex]

With this we have proved our statement to be true for n+1.

In conlusion, for every natural [tex]n[/tex].

[tex]\sum^{n}_{i=1}\frac{1}{(2i-1)(2i+1)} =\frac{n}{2n+1}[/tex]

Answer:

(a) 65

(b) -26

(c) 54.6

(d) -54.6

Step-by-step explanation:

(a) [tex]f(5)=8(5)+5(5)=40+25=65[/tex]

(b) [tex]f(-2)=8(-2)+5(-2)=-16-10=-26[/tex]

(c) [tex]f(4.2)=8(4.2)+5(4.2)=33.6+21=54.6[/tex]

(d) [tex]f(-4.2)=8(-4.2)+5(-4.2)=-33.6-21=-54.6[/tex]

Answer: Our required probability is 0.406.

Step-by-step explanation:

Since we have given that

Probability of selecting an adult over 40 years of age with cancer = 0.05

Probability of a doctor correctly diagnosing a person with cancer as having the disease = 0.78

Probability of incorrectly diagnosing a person without cancer as having the disease = 0.06

Let A be the given event i.e. adult over 40 years of age with cancer. P(A) = 0.05.

So, P(A')=1-0.05 = 0.95

Let C be the event that having cancer.

P(C|A)=0.78

P(C|A')=0.06

So, using the Bayes theorem, we get that

[tex]P(A|C)=\dfrac{P(A).P(C|A)}{P(A).P(C|A)+P(A')P(C|A')}\\\\P(A|C)=\dfrac{0.78\times 0.05}{0.78\times 0.05+0.06\times 0.95}\\\\P(A|C)=0.406[/tex]

Hence, our required probability is 0.406.

Answer:

P(A)=1/7

P(B)=1/2

P(C)=1/42

P(A|C)=1/10

P(B|C)=1/10

P(A|B)=1/7

A and B are independent

A and C aren't independent

B and C aren't independent

Step-by-step explanation:

A="b falls in the middle"

- b can fall in seven possible places, but only one is the middle. So, P(A)=1/7

B="c falls to the right of b"

X=i means "b falls in the i-th position"

Y=j means "c falls in the j-th position"

if b falls in the first place, c can fall in the 2nd, 3rd, 4th, 5th, 6th or 7th place.

if b falls in the 2nd place, c can fall in the 3rd, 4th, 5th, 6th or 7th place

 ...

If b falls in the 6th place, c can fall in the 7th place

then:

[tex][tex]P(B)=\displaystyle\sum_{i=1}^{6}( P(X=i)\displaystyle\sum_{j=i+1}^{7} P(Y=j))=\displaystyle\sum_{i=1}^{6}( \frac{1}{7}\displaystyle\sum_{j=i+1}^{7} \frac{1}{6})=\frac{1}{42}\displaystyle\sum_{i=1}^{6}(\displaystyle\sum_{j=i+1}^{7}1)=\frac{6+5+4+3+2+1}{42}=\frac{1}{2}[/tex][/tex]

- if d falls in the 1st place, e falls in the 2nd and f in the 3rd place

- if d falls in the 2nd place, e falls in the 3rd and f in the 4th place

- if d falls in the 3rd place, e falls in the 4th and f in the 5th place

- if d falls in the 4th place, e falls in the 5th and f in the 6th place

- if d falls in the 5th place, e falls in the 6th and f in the 7th place

X=i means "d falls in the i-th position"

Y=j means "e falls in the j-th position"

Z=k means "f falls in the k-th position"

[tex]P(C)=\displaystyle\sum_{i=1}^{5}( P(X=i)P(Y=i+1)P(Z=i+2))=\displaystyle\sum_{i=1}^{5}(\frac{1}{7}\times\frac{1}{6}\times\frac{1}{5})=\frac{1}{210}\displaystyle\sum_{i=1}^{5}(1)=\frac{1}{42}[/tex]

P(A|C)=P(A∩C)/P(C)=?

A∩C:

- d falls in the 1st place, e in the 2nd, f in the 3rd and b in the 4th place

- b falls in the 4th place, d in the 5th place, e in the 6th, f in the 7th place

P(A∩C)=2*(1/7*1/6*1/5*1/4)=1/420

P(A|C)=(1/420)/(1/42)=1/10

P(B|C)=P(B∩C)/P(C)=?

