Find the rule for the geometric sequence with terms \(a_2=12\) and \(a_3=24\).

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.

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

2. Find the frequency corresponding to the fifth harmonic.

3. 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.

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

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

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

4. 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?

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

Find the sum of the first \(11\) terms of the arithmetic sequence \(2, 5, 8, 11, 14, \ldots\)

Find the sum of all even numbers between \(2\) and \(100\), inclusive.

Find the sum of the first five terms of the sequence \(a_j=\dfrac{1}{2}(3)^j\), \(j=0, 1, 2, \ldots\).

Determine whether each of the following infinite geometric series has a sum. If so, find the sum.

1. \(2+4+8+16+\ldots\)

2. \(1+\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+\ldots\)

An auditorium has \(30\) seats in the front row. Each subsequent row has two seats more than the row directly in front of it. If there are \(12\) rows in the auditorium, how many seats are there all together?

Suppose \(\$2000\) is deposited initially (and at the end of each year) into a retirement account that pays \(5\%\) interest compounded annually. What is the total amount in the account at the end of \(10\) years?

Find the first four terms of the sequence defined recursively by \(a_0=1\), \(a_n=3+a_{n-1}\), \(n=1, 2, 3, \ldots\). What type of sequence is generated? Find a rule for the \(n\text{th}\) terms of the sequence that depends only on \(n\).

Find the first five terms of the Fibonacci sequence, defined recursively as follows: \(f_0=1\), \(f_1=1\), \(f_n=f_{n-1}+f_{n-2}\), \(n=2, 3, \ldots\)

Compute each sum.

1. \(\displaystyle\sum_{i=0}^4 i^2\)

2. \(\displaystyle\sum_{i=0}^3 (-1)^i(2i+1)\)

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 patient’s bloodstream is \(25\%\) of the total amount in the bloodstream just after the previous dose was administered.

1. 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 \(a_6\).

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

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

Suppose \(10\) members of a cycling club are practicing for a race. From among these members, how many possibilities are there for the first-place and second-place finishes?

You wish to form a three-digit number, with each digit ranging from \(1\) to \(9\). You may use the same digit more than once. How many three-digit numbers can you make?

How many different four-block towers can be built with a given set of four different colored blocks?

Use factorial notation to calculate the number of ways in which seven people can be arranged in a row for a photograph.

A store manager has six different candy boxes, but spaces for only four of them on the shelf. In how many ways can the boxes be arranged horizontally on the shelf?

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.

A group photograph is taken with four children in the front row and five adults in the back row. How many different photographs are possible?

You have three textbooks on your desk: history (H), English (E), and mathematics (M). You choose two of them to put in your book bag. In how many ways can you do this?

Suppose a cycling club has \(10\) members. From among the members of the club, how many ways are there to select a two-person committee?

A six-member student board is formed with three male students. If there are five male candidates and six female candidates, how many different student boards are possible?

What are the possible outcomes when a six-sided die is rolled and the number on the top face is recorded? What is the event that the number on the top face is even?

A coin is tossed three times, and the sequence of heads and tails that occurs is recorded.

1. What is the sample space for this experiment?

2. What is the event that at least two heads occur?

3. What is the event that exactly two heads occur?

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?

Suppose a coin is tossed three times. What is the probability of obtaining at least two heads?

Decide which of the following pairs of events are mutually exclusive.

1. "Drawing a queen" and "drawing a king" from a deck of \(52\) cards.

2. "Drawing a queen" and "drawing a spade" from a deck of \(52\) cards.

Find the probability of drawing a queen or a king from a deck of \(52\) cards.

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

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

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

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

Suppose you are told the probability of rain today is \(0.6\). What is the probability that it will not rain?

Refer to Figure 2 in Example 7, which lists all the possible outcomes of rolling two dice. What is the probability of rolling a sum of at least \(4\)?

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

1. the Bulls or the Warriors getting the first draft pick.

2. the Clippers not getting the first draft pick.

Consider the expansion of \((a+b)^4\)

1. Write down the variable parts of all the terms that occur in the expansion.

2. What is the sum of the exponents on \(a\) and \(b\) for each term of the expansion?

Replace \(k\) with \(k+1\) in the following:

1. \(3^k-1\)

2. \(\dfrac{k(k+1)(2k+1)}{6}\)

Prove the following formula by induction:
\(2+4+\ldots+2n=n(n+1)\)

Prove the following formula by induction:
\(1+2+3+\ldots+n=\dfrac{n(n+1)}{2}\)

Prove by induction:
\(1+2+2^2+2^3+\ldots +2^{n-1}=2^n-1\)