# For each sequence, decide whether it could be arithmetic, a)  25,  5,  1,  ... b)  25,  19,  13,  ... c)  4,  9,  16,  ... d)  50,  60,  70,  ... e)  frac{1}{2},  3,  18,  ...

Question
Polynomial arithmetic
For each sequence, decide whether it could be arithmetic,
$$a) 25, 5, 1, ...$$
$$b) 25, 19, 13, ...$$
$$c) 4, 9, 16, ...$$
$$d) 50, 60, 70, ...$$
$$e) \frac{1}{2}, 3, 18, ...$$

2020-11-23
Step 1 Given:
$$a) 25, 5, 1, ...b) 25, 19, 13, ...c) 4, 9, 16, ...d) 50, 60, 70, ...e) \frac{1}{2}, 3, 18, ...$$
Step 2 Concept:
Let, the series $$a_{1}, a_{2}, a_{3}, ...$$
When
$$d=a_{2}−a_{1}$$
$$d=a_{3}−a_{2}$$
Then, the series is known as Arithmetic Series and d is called the common difference
When
$$r=\frac{a_{2}}{a_{1}}$$
$$r=\frac{a_{3}}{a_{2}}$$
Then, the series is known as the Geometric Series and r is called the common ratio
Part a
$$25, 5, 1, ...$$
Here, $$r=\frac{a_{2}}{a_{1}}=\frac{25}{5}=5$$
$$r=\frac{a_{3}}{a_{2}}=\frac{5}{1}=5$$
This series is a Geometric Series with a common ratio of 5
Part b
$$25, 19, 13, ...$$
Here, $$d = a_{2} − a_{1} = 19 − 25 = − 6$$
$$d = a_{3} − a_{2} = 13 − 19 = − 6$$
This series is Arithmetic Series with common difference –6
Part c
$$4, 9, 16, ...$$
Here, $$d=a_{2} − a_{1} = 9 − 4 = 5$$
$$d=a_{3} − a_{2} = 16 − 9 = 7$$
This series is not Arithmetic Series because the common difference is not the same
Continuation from the last step:
$$4, 9, 16, ...$$
Here, $$r=\frac{a_{2}}{a_{1}}=\frac{9}{4}$$
$$r=(a_{3})/(a_{2}) = \frac{16}{9}$$
This series is not Geometric Series because the common ratio is not the same
Part d
$$50, 60, 70, ...$$
Here, $$d=a_{2} − a_{1} = 60 − 50 = 10$$
$$d=a_{3} − a_{2} = 70 − 60 = 10$$
This series is Arithmetic Series with a common difference of 10
Part e
$$12, 3, 18, ...$$
Here, $$r=\frac{a_{2}}{a_{1}}=\frac{3}{\frac{1}{2}}=6$$
$$r=\frac{a_{3}}{a_{2}}=\frac{18}{3}=6$$
This series is a Geometric Series with a common ratio of 6

