If the sequence below is an arithmetic sequence then what are the next two numbers?
Sequence: 3.9,

Answer:

Step-by-step explanation:

(a) We will prove by induction that if n is a multiple of 3 then Bob has a winning strategy.

Let n=3

It is given that Alice always plays first.

Then, for the first move, Alice has a choice of removing 1 or 2 stones.

Case 1:

Alice removes 1 stone. Then it is now Bob's turn. There are 2 stones left. Bob has the choice of removing 1 or 2 stones. Then Bob's winning strategy should be to remove 2 stones. Then Bob removes the last stone and wins.

Case 2:

Alice removes 2 stones. Then it is Bob's turn. There is exactly 1 stone left, which Bob removes in his turn. Since he removes the last stone, he wins.

Thus, in either case, for n=3, Bob has a winning strategy. ____ (A)

Now, let us assume that Bob has a winning strategy for n=3p.

We will now check if Bob has a winning strategy for n=3p+3

It is given that Alice plays first.

Case 1:

Alice starts the game by removing 1 stone. Then, Bob has a choice of removing 1 or 2 stones. If he chooses to remove 2 stones, then the number of stones left is 3p+3-3=3p and it is now Alice's turn to play. This is exactly the game when n=3p and we have already assumed that Bob has a winning strategy for n=3p. Thus, in this case Bob has a winning strategy.

Case 2:

Alice starts the game by removing 2 stones. Then, Bob has a choice of removing 1 or 2 stones. If he chooses to remove 1 stone, then the number of stones left is 3p+3-3=3p and it is now Alice's turn to play. This is again exactly the game when n=3p and we have already assumed that Bob has a winning strategy for n=3p. Thus, in this case also, Bob has a winning strategy.

Thus, in either case Bob has a winning strategy for n=3p+3 if he has a winning strategy for n=3p. _______ (B)

Thus, from (A) & (B), using induction, we can say that Bob has a winning strategy if n is a multiple of 3.

(b) We will now prove that if n is not a multiple of 3, then Alice has a winning strategy.

If n is not a multiple of 3, then n can have either of the forms of 3m+1 or 3m+2, m ∈ W.

We will prove the given fact for both the forms simultaneously

Let m=0, i.e., n=1 or n=2

Since Alice starts first, she removes 1 stone, if n=1 or 2 stones if n=2 and thus wins. Thus Alice has a winning strategy if m=0. ______ (A)

Let us assume that Alice has a winning strategy for m=k, i.e., for n=3k+1 & n=3k+2

Now, we will check if Alice has a winning strategy for m=k+1, i.e., for n=3(k+1)+1=3k+4 or n=3(k+1)+2=3k+5

Let n=3k+4

Since Alice plays first, she has a choice to remove 1 or 2 stones.

Note that, if Alice removes 2 stones and in turn Bob removes 2 stones, then the number of stones becomes a multiple of 3 such that it is Alice's turn to play. In that case, Bob will have a winning strategy as shown in the previous part.

Then, Alice should remove 1 stone. Then, it is now Bob's turn and he has a choice of removing 1 or 2 stones.

Case 1:

Bob removes 1 stone. Then there are 3k+4-1-1=3k+2 stones remaining and it is Alice's turn. This is identical to the game where n=3k+2. We have already assumed that Alice has a winning strategy in this case.

Case 2:

Bob removes 2 stones. Then there are 3k+4-1-2=3k+1 stones remaining and it is Alice's turn. This is identical to the game where n=3k+1. We have already assumed that Alice has a winning strategy in this case.

Thus, in either case, Alice has a winning strategy.

Let n=3k+5

Since Alice plays first, she has a choice to remove 1 or 2 stones.

Note that, if Alice removes 1 stone and in turn Bob removes 1 stone, then the number of stones becomes a multiple of 3 such that it is Alice's turn to play. In that case, Bob will have a winning strategy as shown in the previous part.

Then, Alice should remove 2 stones. Then, it is now Bob's turn and he has a choice of removing 1 or 2 stones.

Case 1:

Bob removes 1 stone. Then there are 3k+5-2-1=3k+2 stones remaining and it is Alice's turn. This is identical to the game where n=3k+2. We have already assumed that Alice has a winning strategy in this case.

Case 2:

Bob removes 2 stones. Then there are 3k+5-2-2=3k+1 stones remaining and it is Alice's turn. This is identical to the game where n=3k+1. We have already assumed that Alice has a winning strategy in this case.

Thus, in either case, Alice has a winning strategy.

Then, we can say that Alice has a winning strategy for m=k+1 if she has a winning strategy for m=k. _____ (B)

Then, by induction, from (A) & (B), we can say that if n is not a multiple of 3, then Alice has a winning strategy.

