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1. The starting salary for a job is $43,800 with a guaranteed increase of $1950 per year. Determine

(a) The salary during the fifth year

(b) The total compensation through five full years of employment.

2. In the first two trips baling hay around a large field, a farmer obtains 123 bales and 112 bales, respectively. Each round gets shorter, so the farmer estimates that the same pattern will continue. Estimate the total number of bales made after the farmer takes another six trips around the field.

(a) The salary during the fifth year

(b) The total compensation through five full years of employment.

2. In the first two trips baling hay around a large field, a farmer obtains 123 bales and 112 bales, respectively. Each round gets shorter, so the farmer estimates that the same pattern will continue. Estimate the total number of bales made after the farmer takes another six trips around the field.

Determine whether the sequence is geometric. If so, find the common ratio.

1. 2, 6, 18, 54, 162, . . .

2. 48, −24, 12, −6, . . .

3. 1/5, -3/5, 9/5, -27/5, . . . .

1. 2, 6, 18, 54, 162, . . .

2. 48, −24, 12, −6, . . .

3. 1/5, -3/5, 9/5, -27/5, . . . .

Write the first five terms of the geometric sequence.

1. a1 = 2, r = 15

2. a1 = 6, r = -1/3

1. a1 = 2, r = 15

2. a1 = 6, r = -1/3

Write an expression for the nth term of the geometric sequence. Then find the 10th term of the sequence.

1. a1 = 100, r = 1.05

2. a1 = 5, r = 0.2

3. a1 = 18, a2 = −9

1. a1 = 100, r = 1.05

2. a1 = 5, r = 0.2

3. a1 = 18, a2 = −9

Write an expression for the apparent nth term (an) of the sequence. (Assume that n begins with 1.)

1. −2, 2, −2, 2, −2, . . .

2. −1, 2, 7, 14, 23, . . .

1. −2, 2, −2, 2, −2, . . .

2. −1, 2, 7, 14, 23, . . .

Find the sum of the finite geometric sequence.

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1.

2.

3.

4.

Find the sum of the infinite geometric series.

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1.

2.

3.

4.

A paper manufacturer buys a machine for $120,000. It depreciates at a rate of 30% per year. (In other words, at the end of each year the depreciated value is 70% of what it was at the beginning of the year.)

(a) Find the formula for the nth term of a geometric sequence that gives the value of the machine t full years after it is purchased.

(b) Find the depreciated value of the machine after 5 full years.

2. An investor deposits $800 in an account on the first day of each month for 10 years. The account pays 3%, compounded monthly. What is the balance at the end of 10 years?

(a) Find the formula for the nth term of a geometric sequence that gives the value of the machine t full years after it is purchased.

(b) Find the depreciated value of the machine after 5 full years.

2. An investor deposits $800 in an account on the first day of each month for 10 years. The account pays 3%, compounded monthly. What is the balance at the end of 10 years?

Use mathematical induction to prove the formula for all integers n ≥1.

1. 3 + 5 + 7 + . . . + (2n + 1) = n (n + 2)

1. 3 + 5 + 7 + . . . + (2n + 1) = n (n + 2)

Find a formula for the sum of the first n terms of the sequence. Prove the validity of your formula.

9, 13, 17, 21, . . .

9, 13, 17, 21, . . .

Find the sum using the formulas for the sums of powers of integers.

1.

1.

2.

Write the first five terms of the sequence beginning with the term a1. Then calculate the first and second differences of the sequence. State whether the sequence has a perfect linear model, a perfect quadratic model, or neither

1. a1 = 5

an = an−1 + 5

1. a1 = 5

an = an−1 + 5

Find the binomial coefficient.

1. 6C4

2. 12C3

1. 6C4

2. 12C3

Evaluate using Pascal's Triangle.

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1.

2.

Use the Binomial Theorem to write the expansion of the expression.

1. (x + 4)4

2. (5 + 2z)4

1. (x + 4)4

2. (5 + 2z)4

Determine the number of ways a computer can generate the sum using randomly selected integers from 1 through 14.

1. Two distinct integers whose sum is 7

2. Two distinct integers whose sum is 12

1. Two distinct integers whose sum is 7

2. Two distinct integers whose sum is 12

1. All of the landline telephone numbers in a small town use the same three-digit prefix. How many different telephone numbers are possible by changing only the last four digits?

2. A college student is preparing a course schedule for the next semester. The student may select one of three mathematics courses, one of four science courses, and one of six history courses. How many schedules are possible?