X=i means "d falls in the i-th position"

Y=j means "e falls in the j-th position"

Z=k means "f falls in the k-th position"

V=k means "b falls in the k-th position"

W=k means "c falls in the k-th position"

[tex]P(B\cap C)=\displaystyle\sum_{i=1}^{3} P(X=i)P(Y=i+1)P(Z=i+2)\displaystyle\sum_{j=i+3}^{6}P(V=j)P(W=j+1)[/tex]

[tex]P(B\cap C)=\displaystyle\sum_{i=1}^{3} \frac{1}{7}\times\frac{1}{6}\times\frac{1}{5}(\displaystyle\sum_{j=i+3}^{6}\frac{1}{4}\times\frac{1}{3})=\frac{1}{2520}\displaystyle\sum_{i=1}^{3} \displaystyle\sum_{j=i+3}^{6}1=\frac{1}{420}[/tex]

P(B|C)=(1/420)/(1/42)=1/10

P(A|B)=P(B∩A)/P(B)=?

B∩A:

- b falls in the 4th place and c in the 5th

- b falls in the 4th place and c in the 6th

- b falls in the 4th place and c in the 7th

P(B∩A)=3*(1/7*1/6)=1/14

P(A|B)=(1/14)(1/2)=1/7

If one event is independent of another, P(X∩Y)=P(X)P(Y)

So:

P(A∩B)=1/14=(1/7)*(1/2)=P(A)P(B), A and B are independent

P(A∩C)=1/420≠(1/7)*(1/42)=1/294=P(A)P(C), A and C aren't independent

P(B∩C)=1/420≠(1/2)*(1/42)=1/84=P(A)P(C), B and C aren't independent

Answer: [tex](0.0628,\ 0.1186)[/tex]

Step-by-step explanation:

Given : Significance level : [tex]\alpha:1-0.95=0.05[/tex]

Critical value : [tex]z_{\alpha/2}=\pm1.96[/tex]

Sample size : n= 408

Proportion of new websites registered on the Internet were anonymous :

[tex]\hat{p}=\dfrac{37}{408}\approx0.0907[/tex]

The formula to find the confidence interval for population proportion is given by :-

[tex]\hat{p}\pm z_{\alpha/2}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

i.e. [tex]0.0907\pm (1.96)\sqrt{\dfrac{0.0907(1-0.0907)}{408}}[/tex]

[tex]=0.0907\pm0.0278665515649\\\\\approx 0.0907\pm0.0279\\\\=(0.0907-0.0279,\ 0.0907+0.0279)\\\\=(0.0628,\ 0.1186)[/tex]

Hence,  the 95 percent confidence interval for the proportion of all new websites that were anonymous = [tex](0.0628,\ 0.1186)[/tex]

Step-by-step explanation:

When we need to subtract a number from another number, in that case, we can take the complement of the first number to add it to the second number, the result will be the same. It is because when we take the complement of 9 of that number, it will represent the negative of that number. Hence, by adding the negative of a number we will get the same result as we get after subtraction.

For example:

Subtract 213 from 843

843 - 213 = 630

complement of 9 of 213= 999-213

                                       =786

Now, add 786 and 843

786+843=1629

We got the result in 4 digits so by adding the left-most digit to the right-sided three-digit number of the result, we will get

629+1 = 630

Answer:

19/30

Explanation:

1. Exchange them to a common factor which happens to be 30 for both of them

2. Multiply by that factor on both the top and bottom to get the number equivalent to a fraction of that category

3. Subtract

4. Simplify, however in this case simplification isn't doable.

Answer:

θ = -33.69°

Step-by-step explanation:

For Φ>0 and Φ<0  (in general Φ≠nπ  where n is an integer), sin(Φ) ≠ 0

Dividing both equations:

[tex]\frac{sin(\phi) sin(\theta)}{sin(\phi)cos(\theta)} = tan(\theta) = 0.2/(-0.3)=-2/3\\[/tex]

Therefore:

arctan(θ) = -2/3

  θ = -33.69°

The answer does not depend on the sign of Φ, in fact we just need that the sine does not become zero, which occurs when Φ is equal to an integer times π (radians) or 180 (degrees)

Have a nice day!