Answer:

The 99% confidence interval for the percent of all the students at that college who have at least $1000 in credit card debt is (20.11%, 32.17%).

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence level of [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the z-score that has a p-value of [tex]1 - \frac{\alpha}{2}[/tex].

In a simple random sample of 352 students at a college, 92 reported that they have at least $1000 of credit card debt.

This means that [tex]n = 352, \pi = \frac{92}{352} = 0.2614[/tex]

99% confidence level

So [tex]\alpha = 0.01[/tex], z is the value of Z that has a p-value of [tex]1 - \frac{0.01}{2} = 0.995[/tex], so [tex]Z = 2.575[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2614 - 2.575\sqrt{\frac{0.2614*0.7386}{352}} = 0.2011[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.2614 + 2.575\sqrt{\frac{0.2614*0.7386}{352}} = 0.3217[/tex]

As percent:

0.2011*100% = 20.11%

0.3217*100% = 32.17%.

The 99% confidence interval for the percent of all the students at that college who have at least $1000 in credit card debt is (20.11%, 32.17%).

Answer:

$1,260

Step-by-step explanation:

30% off  

pay 70%

70% = 0.7

1,800 * 0.7 = 1,260

$1,260

Answer:

30%

Step-by-step explanation:

Find the total number of candies in the jar.

8 + 5 + 1 + 6 = 20

The probability of choosing a green candy is 6/20 = 3/10 = .3 = 30%

Answer:

no

Step-by-step explanation:

8÷ 1/2

Copy dot flip

8 * 2/1 = 16

Divide 8 things in half you get 16

1/2 of 8 = 1/2 * 8 = 4

Half of 8 things is 4

They are not the same

Answer:

8 ÷ 1/2 = 16

I disagree.

Step-by-step explanation:

8 ÷ 1/2

8 × 2/1

8 × 2

16

Answer:

32x10^4 and 3.2x10^5

Step-by-step explanation:

Mind marking me brainliest? :)

Answer:

first and the 3rd one

Step-by-step explanation:

Answer:

It's D

Step-by-step explanation:

100%

Answer:

complementary angles

Step-by-step explanation:

Complementary angles add to 90 degrees

Supplementary angles add to 180 degrees

A right angle is 90 degrees so they are complementary angles

Answer:

I cant see the question, sorry bro.

Answer:

First box: 35 Negative Result

Second box : 60 Positive Result

Step-by-step explanation:

Contains organic matter

Sample size : 700

95% positive means  [tex]\frac{95}{100} \times 700 = 665\\[/tex]

5% negative means [tex]\frac{5}{100}\times 700 = 35[/tex]

Does not contain organic matter

Sample size : 300

20% positive means [tex]\frac{20}{100} \times 300 = 60[/tex]

80% negative means [tex]\frac{80}{100}\times 300 = 240[/tex]

Answer:

Just subtract the less no by greater one .you get the answer.

Positive result of sample test gives a positive result than it is 665/1000*100*=66.5% likely that the soil sample contains organic matter.

9514 1404 393

Answer:

  m = 18

Step-by-step explanation:

In an arithmetic progression, the middle term of three is half the sum of the terms on either side.

  a1 +a3 = 2(a2)

  (m +9) +(2m +7) = 2(2m -1)

  3m +16 = 4m -2 . . . . . . collect terms

  18 = m . . . . . . . . . . . . . . add 2-3m

Answer:

Solution given:

perpendicular [P]=15m

base[B]=8m

hypotenuse [H]=17m

rate :4 plants per square metre

no of plants=?

we have

Area of triangular garden: ½(P*B)=½(15*8)=60m²

Now

Total no of plants =rate ×Area =4×60=240

Michael needs 240 plants in the garden

Answer: 405 dollars

Step-by-step explanation:

200 * x = 270

x = 270 * 300/200. x =405 dollars

Answer:

no

explanation:

Side lengths do not adhere to the triangle inequality theorem. Which states that the sum of the side lengths of any 2 sides of a triangle must exceed the length of the third side

Answer:

it is 20

Step-by-step explanation:

well the angles looks the same

Answer:

x=20[vertically opposite angle are equal]

Answer:

Slope = 1/4

Step-by-step explanation:

[tex]slope = \frac{y2-y1}{x2-x1} \\m=\frac{3-2}{3--1} \\m=1/4[/tex]

Answer:

4.right

5. obtuse

6. acute

7. obtuse

8.acute

9. right

10. right

11. acute

12. obtuse

hope this helps

have a good day :)

Step-by-step explanation:

Answer:

only 5 points

Step-by-step explanation:

Answer:

4% per annum

Step-by-step explanation:

Recall :

Simple interest = principal * rate * time

Given that :

