Question 1

Solve the problem.
-A game involves choosing 8 numbers from the numbers 1 through 13. In how many ways can this be done?

Accepted Answer

A)  1267 
B)  1295 
C)  51,891,840 
D)  1287 
A)  1267 
B)  1295 
C)  51,891,840 
D)  1287

Question 2

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest
hundredth.
-$$\mathrm { S } _ { 6 } = 141 , \mathrm { a } _ { 6 } = 41$$

Accepted Answer

A)  $a _ { 1 } = 6 , d = 7$ 
B)  $\mathrm { a } _ { 1 } = - 199 , \mathrm {~d} = 13.67$ 
C)  $a _ { 1 } = - 199 , d = 7$ 
D)  $a _ { 1 } = 6 , d = 13.67$ 
A)  $a _ { 1 } = 6 , d = 7$ 
B)  $\mathrm { a } _ { 1 } = - 199 , \mathrm {~d} = 13.67$ 
C)  $a _ { 1 } = - 199 , d = 7$ 
D)  $a _ { 1 } = 6 , d = 13.67$

Question 3

Decide whether the given sequence is finite or infinite.
-$$\mathrm { a } _ { 1 } = 2 , \mathrm { a } _ { 2 } = 4 ; \text { for } \mathrm { n } \geq 3 , \mathrm { a } _ { \mathrm { n } } = 4 \cdot \mathrm { a } _ { \mathrm { n } - 1 } + 7 \cdot \mathrm { a } _ { \mathrm { n } - 2 }$$

Accepted Answer

A)  Finite 
B)  Infinite 
A)  Finite 
B)  Infinite

Question 4

Solve the problem.
-What are the odds in favor of drawing a number greater than 2 from these cards?

Accepted Answer

A)  3 to 5 
B)  3 to 2 
C)  4 to 1 
D)  4 to 5 
A)  3 to 5 
B)  3 to 2 
C)  4 to 1 
D)  4 to 5

Question 5

Find the probability.
-A bag contains 5 red marbles, 2 blue marbles, and 1 green marble. What is the probability of choosing a marble that is not blue when one marble is drawn from the bag?

Accepted Answer

A)  $\frac { 4 } { 3 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  6 
A)  $\frac { 4 } { 3 }$ 
B)  $\frac { 1 } { 4 }$ 
C)  $\frac { 3 } { 4 }$ 
D)  6

Question 6

Find the nth term of the geometric sequence.
-$$a _ { 1 } = 4 , r = - 3 , n = 5$$

Accepted Answer

A)  $a _ { 5 } = - 108$ 
B)  $a _ { 5 } = 729$ 
C)  $a _ { 5 } = - 60$ 
D)  $a _ { 5 } = 324$ 
A)  $a _ { 5 } = - 108$ 
B)  $a _ { 5 } = 729$ 
C)  $a _ { 5 } = - 60$ 
D)  $a _ { 5 } = 324$

Question 7

It can be shown that $$( 1 + x ) ^ { n } = 1 + n x + \frac { n ( n - 1 ) } { 2 ! } x ^ { 2 } + \frac { n ( n - 1 ) ( n - 2 ) } { 3 ! } x ^ { 3 }$$ . . . is true for any real number n (not just positive
integer values) and any real number x, wher $$| x | < 1$$ . Use this series to approximate the given number to the nearest
thousandth.
-$1.06 ^ { 3 }$

Accepted Answer

A)  $0.837$ 
B)  $1.191$ 
C)  $1.194$ 
D)  $0.840$ 
A)  $0.837$ 
B)  $1.191$ 
C)  $1.194$ 
D)  $0.840$

Question 8

Solve the problem.
-How many 5-card poker hands consisting of 3 aces and 2 kings are possible with an ordinary 52-card deck?