To find theta (θ) given that sin phi (φ) sin theta (θ) = 0.2 and sin phi (φ) cos theta (θ) = -0.3 with sin phi (φ) being positive or negative, one must first eliminate sin phi (φ) by manipulating the given equations, then solve for theta (θ) using trigonometric identities and inverse functions based on the signs of sin and cos.

We have two equations involving sin φ and θ (theta): sin φ sin θ = 0.2 and sin φ cos θ = -0.3. Also, it is given that sin φ > 0 or sin φ < 0. To find θ, first, we need to derive an equation involving only θ by eliminating sin φ. We can do this by squaring and adding both equations.

∑: (sin φ sin θ)^2 + (sin φ cos θ)^2 = 0.2^2 + (-0.3)^2 = 0.04 + 0.09 = 0.13

Using the Pythagorean identity sin^2 θ + cos^2 θ = 1, we can rewrite ∑ as sin^2 φ = 0.13. To solve for θ, we can take either of the initial equations, say sin φ sin θ = 0.2, and substitute sin^2 φ from ∑ giving sin θ = (0.2 / √0.13) or cos θ = (-0.3 / √0.13). Both positive and negative values of sin φ lead to the calculation for different θ values. The actual values of θ are determined by using the arc functions (arcsin, arccos) for both positive and negative scenarios of sin φ, taking into account the range of θ based on the signs of sin and cos.

Answer:

(A) [tex]y=ke^{2t}[/tex] with [tex]k\in\mathbb{R}[/tex].

(B) [tex]y=ke^{2t}/2-1/2[/tex] with [tex]k\in\mathbb{R}[/tex]

(C) [tex]y=k_1e^{2t}+k_2e^{-2t}[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex]

(D) [tex]y=k_1e^{2t}+k_2e^{-2t}+e^{3t}/5[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex],

Step-by-step explanation

(A) We can see this as separation of variables or just a linear ODE of first grade, then [tex]0=y'-2y=\frac{dy}{dt}-2y\Rightarrow \frac{dy}{dt}=2y \Rightarrow  \frac{1}{2y}dy=dt \ \Rightarrow \int \frac{1}{2y}dy=\int dt \Rightarrow \ln |y|^{1/2}=t+C \Rightarrow |y|^{1/2}=e^{\ln |y|^{1/2}}=e^{t+C}=e^{C}e^t} \Rightarrow y=ke^{2t}[/tex]. With this answer we see that the set of solutions of the ODE form a vector space over, where vectors are of the form [tex]e^{2t}[/tex] with [tex]t[/tex] real.

(B) Proceeding and the previous item, we obtain [tex]1=y'-2y=\frac{dy}{dt}-2y\Rightarrow \frac{dy}{dt}=2y+1 \Rightarrow  \frac{1}{2y+1}dy=dt \ \Rightarrow \int \frac{1}{2y+1}dy=\int dt \Rightarrow 1/2\ln |2y+1|=t+C \Rightarrow |2y+1|^{1/2}=e^{\ln |2y+1|^{1/2}}=e^{t+C}=e^{C}e^t \Rightarrow y=ke^{2t}/2-1/2[/tex]. Which is not a vector space with the usual operations (this is because [tex]-1/2[/tex]), in other words, if you sum two solutions you don't obtain a solution.

(C) This is a linear ODE of second grade, then if we set [tex]y=e^{mt} \Rightarrow y''=m^2e^{mt}[/tex] and we obtain the characteristic equation [tex]0=y''-4y=m^2e^{mt}-4e^{mt}=(m^2-4)e^{mt}\Rightarrow m^{2}-4=0\Rightarrow m=\pm 2[/tex] and then the general solution is [tex]y=k_1e^{2t}+k_2e^{-2t}[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex], and as in the first items the set of solutions form a vector space.

(D) Using C, let be [tex]y=me^{3t} [/tex] we obtain that it must satisfies [tex]3^2m-4m=1\Rightarrow m=1/5[/tex] and then the general solution is [tex]y=k_1e^{2t}+k_2e^{-2t}+e^{3t}/5[/tex] with [tex]k_1,k_2\in\mathbb{R}[/tex], and as in (B) the set of solutions does not form a vector space (same reason! as in (B)).  

Answer:

Cost function C(x) == FC + VC*Q

Revenue function R(x) = Px * Q

Profit function P(x) =(Px * Q)-(FC + VC*Q)

P(12000) = -38000 Loss

P(23000) = 28000 profit

Step-by-step explanation:

Total Cost is Fixed cost plus Variable cost multiplied by the produce quantity.  

