Q: In a race with 50 runners and no ties, in how many ways can the first three finishers come in?

$50 \times 49 \times 48 = 117,600$

Q: A single die is rolled twice. What is the probability of rolling a 2 the first time and a 3 the second time?

$\frac{1}{6} \times \frac{1}{6} = \frac{1}{36}$

Question 1

Given the sequence whose general term is $a_n = 3n + 2$, what is the value of the 10th term?

Accepted Answer

$3(10) + 2 = 32$

Question 2

A sequence is defined recursively by $a_1 = 4$ and $a_{n} = 2a_{n-1} + 1$ for $n \geq 2$. What is the value of $a_4$?

Accepted Answer

$a_4 = 2a_3 + 1 = 2(11) + 1 = 23$

Question 3

Which of the following is the correct formula for the sum of the first $n$ terms of an arithmetic sequence with first term $a_1$ and last term $a_n$?

Accepted Answer

$S_n = \frac{n}{2}(a_1 + a_n)$

Question 4

A stadium section has 20 seats in the first row, 23 in the second, increasing by 3 seats each row for a total of 38 rows. How many seats are in the section?

Accepted Answer

$S_{38} = \frac{38}{2}(20 + 127) = 2,869$

Question 5

Which of the following sequences is geometric?

Accepted Answer

$3, 6, 12, 24, ...$

Question 6

Find the eighth term of the geometric sequence with first term $a_1 = 6$ and common ratio $r = 2$.

Accepted Answer

$a_8 = 6 	imes 2^7 = 768$

Question 7

A job pays $44,000$ the first year. Each year for 19 more years, the salary increases by 5%. What is the total lifetime salary over the 20-year period (rounded to the nearest dollar)?

Accepted Answer

$S_{20} = 44,000 \frac{1 - 1.05^{20}}{1 - 1.05} \approx 1,454,902$

Question 8

What is the sum of the infinite geometric series $4 + \frac{8}{3} + \frac{16}{9} + \frac{32}{27} + \cdots$?

Accepted Answer

$9$

Question 9

Which step is NOT required in a proof by mathematical induction?

Accepted Answer

Show the statement is true for all real numbers $n$

Question 10

Evaluate the binomial coefficient $\binom{8}{3}$.

Accepted Answer

$56$

Question 11

Expand $(x+2)^3$ using the Binomial Theorem.

Accepted Answer

$x^3 + 6x^2 + 12x + 8$

Question 12

How many different 4-card hands can be dealt from a deck of 16 different cards?

Accepted Answer

$\binom{16}{4} = 1820$

Question 13

A pizza can be ordered with 3 choices of size, 4 choices of crust, and 6 choices of toppings. How many different one-topping pizzas can be ordered?

Accepted Answer

$3 	imes 4 	imes 6 = 72$

Question 14

In a race with 50 runners and no ties, in how many ways can the first three finishers come in?

Accepted Answer

$50 	imes 49 	imes 48 = 117,600$

Question 15

If you roll a single six-sided die, what is the probability of getting either a 4 or a 5?

Accepted Answer

$\frac{2}{6} = \frac{1}{3}$

Question 16

A single die is rolled twice. What is the probability of rolling a 2 the first time and a 3 the second time?

Accepted Answer

$\frac{1}{6} 	imes \frac{1}{6} = \frac{1}{36}$

Question 17

If you are dealt one card from a standard 52-card deck, what is the probability that you are not dealt a king?

Accepted Answer

$\frac{48}{52} = \frac{12}{13}$

Question 18

A family has four children. What is the probability that all four are boys?

Accepted Answer

$\frac{1}{16}$

Question 19

Which formula gives the number of permutations of $n$ objects taken $r$ at a time?

Accepted Answer

$P(n, r) = \frac{n!}{(n-r)!}$

Sequences, Series, and Probability: Study Notes for College Algebra

Study Guide - Practice Questions

Study Guide - Flashcards