Time = 3 ; simple interest = 84 ; principal = 700

84 = 700 * rate * 3

84 = 2100 * rate

Rate = 84 /2100

= 0.04

= 0.04 * 100% = 4% per annum

Answer:

purple and red has same area

Step-by-step explanation:

Area of purple = 4 x 4 = 16 square cm

Area of red = 2 x 8 = 16 square cm

Area of yellow = 3 x 5 = 15 square cm

Answer:

(d) above [tex]74.1^o[/tex]

Step-by-step explanation:

Given

[tex]\tan w = 3.52[/tex]

Required

Determine w, using a calculator

We have:

[tex]\tan w = 3.52[/tex]

Take arctan of both sides

[tex]w = \tan^{-1}(3.52)[/tex]

Using a calculator, we have:

[tex]w \approx 74.1^o[/tex]

From the given options, (d) is true

Answer:

w = 6ft 8in

Step-by-step explanation:

the proportions will be the same

w/7.5 = 12/13.5

multiply both sides by 7.5

w = 12/13.5 * 7.5

w = 6.6666666667ft

w = 6ft 8in

The original truck was 6.67 feet wide.

"It is a comparison of two or more numbers that indicates their sizes in relation to each other."

"It is an equation in which two ratios are set equal to each other."

For given example,

A model truck is 13.5 inches long 7.5 inches wide.

The ratio of length to width of a model truck would be,

13.5 : 7.5                       ...........................(1)

The original truck was 12 feet long.

This means the original truck was 144 inches long.

Let 'x' be the width (in inches) of the original truck.

So, the ratio of the length to the width of the original truck would be,

144 : x                           .................................(2)

Also, the ratios given by (1) and (2) must be in proportion.

[tex]\Rightarrow \frac{13.5}{7.5} = \frac{144}{x} \\\\\Rightarrow 13.5 \times x = 144 \times 7.5\\\\\Rightarrow \bold{x=80~inches}\\\\\Rightarrow \bold{x=6.67~feet}[/tex]

Therefore, the original truck was 6.67 feet wide.

Learn more about ratio and proportion here:

https://brainly.com/question/26974513

#SPJ2

Answer:

r = 7

Step-by-step explanation:

This is how you solve the equation...

[tex]r^{3} = 49r\\r^{3} / r= 49r / r\\r^{2} = 49\\r = \sqrt{49} \\r = 7[/tex]

Answer:

r = 7 is the answer

Answer:

97,655

Step-by-step explanation:

5(5)^(n-1) = 78,125

5^n = 78,125

n = 7

=> S7 = 5(5^7 -1) / (5-1)

= 5/4 (78, 125 -1)

= 5/4 (78 124) = 97,655

Answer:

The relationship between weeks and the total amount is:

[tex]y=10x[/tex]

Step-by-step explanation:

We have here a linear relationship between weeks and the total amount. The equation of a line is:

[tex]y=mx+b[/tex]

Where:

We need two points to find the slope.

[tex]m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}[/tex]

Let's use any pair points from the plot.

[tex]m=\frac{60-40}{6-4}[/tex]

[tex]m=10[/tex]

And we also see that the y-intercept is 0, therefore the equation will be:

[tex]y=10x[/tex]

I hope it helps you!

Answer:

y=10x and Over 4 weeks, Aurora saves $40

Step-by-step explanation:

=======================================================

Explanation:

There are...

We start with 6, and count down by 1 each time we fill up a slot. We stop once the third slot is filled or accounted for. The countdown is to ensure that we don't pick the same person twice. From here, multiply those values: 6*5*4 = 30*4 = 120

Interestingly, this is equal to 5! = 5*4*3*2*1 = 120 because the 3*2 becomes 6 and that *1 at the end doesn't affect things. Though usually results of permutation problems don't always end up like this. The order matters because a result like ABC is different from BAC, where A,B,C,D,E,F are the six school organizations.

As a slightly longer way to do the problem, you can use the nPr formula which is [tex]_nP_r = \frac{n!}{(n-r)!}[/tex] where n = 6 and r = 3 in this case. The exclamation marks indicate factorial. If you go this route, you should find that one of the steps will involve 6*5*4.

Answer:

4(x - 4) = 4x - 16

Step-by-step explanation:

4(x - A) = 4x - 16

4x - 4A = 4x - 16

Equating Constant terms (terms without x): -4A = -16

                                                                     A = 4

Therefore , 4(x - 4) = 4x - 16

The answer is 4.

In order to find the answer, divide 4 from the each number on the right side. You pull the 4 out from 4x, and 4 from 16 to arrive at 4(x - 4).

Answer:

N'(-2,6)  

Noting Else To Put

Answer: The correct answer is B (-2,6)

Step-by-step explanation: it is show down below