(a)Cost function

C(x) = FC + vc*Q

Where  

FC=Fixed cost

VC=Variable cost

Q=produce quantity

(b)

Revenue function

R(x) = Px * Q

Where  

Px= Sales Price

Q=produce quantity

(c) Profit function

Profit = Revenue- Total cost

P(x) =(Px * Q)-(FC + vc*Q)

(d) We have to replace in the profit function

at 12,000 units

P(12000) =($20 * 12,000)-($110,000 + $14*12,000)

P(12000) = -38000

at 23,000 units

P(x) =($20 * 23,000)-($110,000 + $14*23,000)

P(23000) = 28000

Answer:

(x,y,z)=(2,0,-4)

Step-by-step explanation:

[tex]\left[\begin{array}{ccc|c}2&2&-3&16\\2&-3&2&-4\\4&-1&3&-4\end{array}\right][/tex]

Using the elementary operations

[tex]\left[\begin{array}{ccc|c}2&2&-3&16\\0&-5&5&-20\\0&-5&9&-36\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&-1&4\\0&0&4&-16\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&-1&4\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&1&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&0&-3/2&8\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

[tex]\left[\begin{array}{ccc|c}1&0&0&2\\0&1&0&0\\0&0&1&-4\\\end{array}\right][/tex]

The answer is determined:

x=2

y=0

z=-4

You can check they are correct, by entering in the original formulas.

For this case we propose a rule of three:

110.2 million -------------> 100%

x --------------------------------------> 17.7%

Where the variable "x" represents the number of households (in millions) that watched the finals of a popular reality series.

[tex]x =  \frac {17.7 * 110.2} {100}\\x = \frac {1950.54} {100}\\x = 19.5054[/tex]

Thus, a total of 19.5054 million homes watched the finals of a popular reality series.

Answer:

19.5054 million homes watched the finals of a popular reality series.

Answer:

The age of this sample is 13,417 years.

Step-by-step explanation:

The amount of carbon 14 present in a sample after t years is given by the following equation:

[tex]C(t) = C_{0}e^{-0.00012t}[/tex]

Estimate the age of a sample of wood discoverd by a arecheologist if the carbon level in the sampleis only 20% of it orginal carbon 14 level.

The problem asks us to find the value of t when

[tex]C(t) = 0.2C_{0}[/tex]

So:

[tex]C(t) = C_{0}e^{-0.00012t}[/tex]

[tex]0.2C_{0} = C_{0}e^{-0.00012t}[/tex]

[tex]e^{-0.00012t} = \frac{0.2C_{0}}{C_{0}}[/tex]

[tex]e^{-0.00012t} = 0.2[/tex]

[tex]ln e^{-0.00012t} = ln 0.2[/tex]

[tex]-0.00012t = -1.61[/tex]

[tex]0.00012t = 1.61[/tex]

[tex]t = \frac{1.61}{0.00012}[/tex]

[tex]t = 13,416.7[/tex]

The age of this sample is 13,417 years.

Answer:

0.2 kW-hr

Step-by-step explanation:

First, we are going to transform 100W in kW. We can use a rule of three in which we know that 1000W is equivalent to 1 kW, then how many kW are equivalent to 100W. This is:

1000W ------------- 1 kW

100W -------------- X

Where X is the the number of kW that are equivalent to 100W. Solving for X, we get:

[tex]X=\frac{100W * 1kW}{1000W} =0.1kW[/tex]

Then, for calculate the number of kW-hr we need to multiplicate the number of kW with the number of hours, This is:

0.1 kW * 2 hours = 0.2 kW-hr

Finally, when one 100W light bulbs is used for 2 hours, it used 0.2 kW-hr

Answer:

total mass of 6 samples = 538.896 g

in terms of X = 5.283 g

Step-by-step explanation:

Given:

Number of crude oil samples = 6

Density of each sample = 0.8240 kg/L

Volume filled by each sample = 1.09 × 10⁻⁴ m³

now,

1 m³ = 1000 L

thus,

1.09 × 10⁻⁴ m³ = 1.09 × 10⁻⁴ m³ × 1000 = 0.109 L

also,

Mass = Density × Volume

or

Mass of each sample = 0.8240 × 0.109 = 0.089816 kg

Thus,

total mass of 6 samples = Mass of each sample × 6

or

total mass of 6 samples = 0.089816 kg × 6 = 0.538896 kg

or

total mass of 6 samples = 538.896 g

or

in X = [tex]\frac{\textup{total mass of 6 samples}}{\textup{102}}[/tex]

= 5.283 g

Step-by-step explanation:

Consider the provided information.

