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

$$a_8 = 6$$ \times $$2^7 = 768$

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

Guida di studio - Domande di pratica

Study guide - Flashcard