### Relevant Questions

Determine whether the given sequence could be geometric,arithmetic, or neither.
If possoble, identify the common ratio or difference.
9, 13, 17, 21, ....
a. arithmetic $$d = 4$$
b. geometric $$r = 4$$
c. geometric $$r = \frac{1}{4}$$
d. neither
Write A if the sequence is arithmetic, G if it is geometric, H if it is harmonic, F if Fibonacci, and O if it is not one of the mentioned types. Show your Solution. a. $$\frac{1}{3}, \frac{2}{9}, \frac{3}{27}, \frac{4}{81}, ...$$ b. $$3, 8, 13, 18, ..., 48$$
1. Is the sequence $$0.3, 1.2, 2.1, 3, ...$$ arithmetic? If so find the common difference.
2. An arithmetic sequence has the first term $$a_{1} = -4$$ and common difference $$d = - \frac{4}{3}$$. What is the $$6^{th}$$ term?
3. Write a recursive formula for the arithmetic sequence $$-2, - \frac{7}{2}, -5, - \frac{13}{2} ...$$ and then find the $$22^{nd}$$ term.
4. Write an explicit formula for the arithmetic sequence $$15.6, 15, 14.4, 13.8, ...$$ and then find the $$32^{nd}$$ term.
5. Is the sequence $$- 2, - 1, - \frac{1}{2},- \frac{1}{4},...$$ geometric? If so find the common ratio. If not, explain why.
1. Find each of the requested values for a population with a mean of $$? = 40$$, and a standard deviation of $$? = 8$$ A. What is the z-score corresponding to $$X = 52?$$ B. What is the X value corresponding to $$z = - 0.50?$$ C. If all of the scores in the population are transformed into z-scores, what will be the values for the mean and standard deviation for the complete set of z-scores? D. What is the z-score corresponding to a sample mean of $$M=42$$ for a sample of $$n = 4$$ scores? E. What is the z-scores corresponding to a sample mean of $$M= 42$$ for a sample of $$n = 6$$ scores? 2. True or false: a. All normal distributions are symmetrical b. All normal distributions have a mean of 1.0 c. All normal distributions have a standard deviation of 1.0 d. The total area under the curve of all normal distributions is equal to 1 3. Interpret the location, direction, and distance (near or far) of the following zscores: $$a. -2.00 b. 1.25 c. 3.50 d. -0.34$$ 4. You are part of a trivia team and have tracked your team’s performance since you started playing, so you know that your scores are normally distributed with $$\mu = 78$$ and $$\sigma = 12$$. Recently, a new person joined the team, and you think the scores have gotten better. Use hypothesis testing to see if the average score has improved based on the following 8 weeks’ worth of score data: $$82, 74, 62, 68, 79, 94, 90, 81, 80$$. 5. You get hired as a server at a local restaurant, and the manager tells you that servers’ tips are $42 on average but vary about $$12 (\mu = 42, \sigma = 12)$$. You decide to track your tips to see if you make a different amount, but because this is your first job as a server, you don’t know if you will make more or less in tips. After working 16 shifts, you find that your average nightly amount is$44.50 from tips. Test for a difference between this value and the population mean at the $$\alpha = 0.05$$ level of significance.
The arithmetic mean (average) of two numbers c and d is given by $$\overline{x} = \frac{c+d}{2}$$
The value $$\overline{x}$$ is equidistant between c and d, so the sequence c,$$\overline{x}$$, d is an arithmetic sequence. inserting k equally spaced values between c and d, yields the arithmetic sequence $$c, \overline{X}_{1}, \overline{X}_{2}, \overline{X}_{3}, \overline{X}_{4}, ..., \overline{X}_{k}, d$$. Use this information for Exercise.
Insert four arithmetic means between 19 and 64.
For each of the following sequences: @ identify if the sequence is arithmetic, geometric or quadratic’. Justify your response. @ assuming the first item of each sequence is a1, give an expression for aj. (In other words, find a formula for the i-th term in the sequence). @ if the sequence is arithmetic or geometric, compute the sum of the first 10 terms in the sequence $$i 2,-12, 72, -432, 2592,...$$
$$ii 9, 18, 31, 48, 69, 94,...$$
$$iii 14, 11.5, 9, 6.5, 4, 1.5,...$$
A new thermostat has been engineered for the frozen food cases in large supermarkets. Both the old and new thermostats hold temperatures at an average of $$25^{\circ}F$$. However, it is hoped that the new thermostat might be more dependable in the sense that it will hold temperatures closer to $$25^{\circ}F$$. One frozen food case was equipped with the new thermostat, and a random sample of 21 temperature readings gave a sample variance of 5.1. Another similar frozen food case was equipped with the old thermostat, and a random sample of 19 temperature readings gave a sample variance of 12.8. Test the claim that the population variance of the old thermostat temperature readings is larger than that for the new thermostat. Use a $$5\%$$ level of significance. How could your test conclusion relate to the question regarding the dependability of the temperature readings? (Let population 1 refer to data from the old thermostat.)
(a) What is the level of significance?
State the null and alternate hypotheses.
$$H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}>?_{2}^{2}H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}\neq?_{2}^{2}H0:?_{1}^{2}=?_{2}^{2},H1:?_{1}^{2}?_{2}^{2},H1:?_{1}^{2}=?_{2}^{2}$$
(b) Find the value of the sample F statistic. (Round your answer to two decimal places.)
What are the degrees of freedom?
$$df_{N} = ?$$
$$df_{D} = ?$$
What assumptions are you making about the original distribution?
The populations follow independent normal distributions. We have random samples from each population.The populations follow dependent normal distributions. We have random samples from each population.The populations follow independent normal distributions.The populations follow independent chi-square distributions. We have random samples from each population.
(c) Find or estimate the P-value of the sample test statistic. (Round your answer to four decimal places.)
(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?
At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant. At the ? = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.At the ? = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.
(e) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings.Fail to reject the null hypothesis, there is sufficient evidence that the population variance is larger in the old thermostat temperature readings. Fail to reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.Reject the null hypothesis, there is insufficient evidence that the population variance is larger in the old thermostat temperature readings.
$$\{9=\frac{10}{11}n\}$$
What type of sequence is $$\{9=\frac{10}{11}n\}? asked 2020-11-02 In each of the​ following, list three terms that continue the arithmetic or geometric sequences. Identify the sequences as arithmetic or geometric. a. 2, 6, 18, 54, 162 b. 1, 8 ,15, 22, 29 c. 11, 15, 19, 23, 27 asked 2021-03-07 In an experiment designed to study the effects of illumination level on task performance (“Performance of Complex Tasks Under Different Levels of Illumination,” J. Illuminating Eng., 1976: 235–242), subjects were required to insert a fine-tipped probe into the eyeholes of ten needles in rapid succession both for a low light level with a black background and a higher level with a white background. Each data value is the time (sec) required to complete the task. \(\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{S}{u}{b}{j}{e}{c}{t}&{\left({1}\right)}&{\left({2}\right)}&{\left({3}\right)}&{\left({4}\right)}&{\left({5}\right)}&{\left({6}\right)}&{\left({7}\right)}&{\left({8}\right)}&{\left({9}\right)}\backslash{h}{l}\in{e}{B}{l}{a}{c}{k}&{25.85}&{28.84}&{32.05}&{25.74}&{20.89}&{41.05}&{25.01}&{24.96}&{27.47}\backslash{h}{l}\in{e}{W}{h}{i}{t}{e}&{18.28}&{20.84}&{22.96}&{19.68}&{19.509}&{24.98}&{16.61}&{16.07}&{24.59}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$