Accepted Answer

A)  288 
B)  24 
C)  12 
D)  6 
A)  288 
B)  24 
C)  12 
D)  6

Question 9

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
-$$a _ { 5 } = 4 , d = - 3 , n = 5$$

Accepted Answer

A)  $0,16,13,10,7$ 
B)  $20,16,12,8,4$ 
C)  $- 16 , - 13 , - 10 , - 7 , - 4$ 
D)  $16,13,10,7,4$ 
A)  $0,16,13,10,7$ 
B)  $20,16,12,8,4$ 
C)  $- 16 , - 13 , - 10 , - 7 , - 4$ 
D)  $16,13,10,7,4$

Question 10

Solve the problem.
-A fair spinner has regions numbered 1 through 18. What is the probability that the spinner will stop on an even number or a multiple of 3?

Accepted Answer

A)  15 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  1 
A)  15 
B)  $\frac { 1 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  1

Question 11

Evaluate the series, if it converges.
-$$- 10 - \frac { 25 } { 2 } - \frac { 125 } { 8 } - \frac { 625 } { 32 } + \ldots$$

Accepted Answer

A)  $- 50$ 
B)  Does not converge 
C)  $- 40$ 
D)  50 
A)  $- 50$ 
B)  Does not converge 
C)  $- 40$ 
D)  50

Question 12

Find the common difference for the arithmetic sequence.
-$5,8,11,14 , \ldots$

Accepted Answer

A)  $2.25$ 
B)  3 
C)  35 
D)  5 
A)  $2.25$ 
B)  3 
C)  35 
D)  5

Question 13

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence.
Round to the nearest thousandth, if necessary.
-Find the sum of all the integers from -44 to 34.

Accepted Answer

A)  -385 
B)  -395 
C)  -429 
D)  -351 
A)  -385 
B)  -395 
C)  -429 
D)  -351

Question 14

Evaluate the sum.
-$\sum _ { j = 800 } ^ { 5000 } j$

Accepted Answer

A)  16,822,900 
B)  19,142,900 
C)  12,182,900 
D)  9,862,900 
A)  16,822,900 
B)  19,142,900 
C)  12,182,900 
D)  9,862,900

Question 15

Evaluate the series, if it converges.
-$$\sum _ { \mathrm { i } = 1 } ^ { \infty } 45 \left( \frac { 9 } { 4 } \right) ^ { \mathrm { i } - 1 }$$

Accepted Answer

A)  $- 36$ 
B)  Does not converge. 
C)  36 
D)  81 
A)  $- 36$ 
B)  Does not converge. 
C)  36 
D)  81

Question 16

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth.
-$$a _ { 2 } = - 48 , a _ { 4 } = - 3$$

Accepted Answer

A)  $\mathrm { a } _ { 1 } = - 192 , \mathrm { r } = 0.25$ 
B)  $\mathrm { a } _ { 1 } = - 3 , \mathrm { r } = 4$ 
C)  $\mathrm { a } _ { 1 } = - 768 , \mathrm { r } = 0.25$ 
D)  $a _ { 1 } = - 12 , r = 4$ 
A)  $\mathrm { a } _ { 1 } = - 192 , \mathrm { r } = 0.25$ 
B)  $\mathrm { a } _ { 1 } = - 3 , \mathrm { r } = 4$ 
C)  $\mathrm { a } _ { 1 } = - 768 , \mathrm { r } = 0.25$ 
D)  $a _ { 1 } = - 12 , r = 4$

Question 17

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest
hundredth.
-$$S _ { 4 } = - 38 , a _ { 4 } = - 17$$

Accepted Answer

A)  $\mathrm { a } _ { 1 } = 49 , \mathrm {~d} = - 8.5$ 
B)  $\mathrm { a } _ { 1 } = 49 , \mathrm {~d} = - 5$ 
C)  $a _ { 1 } = - 2 , d = - 5$ 
D)  $a _ { 1 } = - 2 , d = - 8.5$ 
A)  $\mathrm { a } _ { 1 } = 49 , \mathrm {~d} = - 8.5$ 
B)  $\mathrm { a } _ { 1 } = 49 , \mathrm {~d} = - 5$ 
C)  $a _ { 1 } = - 2 , d = - 5$ 
D)  $a _ { 1 } = - 2 , d = - 8.5$

