Jay Abramson

Review Exercises

Sequences and Their Notation

1.

Write the first four terms of the sequence defined by the recursive formula $a_{1} = 2, a_{n} = a_{n - 1} + n .$

2.

Evaluate $\frac{6!}{(5 - 3)! 3!} .$

3.

Write the first four terms of the sequence defined by the explicit formula $a_{n} = 10^{n} + 3.$

4.

Write the first four terms of the sequence defined by the explicit formula $a_{n} = \frac{n!}{n (n + 1)} .$

Arithmetic Sequences

5.

Is the sequence $\frac{4}{7}, \frac{47}{21}, \frac{82}{21}, \frac{39}{7}, ...$ arithmetic? If so, find the common difference.

6.

Is the sequence $2, 4, 8, 16, ...$ arithmetic? If so, find the common difference.

7.

An arithmetic sequence has the first term $a_{1} = 18$ and common difference $d = - 8.$ What are the first five terms?

8.

An arithmetic sequence has terms $a_{3} = 11.7$ and $a_{8} = - 14.6.$ What is the first term?

9.

Write a recursive formula for the arithmetic sequence $- 20, - 10, 0, 10,…$

10.

Write a recursive formula for the arithmetic sequence $0, - \frac{1}{2}, - 1, - \frac{3}{2}, \dots,$ and then find the 31^st term.

11.

Write an explicit formula for the arithmetic sequence $\frac{7}{8}, \frac{29}{24}, \frac{37}{24}, \frac{15}{8}, \dots$

12.

How many terms are in the finite arithmetic sequence $12, 20, 28, \dots, 172 ?$

Geometric Sequences

13.

Find the common ratio for the geometric sequence $2.5, 5, 10, 20, \dots$

14.

Is the sequence 4, 16, 28, 40 … geometric? If so find the common ratio. If not, explain why.

15.

A geometric sequence has terms $a_{7} = 16, 384$ and $a_{9} = 262, 144$ . What are the first five terms?

16.

A geometric sequence has the first term $a_{1} = - 3$ and common ratio $r = \frac{1}{2} .$ What is the 8^th term?

17.

What are the first five terms of the geometric sequence $a_{1} = 3, a_{n} = 4 \cdot a_{n - 1} ?$

18.

Write a recursive formula for the geometric sequence $1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \dots$

19.

Write an explicit formula for the geometric sequence $- \frac{1}{5}, - \frac{1}{15}, - \frac{1}{45}, - \frac{1}{135}, \dots$

20.

How many terms are in the finite geometric sequence $- 5, - \frac{5}{3}, - \frac{5}{9}, \dots, - \frac{5}{59, 049} ?$

Series and Their Notation

21.

Use summation notation to write the sum of terms $\frac{1}{2} m + 5$ from $m = 0$ to $m = 5.$

22.

Use summation notation to write the sum that results from adding the number $13$ twenty times.

23.

Use the formula for the sum of the first $n$ terms of an arithmetic series to find the sum of the first eleven terms of the arithmetic series 2.5, 4, 5.5, … .

24.

A ladder has $15$ tapered rungs, the lengths of which increase by a common difference. The first rung is 5 inches long, and the last rung is 20 inches long. What is the sum of the lengths of the rungs?

25.

Use the formula for the sum of the first n terms of a geometric series to find $S_{9}$ for the series $12, 6, 3, \frac{3}{2}, \dots$

26.

The fees for the first three years of a hunting club membership are given in Table 1. If fees continue to rise at the same rate, how much will the total cost be for the first ten years of membership?

Year	Membership Fees
1	$1500
2	$1950
3	$2535

Table 1

27.

Find the sum of the infinite geometric series $\sum_{k = 1}^{\infty} 45 \cdot {(- \frac{1}{3})}^{k - 1} .$

28.

A ball has a bounce-back ratio of $\frac{3}{5}$ the height of the previous bounce. Write a series representing the total distance traveled by the ball, assuming it was initially dropped from a height of 5 feet. What is the total distance? (Hint: the total distance the ball travels on each bounce is the sum of the heights of the rise and the fall.)

29.

Alejandro deposits $80 of his monthly earnings into an annuity that earns 6.25% annual interest, compounded monthly. How much money will he have saved after 5 years?

30.

The twins Hoa and Binh both opened retirement accounts on their 21^st birthday. Hoa deposits $4,800.00 each year, earning 5.5% annual interest, compounded monthly. Binh deposits $3,600.00 each year, earning 8.5% annual interest, compounded monthly. Which twin will earn the most interest by the time they are $55$ years old? How much more?

Counting Principles

31.

How many ways are there to choose a number from the set ${- 10, - 6, 4, 10, 12, 18, 24, 32}$ that is divisible by either $4$ or $6 ?$

32.

In a group of $20$ musicians, $12$ play piano, $7$ play trumpet, and $2$ play both piano and trumpet. How many musicians play either piano or trumpet?

33.

How many ways are there to construct a 4-digit code if numbers can be repeated?

34.

A palette of water color paints has 3 shades of green, 3 shades of blue, 2 shades of red, 2 shades of yellow, and 1 shade of black. How many ways are there to choose one shade of each color?

35.

Calculate $P (18, 4) .$

36.

In a group of $5$ first-year, $10$ second-year, $3$ third-year, and $2$ fourth-year students, how many ways can a president, vice president, and treasurer be elected?

37.

Calculate $C (15, 6) .$

38.

A coffee shop has 7 Guatemalan roasts, 4 Cuban roasts, and 10 Costa Rican roasts. How many ways can the shop choose 2 Guatemalan, 2 Cuban, and 3 Costa Rican roasts for a coffee tasting event?

39.

How many subsets does the set ${1, 3, 5, \dots, 99}$ have?

40.

A day spa charges a basic day rate that includes use of a sauna, pool, and showers. For an extra charge, guests can choose from the following additional services: massage, body scrub, manicure, pedicure, facial, and straight-razor shave. How many ways are there to order additional services at the day spa?

41.

How many distinct ways can the word DEADWOOD be arranged?

42.

How many distinct rearrangements of the letters of the word DEADWOOD are there if the arrangement must begin and end with the letter D?

Binomial Theorem

43.

Evaluate the binomial coefficient $(\begin{matrix} 23 \\ 8 \end{matrix}) .$

44.

Use the Binomial Theorem to expand ${(3 x + \frac{1}{2} y)}^{6} .$

45.

Use the Binomial Theorem to write the first three terms of ${(2 a + b)}^{17} .$

46.

Find the fourth term of ${(3 a^{2} - 2 b)}^{11}$ without fully expanding the binomial.

Probability

For the following exercises, assume two die are rolled.

47.

Construct a table showing the sample space.

48.

What is the probability that a roll includes a $2 ?$

49.

What is the probability of rolling a pair?

50.

What is the probability that a roll includes a 2 or results in a pair?

51.

What is the probability that a roll doesn’t include a 2 or result in a pair?

52.

What is the probability of rolling a 5 or a 6?

53.

What is the probability that a roll includes neither a 5 nor a 6?

For the following exercises, use the following data: An elementary school survey found that 350 of the 500 students preferred soda to milk. Suppose 8 children from the school are attending a birthday party. (Show calculations and round to the nearest tenth of a percent.)

54.

What is the percent chance that all the children attending the party prefer soda?

55.

What is the percent chance that at least one of the children attending the party prefers milk?

56.

What is the percent chance that exactly 3 of the children attending the party prefer soda?

57.

What is the percent chance that exactly 3 of the children attending the party prefer milk?