If Fury is the director of SHIELD then Hill and Coulson are SHIELD agents"

For the condition statement [tex]p \rightarrow q[/tex] or equivalent "If p then q"

Part (A) Write the contrapositive.

Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.

Contrapositive: If Hill and Coulson are not SHIELD agents, then Fury is not the director of SHIELD.

Part (b) Write the converse.

Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.

Converse: If Hill and Coulson are SHIELD agents, then Fury is the director of SHIELD.

Part (c) Write the inverse.

Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.

Inverse: If Fury is not the director of SHIELD then Hill and Coulson are not SHIELD agents

Part (D) Write the negation.

Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.

Negation: Fury is the director of SHIELD and Hill and Coulson are not SHIELD agents"

Step-by-step explanation:

Consider the provided information.

If Fury is the director of SHIELD then Hill and Coulson are SHIELD agents." For the condition statement  or equivalent "If p then q"

The rule for Converse is: Interchange the two statements.

The rule for Inverse is: Negative both statements.

The rule for Contrapositive is: Negative both statements and interchange them.

The rule for Negation is: If p then q" the negation will be: p and not q.

- Part (A) Write the contrapositive.

.Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.

Contrapositive: If Hill and Coulson are not SHIELD agents, then Fury is not the director of SHIELD.

- Part (b) Write the converse.

 .Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.

 .Converse: If Hill and Coulson are SHIELD agents, then Fury is the director of SHIELD.

- Part (c) Write the inverse.

 .Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.

 .Inverse: If Fury is not the director of SHIELD then Hill and Coulson are not SHIELD agents

- Part (D) Write the negation.

 .Here p is Fury is the director of SHIELD, and q is Hill and Coulson are SHIELD agents.

 .Negation: Fury is the director of SHIELD and Hill and Coulson are not SHIELD agents"

Answer:   99 minutes

Step-by-step explanation:

Given: While completing a race, Edward spent 54 minutes walking.

The ratio of time walking to jogging was 6:5 i.e. [tex]\dfrac{6}{5}[/tex]     (1)

Let x be the time taken ( in minutes ) by him for jogging.

then, the ratio of time walking to jogging will be [tex]\dfrac{54}{x}[/tex]  (2)

From (1) and (2), we have

[tex]\dfrac{6}{5}=\dfrac{54}{x}\\\\\Rightarrow\ 6x=54\times5\\\\\Rightarrow\ x=\dfrac{54\times5}{6}=45[/tex]  

So, the number of minutes he took for jogging = 45 minutes

Now, the total time he spent on completing the race= 54+45=99 minutes

Answer:

Step-by-step explanation:

Given are the observations are on stabilized viscosity (cP) for specimens of a certain grade of asphalt with 18% rubber added:

2767 2924 3042 2844 2895

No of items = 5

If written in ascending order the order would be

2767   2844   2895   2924   3042

Hence median is the middle value in the ordered row = 2895

Mean = sum/5

=[tex]\frac{14472}{5} =2894.4[/tex]

To calculate the time Izzy takes to run to school on rainy days, we use the distance and speed to find that she runs 1.7 miles in approximately 13.6 minutes. We are unable to calculate the distance through the woods or the time saved without further information.

Calculation of Time and Distance

To calculate the time Izzy takes to run to school on rainy days, we use the following formula:

Time (in hours) = Distance (in miles) / Speed (in miles per hour)

Izzy runs a total distance of 1.2 miles on West St and then 0.5 miles on North Ave, summing up to 1.7 miles. Given Izzy's speed is 7.5 miles per hour, the time taken to run to school on rainy days can be calculated as:

Time = (1.2 + 0.5) miles / 7.5 mph = 1.7 / 7.5

To find the time in minutes, multiply the time in hours by 60:

Time in minutes = (1.7 / 7.5)  imes 60
= 13.6 minutes (approximately)

As the dashed path through the woods on dry days is not described in the question, we cannot calculate the exact distance Izzy is traveling through the woods. Without this information, we also cannot calculate the time saved by cutting through the woods.

