Question 1

Solve the problem.
-An ordinary die is tossed. What are the odds in favor of the die showing a 4 ?

Accepted Answer

A)  2 to 3 
B)  1 to 6 
C)  1 to 4 
D)  1 to 5 
A)  2 to 3 
B)  1 to 6 
C)  1 to 4 
D)  1 to 5

Question 2

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
-$$2,6,10,14,18 , \ldots$$

Accepted Answer

A)  $\mathrm { a } _ { \mathrm { n } } = 2 ( 4 ) ^ { \mathrm { n } - 1 } ; \mathrm { a } _ { 6 } = 32,768$ 
B)  $a _ { n } = n + 4 ; a _ { 6 } = 10$ 
C)  $a _ { n } = 4 n - 2 ; a _ { 6 } = 22$ 
D)  $a _ { n } = 2 n - 4 ; a _ { 6 } = 8$ 
A)  $\mathrm { a } _ { \mathrm { n } } = 2 ( 4 ) ^ { \mathrm { n } - 1 } ; \mathrm { a } _ { 6 } = 32,768$ 
B)  $a _ { n } = n + 4 ; a _ { 6 } = 10$ 
C)  $a _ { n } = 4 n - 2 ; a _ { 6 } = 22$ 
D)  $a _ { n } = 2 n - 4 ; a _ { 6 } = 8$

Question 3

Evaluate the sum.
-$\sum _ { i = 3 } ^ { 14 } ( i + 7 )$

Accepted Answer

A)  183 
B)  169 
C)  172 
D)  186 
A)  183 
B)  169 
C)  172 
D)  186

Question 4

Write the indicated term of the binomial expansion.
-$( 3 x - 2 y ) ^ { 12 } ; \quad 11$ th term

Accepted Answer

A)  $202,752 \mathrm { x } ^{10} \mathrm { y }^ {2}$ 
B)  $608,256 x ^ { 2 } y ^ { 10 }$ 
C)  $- 304,128 x ^ { 10 } y ^ { 2 }$ 
D)  $- 304,128 x ^ { 2 } y ^{11}$ 
A)  $202,752 \mathrm { x } ^{10} \mathrm { y }^ {2}$ 
B)  $608,256 x ^ { 2 } y ^ { 10 }$ 
C)  $- 304,128 x ^ { 10 } y ^ { 2 }$ 
D)  $- 304,128 x ^ { 2 } y ^{11}$

Question 5

Find a general term an for the geometric sequence.
-$$a _ { 1 } = \frac { 1 } { 5 } , r = - 5$$

Accepted Answer

A)  $a _ { n } = \left( \frac { 1 } { 5 } \right) ^ { n - 1 } - \frac { 6 } { 25 }$ 
B)  $a _ { n } = \frac { 1 } { 5 } - \frac { 6 } { 25 } ( n - 1 )$ 
C)  $a _ { n } = \frac { 1 } { 5 } \cdot ( - 5 ) ^ { n - 1 }$ 
D)  $a _ { n } = \frac { 1 } { 5 } - \frac { 1 } { 5 } ( n - 1 )$ 
A)  $a _ { n } = \left( \frac { 1 } { 5 } \right) ^ { n - 1 } - \frac { 6 } { 25 }$ 
B)  $a _ { n } = \frac { 1 } { 5 } - \frac { 6 } { 25 } ( n - 1 )$ 
C)  $a _ { n } = \frac { 1 } { 5 } \cdot ( - 5 ) ^ { n - 1 }$ 
D)  $a _ { n } = \frac { 1 } { 5 } - \frac { 1 } { 5 } ( n - 1 )$

Question 6

Write the indicated term of the binomial expansion.
-$( 4 x + 7 ) ^ { 3 } ; \quad 3 r d$ term

Accepted Answer

A)  $1176 \mathrm { x }$ 
B)  $588 x$ 
C)  $336 x ^ { 2 }$ 
D)  49 
A)  $1176 \mathrm { x }$ 
B)  $588 x$ 
C)  $336 x ^ { 2 }$ 
D)  49

Question 7

Write the binomial expansion of the expression.
-$$\left( \frac { 1 } { x } - \sqrt { 13 } y \right) ^ { 3 }$$

Accepted Answer

A)  $\frac { 1 } { x ^ { 3 } } + \frac { 3 \sqrt { 13 } y } { x ^ { 2 } } + \frac { 39 y ^ { 2 } } { x } + 13 \sqrt { 13 } y ^ { 3 }$ 
B)  $\frac { 1 } { x ^ { 3 } } - \frac { 3 \sqrt { 13 } y } { x ^ { 2 } } - \frac { 39 y ^ { 2 } } { x } - 13 \sqrt { 13 } y ^ { 3 }$ 
C)  $\frac { 1 } { x ^ { 3 } } - \frac { 3 \sqrt { 13 } y } { x ^ { 2 } } + \frac { 117 y ^ { 2 } } { x } - 13 \sqrt { 13 } y ^ { 3 }$ 
D)  $\frac { 1 } { x ^ { 3 } } - \frac { 3 \sqrt { 13 } y } { x ^ { 2 } } + \frac { 39 y ^ { 2 } } { x } - 13 \sqrt { 13 } y ^ { 3 }$ 
A)  $\frac { 1 } { x ^ { 3 } } + \frac { 3 \sqrt { 13 } y } { x ^ { 2 } } + \frac { 39 y ^ { 2 } } { x } + 13 \sqrt { 13 } y ^ { 3 }$ 
B)  $\frac { 1 } { x ^ { 3 } } - \frac { 3 \sqrt { 13 } y } { x ^ { 2 } } - \frac { 39 y ^ { 2 } } { x } - 13 \sqrt { 13 } y ^ { 3 }$ 
C)  $\frac { 1 } { x ^ { 3 } } - \frac { 3 \sqrt { 13 } y } { x ^ { 2 } } + \frac { 117 y ^ { 2 } } { x } - 13 \sqrt { 13 } y ^ { 3 }$ 
D)  $\frac { 1 } { x ^ { 3 } } - \frac { 3 \sqrt { 13 } y } { x ^ { 2 } } + \frac { 39 y ^ { 2 } } { x } - 13 \sqrt { 13 } y ^ { 3 }$