3. A geneticist is using gel electrophoresis to analyze five DNA samples. The geneticist treats each sample with a different restriction enzyme and then injects it into one of five wells formed in a bed of gel. In how many orders can the geneticist inject the five samples into the wells?

2. A college student is preparing a course schedule for the next semester. The student may select one of three mathematics courses, one of four science courses, and one of six history courses. How many schedules are possible?

3. A geneticist is using gel electrophoresis to analyze five DNA samples. The geneticist treats each sample with a different restriction enzyme and then injects it into one of five wells formed in a bed of gel. In how many orders can the geneticist inject the five samples into the wells?

Simplify the factorial expression.

1. 3! / 5!

2. 7! / 3! · 4!

3. (n - 1)! / (n + 1)!

1. 3! / 5!

2. 7! / 3! · 4!

3. (n - 1)! / (n + 1)!

1. There are 10 bicyclists entered in a race. In how many different ways can the top three places be decided?

2. In how many different ways can a jury of 12 people be randomly selected from a group of 32 people?

2. In how many different ways can a jury of 12 people be randomly selected from a group of 32 people?

A local sandwich shop offers five different breads, four different meats, three different cheeses, and six different vegetables. A customer can choose one bread, one or no meat, one or no cheese, and up to three vegetables. Find the total number of combinations of sandwiches possible.

1. A drawer contains six white socks, two blue socks, and two gray socks.

(a) What is the probability of randomly selecting one blue sock?

(b) What is the probability of randomly selecting one white sock?

(a) What is the probability of randomly selecting one blue sock?

(b) What is the probability of randomly selecting one white sock?

1. At a university, 31% of the students are freshmen, 26% are sophomores, 25% are juniors, and 18% are seniors. One student receives a cash scholarship randomly by lottery. Find the probability that the scholarship winner is as described.

(a) A junior or senior

(b) A freshman, sophomore, or junior

2. In a survey, a sample of college students, faculty members, and administrators were asked whether they favor a proposed increase in the annual activity fee to enhance student life on campus. The table lists the results of the survey.

(a) A junior or senior

(b) A freshman, sophomore, or junior

2. In a survey, a sample of college students, faculty members, and administrators were asked whether they favor a proposed increase in the annual activity fee to enhance student life on campus. The table lists the results of the survey.

Find the probability that a person selected at random from the sample is as described.

(a) A person who opposes the proposal

(b) A student

(c) A faculty member who favors the proposal

1. You toss a six-sided die four times. What is the probability of getting four 5's?

2. You toss a six-sided die six times. What is the probability of getting each number exactly once?

3. You draw one card at random from a standard deck of 52 playing cards. What is the probability that the card is not a club?

4. You toss a coin five times. What is the probability of getting at least one tail?

2. You toss a six-sided die six times. What is the probability of getting each number exactly once?

3. You draw one card at random from a standard deck of 52 playing cards. What is the probability that the card is not a club?

4. You toss a coin five times. What is the probability of getting at least one tail?

1. To rewrite the expression 3/x5 using negative exponents, move x5 to the ________ and change the sign of the exponent.

2. When dividing fractions, multiply by the ________.

2. When dividing fractions, multiply by the ________.

Factor the expression.

1. 2x(x + 2)−1/2 + (x + 2)1/2

2. x2(x2 + 1)−5 − (x2 + 1)−4

1. 2x(x + 2)−1/2 + (x + 2)1/2

2. x2(x2 + 1)−5 − (x2 + 1)−4

Complete the factored form of the expression.

1. (5x + 3) / 4 = 1/4 ( )

2. 7x2 / 10 = 7/10 ( )

3. 2/3 x2 + 1/3 x + 5 = 1/3 ( )

1. (5x + 3) / 4 = 1/4 ( )

2. 7x2 / 10 = 7/10 ( )

3. 2/3 x2 + 1/3 x + 5 = 1/3 ( )

Insert the required factor in the parentheses.

1. x2(x3 − 1)4 = ( )(x3 − 1)4(3x2)

2. x(1 − 2x2)3 = ( )(1 − 2x2)3(−4x)

3. (4x + 6) / (x2 + 3x + 7)3 = ( ) 1 / (x2 + 3x + 7)3 (2x + 3)

1. x2(x3 − 1)4 = ( )(x3 − 1)4(3x2)

2. x(1 − 2x2)3 = ( )(1 − 2x2)3(−4x)

3. (4x + 6) / (x2 + 3x + 7)3 = ( ) 1 / (x2 + 3x + 7)3 (2x + 3)

Show that the two expressions are equivalent.