Answer:

n = 60

Step-by-step explanation:

GIVEN DATA:

Total sum of consecutive number is 1800

sum of n number is given as

[tex] sum = \frac{ n(n+1)}{2}[/tex]

where n is positive number and belong to natural number i.e 1,2,3,4,...

from the data given we have[tex]1800 = \frac{n(n+1)}{2}[/tex]

solving for n we get

[/tex]n^2 + n -3600 = 0[/tex]

n = 59.5, -60.5

therefore n = 60

Step-by-step explanation:

Proof for i)

We will prove by mathematical induction that, for every natural [tex]n\geq 4[/tex], the number of diagonals of a convex polygon with n vertices is [tex]\frac{n(n-3)}{2}[/tex].

In this proof we will use the expression d(n) to denote the number of diagonals of a convex polygon with n vertices

Base case:

First, observe that:, for n=4, the number of diagonals is

[tex]2=\frac{n(n-3)}{2}[/tex]

Inductive hypothesis:  

Given a natural [tex]n \geq 4[/tex],

[tex]d(n)=\frac{n(n-3)}{2}[/tex]

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that, given a convex polygon with n vertices, wich we will denote by P(n), if we add a new vertix (transforming P(n) into a convex polygon with n+1 vertices, wich we will denote by P(n+1)) we have that:

Because of these observation we know that, for every [tex]n\geq 4[/tex],

[tex]d(n+1)=d(n)+1+(n-2)=d(n)+n-1[/tex]

Therefore:

[tex]d(n+1)=d(n)+n-1=\frac{n(n-3)}{2}+n-1=\frac{n^2-3n+2n-2}{2}=\frac{n^2-n-2}{2}=\frac{(n+1)(n-2)}{2}[/tex]

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural [tex]n \geq 4[/tex],

[tex]d(n)=\frac{n(n-3)}{2}[/tex]

Proof for ii)

Observe that:

Then, the statement is not true for n=1,2,3.

We will prove by mathematical induction that, for every natural [tex]n \geq 4[/tex],

[tex]2n<n![/tex].

Base case:

For n=4, [tex]2n=8<24=n![/tex]

Inductive hypothesis:  

Given a natural [tex]n \geq 4[/tex], [tex]2n<n![/tex]

Now, we will assume the induction hypothesis and then use this assumption, involving n, to prove the statement for n + 1.

Inductive step:

Observe that,

[tex]n!+2\leq (n+1)! \iff n!+2\leq n!(n+1) \iff 1+\frac{2}{n!}\leq n+1 \iff 2\leq n*n![/tex]

wich is true as we are assuming [tex]n\geq 4[/tex]. Therefore:

[tex]2(n+1)=2n+2<n!+2\leq (n+1)![/tex]

With this we have proved our statement to be true for n+1.    

In conlusion, for every natural [tex]n \geq 4[/tex],

[tex]2n<n![/tex]

Answer:

2 500 000 rad.

Step-by-step explanation:

Mega is the metric prefix for [tex]10^{6}[/tex], therefore you just need to multiply by 1 000 000 to find the value in rads.

Answer:

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Step-by-step explanation:

To find : Convert [tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}[/tex] into [tex]\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Solution :

We convert units one by one,

[tex]1\text{ m}^2=10.7639\text{ ft}^2[/tex]

[tex]1\text{ sec}=\frac{1}{3600}\text{ hour}[/tex]

[tex]1\text{ cal}=0.003968\text{ BTU}[/tex]

Converting temperature unit,

[tex]^\circ C\times \frac{9}{5}+32=^\circ F[/tex]

[tex]1^\circ C\times \frac{9}{5}+32=33.8^\circ F[/tex]

So, [tex]1^\circ C=33.8^\circ F[/tex]

Substitute all the values in the unit conversion,

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=\frac{0.003968}{10.7639\times \frac{1}{3600}\times 33.8}\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=\frac{0.003968}{0.101061}\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

[tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Therefore, The conversion of unit is [tex]1\ \frac{\text{cal}}{m^2\times sec\times ^\circ C}=0.03926\frac{\text{BTU}}{ft^2\times hr\times ^\circ F}[/tex]

Answer:

[tex]\frac{\pi}{43082}\text{ radians per second}[/tex]

Step-by-step explanation:

Given,

Time taken in one rotation of earth = 23 hours, 56 minutes and 4 seconds.

