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Precalculus

More Topics in Algebra

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1
Sequences

A string of a musical stringed instrument vibrates in many modes, called *harmonics*. Associated with each harmonic is its *frequency*, which is expressed in units of cycles per second, or Hertz (Hz). The harmonic with the lowest frequency is known as the fundamental mode; all the other harmonics are known as overtones. Table 4 lists the frequencies corresponding to the first four harmonics of a string of a certain instrument.

What type of sequence do the frequencies form: arithmetic, geometric, or neither?

Find the frequency corresponding to the fifth harmonic.

Find the frequency corresponding to the \(n\text{th}\) harmonic.

When scientists conduct tests using DNA, they often need larger samples of DNA than can be readily obtained. A method of duplicating DNA, known as Polymerase Chain Reaction (PCR), was invented in 1983 by biochemist Kary Mullis, who later won the Nobel Prize in Chemistry for his work. After each cycle of the PCR process, the number of DNA fragments doubles. This procedure has a number of applications, including diagnosis of genetic diseases and investigation of criminal activities.

What type of sequence is generated by the repeated application of the PCR process?

How many DNA fragments will there be after five cycles of the PCR process?

How many DNA fragments will there be after \(n\) cycles of the PCR process?

Laboratories typically require millions of DNA fragments to conduct proper tests. How many cycles of the PCR process are needed to produce one million fragments?

Each cycle of the PCR process takes approximately \(30\) minutes. How long will it take to generate one million DNA fragments?

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2
Sums of Terms of Sequences

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3
General Sequences and Series

an initial dose of \(40\) milligrams of the pain reliever acetaminophen is given to a patient. Subsequent doses of \(20\) milligrams each are administered every \(5\) hours. Just before each \(20\)-milligram dose is given, the amount of acetaminophen in the patientÃ¢â‚¬â„¢s bloodstream is \(25\%\) of the total amount in the bloodstream just after the previous dose was administered.

Let \(a_0\) represent the initial amount of the drug in the bloodstream and, for \(n\geq 1\), let \(a_n\) represent the amount in the bloodstream after the \(n\)th \(20\)-milligram dose is given. Make a table of values for \(a_0\) through \(a_6\).

Plot the values you tabulated in part (a). What do you observe?

With the aid of the values you tabulated in part (a), find a recursive definition of \(a_n\).

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4
Counting Methods

A fast food restaurant holds a promotion in which each customer scratched off three boxes on a ticket. Each box contains a picture of one of five items: a burger, a bag of fries, a shake, a pie, or a salad. The item in the first box is free. The item in the second box can be purchased at a \(50\%\) discount, and the item in the third box can be bought at \(25\%\) off. Using permutation notation, determine the number of different tickets that are possible if each picture can be used only once per ticket.

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5
Probability

During a certain episode, the television game show *Wheel of Fortune* had a wheel with \(24\) sectors. One sector was marked "Trip to Hawaii" and two of the other sectors were marked "Bonus." What is the probability of winning a trip to Hawaii, assuming the wheel is equally likely to stop at any one of the \(24\) sectors?

Figure 2 (see book) shows all the possibilities for the numbers on the top faces when rolling a pair of dice. Find the following:

The probability of rolling a sum of \(5\)

The probability of rolling a sum of \(6\) or \(7\)

The probability of rolling a sum of \(13\)

Every spring, the National Basketball Association holds a lottery to determine which team will get first pick of its number 1 draft choice from a pool of college players. The teams with poorer records have a higher chance of winning the lottery than those with better records. Table 1 (see book) lists the percentage chance that each team had of getting first pick of its number 1 draft choice for the year 2002. Find the probability of

the Bulls or the Warriors getting the first draft pick.

the Clippers

*not*getting the first draft pick.