1. 4x2 + 6y2 / 10 x2 / (1/4) + 3y2/5

2. 4x2/14 - 2y2 = 2x2/7 - y2 / (1/2)

3. 25x2 / 36 + 4y2 / 9 = x2 / (36/25) + y2 / (9/4)

1. 4x2 + 6y2 / 10 x2 / (1/4) + 3y2/5

2. 4x2/14 - 2y2 = 2x2/7 - y2 / (1/2)

3. 25x2 / 36 + 4y2 / 9 = x2 / (36/25) + y2 / (9/4)

Describe and correct the error.

Rewrite the expression using negative exponents.

1. 7 / (x + 3)5

2. (2 - x) / (x + 1)3/2

3. 2x5 / (3x + 5)4

4. (x + 1) / x (6 - x)1/2

1. 7 / (x + 3)5

2. (2 - x) / (x + 1)3/2

3. 2x5 / (3x + 5)4

4. (x + 1) / x (6 - x)1/2

Rewrite the fraction as a sum of two or more terms.

1. (x2 + 6x + 12) / 3x

2. (x3 - 5x2 + 4) / x2

3. (4x3 - 7x2 + 1) / x1/3

1. (x2 + 6x + 12) / 3x

2. (x3 - 5x2 + 4) / x2

3. (4x3 - 7x2 + 1) / x1/3

Simplify the expression.

1.

1.

2.

3.

(a) Verify that y1 = y2 analytically.

y1 = x2(1/3) (x2 + 1)âˆ’2/3(2x) + (x2 + 1)1/3(2x)

y2 = 2x(4x2 + 3) / 3(x2 + 1)2/3

(b) Complete the table and demonstrate the equality in part (a) numerically.

y1 = x2(1/3) (x2 + 1)âˆ’2/3(2x) + (x2 + 1)1/3(2x)

y2 = 2x(4x2 + 3) / 3(x2 + 1)2/3

(b) Complete the table and demonstrate the equality in part (a) numerically.

(c) Use a graphing utility to verify the equality in part (a) graphically.

An athlete has set up a course in which she is dropped off by a boat 2 miles from the nearest point on shore. Once she reaches the shore, she must run to a point 4 miles down the coast and 2 miles inland (see figure). She can swim 2 miles per hour and run 6miles per hour. The time t (in hours) required for her to complete the course can be approximated by the model

t = âˆš(x2 + 4)/2 + âˆš{(4 âˆ’ x)2 + 4}/6

Where x is the distance (in miles) down the coast from her starting point to the point at which she leaves the water to start her run.

t = âˆš(x2 + 4)/2 + âˆš{(4 âˆ’ x)2 + 4}/6

Where x is the distance (in miles) down the coast from her starting point to the point at which she leaves the water to start her run.

(a) Use a table to approximate the distance down the coast that will yield the minimum amount of time required for the athlete to complete the course.

(b) The expression below was obtained using calculus. It can be used to find the minimum amount of time required for the tri athlete to reach the finish line. Simplify the expression.

1/2x(x2 + 4)-1/2 + 1/6 [(x - 4) (x2 - 8x + 20)-1/2]

1. Write a paragraph explaining to a classmate why

1 / [(x - 2)1/2 + x4 ≠ (x - 2)-1/2 + x-4

2. You are taking a course in calculus, and for one of the homework problems you obtain the following answer.

2/3x (2x - 3)3/2 - 2/15 (2x - 3)5/2

The answer in the back of the book is

2/5 (2x - 3)3/2 (x + 1).

Show how the second answer can be obtained from the first. Then use the same technique to simplify the expression

2/3 x (4 + x)3/2 - 2/15 (4 + x)5/2.

1 / [(x - 2)1/2 + x4 ≠ (x - 2)-1/2 + x-4

2. You are taking a course in calculus, and for one of the homework problems you obtain the following answer.

2/3x (2x - 3)3/2 - 2/15 (2x - 3)5/2

The answer in the back of the book is

2/5 (2x - 3)3/2 (x + 1).

Show how the second answer can be obtained from the first. Then use the same technique to simplify the expression

2/3 x (4 + x)3/2 - 2/15 (4 + x)5/2.

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