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Chapter 17: Problem 11

Find \(\bar{a}\). $$ a_{i}=i / 2 \text { and } i=1, \ldots, 5 $$

Short Answer

Expert verified
Answer: The mean of the sequence is \(\frac{3}{2}\).

Step by step solution

01

Write out the terms of the sequence

To find the terms of the sequence, we simply substitute each value of \(i\) into \(a_i = \frac{i}{2}\). This gives us: $$ a_1 = \frac{1}{2}, a_2 = \frac{2}{2}, a_3 = \frac{3}{2}, a_4 = \frac{4}{2}, a_5 = \frac{5}{2} $$
02

Calculate the sum of the sequence

Now, we need to find their sum: $$ S = a_1 + a_2 + a_3 + a_4 + a_5 = \frac{1}{2} + \frac{2}{2} + \frac{3}{2} + \frac{4}{2} + \frac{5}{2} $$ Combining the terms, we get: $$ S = \frac{1+2+3+4+5}{2} = \frac{15}{2} $$
03

Find the mean of the sequence

Finally, we find the mean by dividing the sum of the sequence by the number of terms. Since there are 5 terms in the sequence, we get: $$ \bar{a} = \frac{S}{n} = \frac{\frac{15}{2}}{5} = \frac{15}{2} \cdot \frac{1}{5} $$ Simplifying, we find: $$ \bar{a} = \frac{15}{10} = \frac{3}{2} $$ So, the mean of the sequence \(\bar{a}\) is \(\frac{3}{2}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Arithmetic Sequence
An arithmetic sequence is a collection of numbers in which the difference between consecutive terms is constant. In simpler terms, if you take any term in the sequence and subtract its previous term, you will always get the same number. This difference is known as the "common difference."
For example, consider the sequence given in the exercise:
  • The terms are expressed through the formula: \(a_i = \frac{i}{2}\).
  • As you replace \(i\) with sequential integers from 1 to 5, you get the terms: \(\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}\).
The common difference here is \(\frac{1}{2}\), which is consistent across terms because each term increases by \(\frac{1}{2}\) as the sequence progresses.
Sequence Summation
Sequence summation is the process of adding all the terms in a sequence to form a single total. It's an essential concept, especially when calculating the mean of a sequence.
For the sequence in the exercise, the terms are
  • \(\frac{1}{2}\), \(1\), \(\frac{3}{2}\), \(2\), \(\frac{5}{2}\).
  • The sum of these terms can be simply calculated as:\[ S = \frac{1}{2} + 1 + \frac{3}{2} + 2 + \frac{5}{2} \].
By converting these fractions to have a common denominator and then adding them all up, we end up with: \(S = \frac{15}{2}\). This sum is crucial for the next step of finding the average or mean.
Average of Sequence
The average (or mean) of a sequence is a measure of central tendency that tells us the "central" or "typical" value of the set of numbers. The mean is found by dividing the total sum of all terms by the number of terms in the sequence.
For our sequence, the sum was calculated to be \(\frac{15}{2}\).
  • The sequence consists of 5 terms: \(\frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}\).
  • To find the mean, you divide the sum by the total number of terms, which is 5:\[ \bar{a} = \frac{\frac{15}{2}}{5} \].
Performing this division gives us the mean as \(\frac{3}{2}\). This value tells us the typical value of the sequence.

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Most popular questions from this chapter

If your music player has four playlists, Rock (233 songs), Hip-Hop (157 songs), Jazz (107 songs) and Latin ( 258 songs), and you select the shuffle mode, what is the probability, given as a percentage, of starting with a song from (a) Rock (b) Hip-Hop (c) Jazz (d) Latin.

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A city is divided into 4 voting precincts, \(A, B, C,\) and \(D\). Table 17.20 shows the results of mayoral election held for two candidates, a Republican and a Democrat.$$ \begin{array}{c|c|c|c} \hline \text { Precinct } & \text { Number voters } & \text { Republican } & \text { Democrat } \\ \hline \mathrm{A} & 10,000 & 4,200 & 5,800 \\ \mathrm{~B} & 15,000 & 7,100 & 7,900 \\ \mathrm{C} & 17,000 & 8,200 & 8,800 \\ \mathrm{D} & 18,000 & 12,400 & 5,600 \\ \hline \end{array}$$ Assuming random selection, what is the probability, given as a percentage, that a voter: (a) Lives in precinct \(B ?\) (b) Is a Republican? (c) \(\operatorname{Both}(\) a) and \((\mathrm{b})\) (d) Is Republican given that he or she lives in precinct \(B ?\) (e) Lives in precinct \(B\) given that he or she is Republican?

A high-tech company makes silicon wafers for computer chips, and tests them for defects. The test identifies \(90 \%\) of all defective wafers, and misses the remaining \(10 \% .\) In addition, it misidentifies \(20 \%\) of all non- defective wafers as being defective. (a) Suppose 5000 wafers are made. Of the \(5 \%\) of these wafers that contain defects, how many are correctly identified by the test as being defective? (b) How many of the non-defective wafers are incorrectly identified by the test as being defective? (c) What is the probability, given as a percentage, that a wafer identified as defective is actually defective?
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