Since, 1 minute = 60 seconds and 1 hour = 3600 seconds,

⇒ Time taken in one rotation of earth = (23 × 3600 + 56 × 60 + 4) seconds

= 86164  seconds,

Now,  the number of radians in one rotation = 2π,

That is, 86164 seconds = 2π radians

[tex]\implies 1\text{ second }=\frac{2\pi}{86164}=\frac{\pi}{43082}\text{ radians}[/tex]

Hence, the number of radians in one second is [tex]\frac{\pi}{43082}[/tex]

The Earth completes a 2π radian rotation about its axis in 23 hours, 56 minutes, and 4 seconds. After converting this time to 86,164 seconds, the number of radians the Earth rotates in one second can be calculated by dividing 2π by 86,164, giving a result of approximately 0.00007292115 radians.

The Earth completes one full rotation about its axis in 23 hours, 56 minutes and 4 seconds. This rotation can be converted into radians, using the principle that one complete rotation is equivalent to 2π radians. So first, convert the rotation time into seconds: (23 x 60 x 60) + (56 x 60) + 4 = 86,164 seconds. Therefore, the Earth rotates through 2π radians in this time.

Now, we want to find out how many radians the Earth rotates in one second. To calculate this, divide 2π (which represent a full rotation in radians), by the total number of seconds in one rotation: 2π/86,164. This will give you approximately 0.00007292115 radians, which is the angular velocity or the number of radians the Earth rotates in one second.

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Answer:

time = 28 years

Step-by-step explanation:

Given,

principal amount = $10,000

rate = 4%

total amount = $30,000

According to compound interest formula

[tex]A\ =\ P(1+r)^t[/tex]

where, A = total amount

            P = principal amount

            r = rate

            t = time in years

so, from the question we can write,

[tex]30000\ =\ 10000(1+0.04)^t[/tex]

[tex]=>\ \dfrac{30000}{10000}\ =\ (1+0.04)^t[/tex]

[tex]=>\ 3\ =\ (1.04)^t[/tex]

by taking log on both sides, we will get

=> log3 = t.log(1.04)

[tex]=>\ t\ =\ \dfrac{log3}{log1.04}[/tex]

=> t = 28.01

So, the time taken to get the amount from 10000 to 30000 is 28 years.

Answer: 433

Step-by-step explanation:

The given sequence : 19​,42​,65​,88​,...

Here we can see that the difference in each of the two consecutive terms is 23.  [88-65=23, 65-42=23, 42-19=23]

i.e. it has a common difference of 23.

Therefore, it is an arithmetic sequence .

In an arithmetic​ sequence, the nth term an is given by the formula[tex]A_n=a_1+(n-1)d[/tex] , where [tex]a_1[/tex] is the first term and d is the common difference.​

For the given sequence , [tex]a_1=19[/tex]  and [tex]d=23[/tex]

Then,  to find the 19th term of  the sequence, we put n= 19 in the above formula:-

[tex]A_{19}=19+(19-1)(23)=19+(18)(23)=19+414+433[/tex]

Hence, the 19th term of  the sequence = 433

Final answer:

To find the 19th term of the arithmetic sequence 19, 42, 65, 88, ..., the common difference (23) is determined from the sequence and applied in the arithmetic sequence formula. Substituting the values into the formula, the 19th term is calculated to be 433.

Explanation:

To find the 19th term, we must first determine the common difference, d, of the sequence. Observing the given sequence, we see that the difference between consecutive terms is 42 - 19 = 23. Therefore, the common difference is 23.

Next, we apply the formula for the nth term of an arithmetic sequence which is An = a1 + (n-1)d. Here, a1 is the first term, n is the term number, and d is the common difference.

Substituting the values for the 19th term, we have: A19 = 19 + (19-1) × 23 = 19 + 18 × 23 = 19 + 414 = 433. Therefore, the 19th term of the sequence is 433.

Answer:

1880 milliliter (mL) = 3.97 pints (pts)

Step-by-step explanation:

This problem can be solved as a rule of three problem.

In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.

When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too.

When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease.

Unit conversion problems, like this one, is an example of a direct relationship between measures.

