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Step2247 [10]
3 years ago
13

What is wrong with the following proof that for every integer n, there is an integer k such that n < k < n + 2? Suppose n

is an arbitrary integer. Therefore k = n + 1.
a)Nothing.

b)The "therefore" phrase is illogical. The fact that n is an integer does not force us to define k as n + 1.

c)The proof writer mentally assumed the conclusion. He wrote "suppose n is an arbitrary integer", but was really thinking "suppose n is an arbitrary integer, and suppose that for this n, there exists an integer k that satisfies n < k < n+2." Under those assumptions, it follows indeed that k must be n + 1, which justifies the word "therefore": but of course assuming the conclusion destroyed the validity of the proof.

d)Since the theorem required the condition that k satisfies n < k < n + 2, a complete proof should at least mention that the chosen k satisfies that inequality, even if there is nothing to prove algebraically.
Mathematics
1 answer:
expeople1 [14]3 years ago
4 0

Answer:

c)The proof writer mentally assumed the conclusion. He wrote "suppose n is an arbitrary integer", but was really thinking "suppose n is an arbitrary integer, and suppose that for this n, there exists an integer k that satisfies n < k < n+2." Under those assumptions, it follows indeed that k must be n + 1, which justifies the word "therefore": but of course assuming the conclusion destroyed the validity of the proof.

Step-by-step explanation:

when we claim something as a hypothesis we can only conclude with therefore at the end of the proof. so assuming the conclusion nulify the proof from the beginning

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Find two consecutive positive even integers such that the square of the first decreased by 64 equals 3 times the second.
IRINA_888 [86]

Answer:

10 and 12

Step-by-step explanation:

let the consecutive even integers be n and n + 2 , then

n² - 64 = 3(n + 2) ← distribute parenthesis

n² - 64 = 3n + 6 ( subtract 3n + 6 from both sides )

n² - 3n - 70 = 0 ← in standard form

(n - 10)(n + 7) = 0 ← in factored form

Equate each factor to zero and solve for n

n - 10 = 0 ⇒ n = 10

n + 7 = 0 ⇒ n = - 7

Since n must be a positive even integer then n = 10 and n + 2 = 10 + 2 = 12

The 2 numbers are 10 and 12

8 0
3 years ago
Select the correct solution set. 50 ≥ 15x
Semenov [28]
The answer (i think) is 3 <span>≥</span> x. hope i helped
7 0
3 years ago
Simplify. Assume all variables are non-zero. HELP ASAP!
Alexandra [31]

Answer:

D

Step-by-step explanation:

((p^4*q)/p^8)^2.

p^4/p^8=p^(4-8)=p^-4=1/p^4

(q/p^4)^2=(q^2/p^8)

4 0
3 years ago
23. A certain type of gasoline is supposed to have a mean octane rating of at least 90. Suppose measurements are taken on of 5 r
Nostrana [21]

Answer:

a) p_v =P(Z          

b) Since p_v we can reject the null hypothesis at the significance level given. So based on this we can commit type of Error I.

Because Type I error " is the rejection of a true null hypothesis".

c) P(\bar X >90)=1-P(Z

1-P(Z

d) For this case we need a z score that accumulates 0.02 of the area on the right tail and 0.98 on the left tail and this value is z=2.054

And we can use this formula:

2.054=\frac{90-89}{\frac{0.8}{\sqrt{n}}}

And if we solve for n we got:

n = (\frac{2.054 *0.8}{1})^2=2.7 \approx 3

Step-by-step explanation:

Previous concepts  and data given  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

We can calculate the sample mean and deviation with the following formulas:

\bar X = \frac{\sum_{i=1}^n X_i}{n}

s=\sqrt{\sum_{i=1}^n (X_i -\bar X)^2}{n-1}

\bar X=88.96 represent the sample mean    

s=1.011 represent the sample standard deviation  

n=5 represent the sample selected  

\alpha significance level    

State the null and alternative hypotheses.    

We need to conduct a hypothesis in order to check if the mean is less than 90, the system of hypothesis would be:    

Null hypothesis:\mu \geq 90    

Alternative hypothesis:\mu < 90    

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:    

z=\frac{\bar X-\mu_o}{\frac{\sigma}{\sqrt{n}}}  (1)    

z-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".    

Calculate the statistic  

We can replace in formula (1) the info given like this:    

z=\frac{88.96-90}{\frac{0.8}{\sqrt{5}}}=-2.907      

Part a

P-value  

First we need to calculate the degrees of freedom given by:

df=n-1=5-1 = 4

Then since is a left tailed test the p value would be:    

p_v =P(Z    

Part b

Since p_v we can reject the null hypothesis at the significance level given. So based on this we can commit type of Error I.

Because Type I error " is the rejection of a true null hypothesis".

Part c

We want this probability:

P(\bar X

The best way to solve this problem is using the normal standard distribution and the z score given by:

z=\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}

If we apply this formula to our probability we got this:

P(\bar X >90)=1-P(Z

1-P(Z

Part d

For this case we need a z score that accumulates 0.02 of the area on the right tail and 0.98 on the left tail and this value is z=2.054

And we can use this formula:

2.054=\frac{90-89}{\frac{0.8}{\sqrt{n}}}

And if we solve for n we got:

n = (\frac{2.054 *0.8}{1})^2=2.7 \approx 3

7 0
3 years ago
You are taking a​ multiple-choice test that has 44 questions. each of the questions has 55 answer​ choices, with one correct ans
Aleks [24]
Since there are 55 choices, we have to have 55 options, and for 44 questions, we multiply how many options there are in 1 question by 44 to get 55*44=2420
6 0
3 years ago
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