Question 18

Evaluate the expression.
-$${ } _ { 12 } \mathrm { C } _ { 10 }$$

Accepted Answer

A)  $239,500,800$ 
B)  132 
C)  66 
D)  $\frac { 3 } { 5 }$ 
A)  $239,500,800$ 
B)  132 
C)  66 
D)  $\frac { 3 } { 5 }$

Question 19

Evaluate the series, if it converges.
-$$\sum _ { i = 1 } ^ { \infty } 46 \left( - \frac { 3 } { 10 } \right) ^ { i - 1 }$$

Accepted Answer

A)  $\frac { 460 } { 13 }$ 
B)  Does not converge. 
C)  $\frac { 10 } { 13 }$ 
D)  $- \frac { 138 } { 13 }$ 
A)  $\frac { 460 } { 13 }$ 
B)  Does not converge. 
C)  $\frac { 10 } { 13 }$ 
D)  $- \frac { 138 } { 13 }$

Question 20

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$$

Accepted Answer

Answers will vary. One possible proof fo

Solve the problem. -A game involves choosing 8 numbers from the numbers 1 through 13. In how many ways can this be done?

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest hundredth. - $\mathrm { S } _ { 6 } = 141 , \mathrm { a } _ { 6 } = 41$

Decide whether the given sequence is finite or infinite. - $\mathrm { a } _ { 1 } = 2 , \mathrm { a } _ { 2 } = 4 ; \text { for } \mathrm { n } \geq 3 , \mathrm { a } _ { \mathrm { n } } = 4 \cdot \mathrm { a } _ { \mathrm { n } - 1 } + 7 \cdot \mathrm { a } _ { \mathrm { n } - 2 }$

Solve the problem. -What are the odds in favor of drawing a number greater than 2 from these cards?

Find the probability. -A bag contains 5 red marbles, 2 blue marbles, and 1 green marble. What is the probability of choosing a marble that is not blue when one marble is drawn from the bag?

Find the nth term of the geometric sequence. - $a _ { 1 } = 4 , r = - 3 , n = 5$

Solve the problem. -How many 5-card poker hands consisting of 3 aces and 2 kings are possible with an ordinary 52-card deck?

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $a _ { 5 } = 4 , d = - 3 , n = 5$

Solve the problem. -A fair spinner has regions numbered 1 through 18. What is the probability that the spinner will stop on an even number or a multiple of 3?

Evaluate the series, if it converges. - $- 10 - \frac { 25 } { 2 } - \frac { 125 } { 8 } - \frac { 625 } { 32 } + \ldots$

Find the common difference for the arithmetic sequence. - $5,8,11,14 , \ldots$

Use the sequence feature of a graphing calculator to evaluate the sum of the first 10 terms of the arithmetic sequence. Round to the nearest thousandth, if necessary. -Find the sum of all the integers from -44 to 34.

Evaluate the sum. - $\sum _ { j = 800 } ^ { 5000 } j$

Evaluate the series, if it converges. - $\sum _ { \mathrm { i } = 1 } ^ { \infty } 45 \left( \frac { 9 } { 4 } \right) ^ { \mathrm { i } - 1 }$

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - $a _ { 2 } = - 48 , a _ { 4 } = - 3$

Find the first term and the common difference for the arithmetic sequence. Round approximations to the nearest hundredth. - $S _ { 4 } = - 38 , a _ { 4 } = - 17$

Evaluate the expression. - ${ } _ { 12 } \mathrm { C } _ { 10 }$

Evaluate the series, if it converges. - $\sum _ { i = 1 } ^ { \infty } 46 \left( - \frac { 3 } { 10 } \right) ^ { i - 1 }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $1 ^ { 2 } + 4 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 3 n - 2 ) ^ { 2 } = \frac { n \left( 6 n ^ { 2 } - 3 n - 1 \right) } { 2 }$

Review of Basic Concepts

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Filters

Exam 12: Further Topics in Algebra