1 milliliter (mL) is equal to 0.002 pints. How many pints are 1880 milliliter (mL)? We have the following rule of three

1 mL - 0.002 pints

1880 mL - x pints

x = 1880*0.002

x = 3.97 pints

There are 3.97 pints in 1880 milliliters.

Answer:

96%

Step-by-step explanation:

Conditional probability is defined as:  

P(A|B) = P(A∩B) / P(B)  

Or, in English:  

Probability that A occurs, given that B has occurred = Probability that both A and B occur / Probability that B occurs

We want to find the probability that a woman uses an ATM to get cash, given that she uses an ATM to check her balance.

P(withdraws cash | checks account)

Using the definition of condition probability, this equals:

P = P(withdraws cash AND checks account) / P(checks account)

We know that P(checks account) = 0.28.

But we don't know what P(withdraws cash AND checks account) is.  To find that, we need to use the definition of P(A∪B):

P(A∪B) = P(A) + P(B) − P(A∩B)

This says that the probability of A or B occurring (or both) is the probability of A occurring plus the probability of B occurring minus the probability of both A and B occurring.

P(withdraws cash OR checks account) = P(withdraws cash) + P(checks account) − P(withdraws cash AND checks account)

0.95 = 0.94 + 0.28 − P(withdraws cash AND checks account)

P(withdraws cash AND checks account) = 0.27

Therefore:

P = 0.27 / 0.28

P ≈ 0.96

Final answer:

The probability that a woman who checks her account balance at an ATM also withdraws cash is approximately 96.43%.

Explanation:

To solve the problem, we can apply the probability rule for conditional probability. We are provided with the following probabilities:

Using this information, we're interested in finding the probability that a user who checks their account balance also withdraws cash, represented as P(Cash|Balance).

The formula for conditional probability is:

P(A|B) = P(A ∩ B) / P(B)

Where A and B are two events, and P(A|B) is the conditional probability of A given B.

Using the inclusion-exclusion principle, we can express P(Cash ∩ Balance) as:

P(Cash ∩ Balance) = P(Cash) + P(Balance) - P(Cash ∪ Balance)

Substitute the given probabilities:

P(Cash ∩ Balance) = 0.94 + 0.28 - 0.95 = 0.27

The probability that a woman who checks her balance also gets cash (P(Cash|Balance)) is:

P(Cash|Balance) = P(Cash ∩ Balance) / P(Balance)

P(Cash|Balance) = 0.27 / 0.28 ≈ 0.9643

Therefore, the probability is approximately 96.43%.

By considering the given information we have ,

[tex]H_0: \mu = 10\\\\ H_a: \mu\neq10[/tex]

Since, the alternative hypothesis is two tailed so the test is a two-tailed test.

Given : Population mean : [tex]\mu=10[/tex]

Standard deviation: [tex]\sigma= 3[/tex]

Sample size : n=25 , whihc is less than 30 so the sample is small and we use t-test.

Sample mean : [tex]\overline{x}=11.3[/tex]

Significance level : [tex]\alpha= 0.5[/tex]

Formula to find t-test statistic is given by :-

[tex]t=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex]

i.e. [tex]t=\dfrac{11.3-10}{\dfrac{3}{\sqrt{25}}}\approx2.17[/tex]

By using the standard normal distribution table,

The p-value corresponds 2.17 (two-tailed)=0.0300068

Since , the p-value is less than the significance level, so we reject the null hypothesis.

Hence, we conclude that there are enough evidence to to support the claim that movie rental rates in Yuma, Arizona, were different from those of the country as a whole .

A hypothesis test is conducted to see if the average movie rental rate in Yuma, Arizona, is statistically different from the national average. This problem is resolved in several steps including stating hypotheses, formulating an analysis plan, analyzing sample data, and interpreting the results. The rental rate is then compared to a critical value determined by the significance level (α = .05).

The subject here is a hypothesis testing problem related to the mean rental rate of DVDs in Yuma, Arizona. Blockbuster found that the average nation-wide movie rental rate was 10 movies per year with a standard deviation of 3. In Yuma, a sample of 25 people resulted in a mean rental rate of 11.3 movies per year. The company wanted to check whether this difference was significant or not. So, they used an alpha level of .05 to conduct this hypothesis test.

Here are the steps of the hypothesis test:

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