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ruslelena [56]
3 years ago
13

Determine the first four terms of the sequence in which the nth term is a_n=(n+1)!/(n+2)!

Mathematics
1 answer:
Vilka [71]3 years ago
4 0

Answer: Choice B) 1/3, 1/4, 1/5, 1/6

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

Plug in n = 1 to find the first term

a_n = ( (n+1)! )/( (n+2)! )

a_1 = ( (1+1)! )/( (1+2)! )

a_1 = ( 2! )/( 3! )

a_1 = ( 2*1 )/( 3*2*1 )

a_1 = 2/6

a_1 = 1/3

The first term is 1/3. Optionally you can stop here because only choice B has 1/3 listed as the first term, so this must be the answer. However, I'm going to keep going to show how to find the three other terms. This will help confirm why choice B is the answer, and it will be handy for those times when you aren't given multiple choice answers.

------------

Plug in n = 2

a_n = ( (n+1)! )/( (n+2)! )

a_2 = ( (2+1)! )/( (2+2)! )

a_2 = ( 3! )/( 4! )

a_2 = ( 3*2*1 )/( 4*3*2*1 )

a_2 = 6/24

a_2 = 1/4

The second term is 1/4

------------

Plug in n = 3

a_n = ( (n+1)! )/( (n+2)! )

a_3 = ( (3+1)! )/( (3+2)! )

a_3 = ( 4! )/( 5! )

a_3 = ( 4*3*2*1 )/( 5*4*3*2*1 )

a_3 = 24/120

a_3 = 1/5

The third term is 1/5

------------

Plug in n = 4

a_n = ( (n+1)! )/( (n+2)! )

a_4 = ( (4+1)! )/( (4+2)! )

a_4 = ( 5! )/( 6! )

a_4 = ( 5*4*3*2*1 )/( 6*5*4*3*2*1 )

a_4 = 120/720

a_4 = 1/6

The fourth term is 1/6

------------

The first four terms are: 1/3, 1/4, 1/5, 1/6, so that confirms why choice B is the answer.

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Answer:

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Step-by-step explanation:

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Answer:

t=\frac{2.35-1.83}{\sqrt{\frac{0.88^2}{10}+\frac{0.54^2}{7}}}}=1.507  

p_v =P(t_{(15)}>1.507)=0.076

If we compare the p value and the significance level given \alpha=0.01 we see that p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and the group CTS, NOT have a significant higher mean compared to the Normal group at 1% of significance.

Step-by-step explanation:

1) Data given and notation

\bar X_{CTS}=2.35 represent the mean for the sample CTS

\bar X_{N}=1.83 represent the mean for the sample Normal

s_{CTS}=0.88 represent the sample standard deviation for the sample of CTS

s_{N}=0.54 represent the sample standard deviation for the sample of Normal

n_{CTS}=10 sample size selected for the CTS

n_{N}=7 sample size selected for the Normal

\alpha=0.01 represent the significance level for the hypothesis test.

t would represent the statistic (variable of interest)

p_v represent the p value for the test (variable of interest)

State the null and alternative hypotheses.

We need to conduct a hypothesis in order to check if the mean for the group CTS is higher than the mean for the Normal, the system of hypothesis would be:

Null hypothesis:\mu_{CTS} \leq \mu_{N}

Alternative hypothesis:\mu_{CTS} > \mu_{N}

If we analyze the size for the samples both are less than 30 so for this case is better apply a t test to compare means, and the statistic is given by:

t=\frac{\bar X_{CTS}-\bar X_{N}}{\sqrt{\frac{s^2_{CTS}}{n_{CTS}}+\frac{s^2_{N}}{n_{N}}}} (1)

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other".

Calculate the statistic

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

t=\frac{2.35-1.83}{\sqrt{\frac{0.88^2}{10}+\frac{0.54^2}{7}}}}=1.507  

P-value

The first step is calculate the degrees of freedom, on this case:

df=n_{CTS}+n_{N}-2=10+7-2=15

Since is a one side right tailed test the p value would be:

p_v =P(t_{(15)}>1.507)=0.076

We can use the following excel code to calculate the p value in Excel:"=1-T.DIST(1.507,15,TRUE)"

Conclusion

If we compare the p value and the significance level given \alpha=0.01 we see that p_v>\alpha so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and the group CTS, NOT have a significant higher mean compared to the Normal group at 1% of significance.

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