Question 8

Find a formula for the nth term of the arithmetic sequence shown in the graph.
-

Accepted Answer

A)  $a _ { n } = n + 2$ 
B)  $a _ { n } = n + 1$ 
C)  $a _ { n } = n - 2$ 
D)  $a _ { n } = n - 1$ 
A)  $a _ { n } = n + 2$ 
B)  $a _ { n } = n + 1$ 
C)  $a _ { n } = n - 2$ 
D)  $a _ { n } = n - 1$

Question 9

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or
diverges.
-Suppose that certain bacteria can double their size and divide every 30 minutes. Write a recursive sequence that describes this growth where each value of $\mathrm { n }$ represents a 30 -minute interval. Let a 1 $= 436$ represent the initial number of bacteria present.

Accepted Answer

A)  $a _ { 1 } = 436 ; a _ { n } = \frac { 1 } { 2 } a _ { n - 1 }$ for $n > 1$. 
B)  $a _ { n } = 2 a _ { 1 }$ for all values of $n$. 
C)  $a _ { 1 } = 436 ; a _ { n } = 2 a _ { n - 1 }$ for $n > 1$. 
D)  $a _ { 1 } = 436 ; a _ { n } = 2 a _ { n } + 1$ for $n > 1$. 
A)  $a _ { 1 } = 436 ; a _ { n } = \frac { 1 } { 2 } a _ { n - 1 }$ for $n > 1$. 
B)  $a _ { n } = 2 a _ { 1 }$ for all values of $n$. 
C)  $a _ { 1 } = 436 ; a _ { n } = 2 a _ { n - 1 }$ for $n > 1$. 
D)  $a _ { 1 } = 436 ; a _ { n } = 2 a _ { n } + 1$ for $n > 1$.

Question 10

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$2 \cdot 4 + 3 \cdot 5 + 4 \cdot 6 + \ldots + ( n + 1 ) ( n + 3 ) = \frac { n \left( 2 n ^ { 2 } + 15 n + 31 \right) } { 6 }$$

Accepted Answer

Answers will vary. One possible proof fo

Question 11

List the elements in the sample space of the experiment.
-A group of 13 people are assigned numbers 1 through 13 . List the sample space of the event choosing a person with a number 5 or less.

Accepted Answer

A)  $\{ 1,2,3,4,5 \}$ 
B)  $\{ 1 \}$ 
C)  $\{ 1,2,3,4 \}$ 
D)  $\{ 13 \}$ 
A)  $\{ 1,2,3,4,5 \}$ 
B)  $\{ 1 \}$ 
C)  $\{ 1,2,3,4 \}$ 
D)  $\{ 13 \}$

Question 12

Evaluate the sum.
-$\sum _ { \mathrm { i } = 1 } ^ { 8 } ( \mathrm { i } + 4 )$

Accepted Answer

A)  40 
B)  68 
C)  56 
D)  32 
A)  40 
B)  68 
C)  56 
D)  32

Question 13

List the elements in the sample space of the experiment.
-A box contains 3 blue cards numbered 1 through 3 , and 4 green cards numbered 1 through 4 . List the sample space of picking a blue card followed by a green card.

Accepted Answer

A)  $\{ 12 \}$ 
B)  $\{ ( 1,1 ) , ( 1,2 ) , ( 1,3 ) , ( 1,4 ) , ( 2,1 ) , ( 2,2 ) , ( 2,3 ) , ( 2,4 ) , ( 3,1 ) , ( 3,2 ) , ( 3,3 ) , ( 3,4 ) \}$ 
C)  $\{ 7 \}$ 
D)  $\{ ( 1,1 ) , ( 1,2 ) , ( 1,3 ) , ( 2,1 ) , ( 2,2 ) , ( 2,3 ) , ( 3,1 ) , ( 3,2 ) , ( 3,3 ) , ( 4,1 ) , ( 4,2 ) , ( 4,3 ) \}$ 
A)  $\{ 12 \}$ 
B)  $\{ ( 1,1 ) , ( 1,2 ) , ( 1,3 ) , ( 1,4 ) , ( 2,1 ) , ( 2,2 ) , ( 2,3 ) , ( 2,4 ) , ( 3,1 ) , ( 3,2 ) , ( 3,3 ) , ( 3,4 ) \}$ 
C)  $\{ 7 \}$ 
D)  $\{ ( 1,1 ) , ( 1,2 ) , ( 1,3 ) , ( 2,1 ) , ( 2,2 ) , ( 2,3 ) , ( 3,1 ) , ( 3,2 ) , ( 3,3 ) , ( 4,1 ) , ( 4,2 ) , ( 4,3 ) \}$

Question 14

Solve the problem.
-A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing an ace or 9?

Accepted Answer

A)  $\frac { 2 } { 13 }$ 
B)  $\frac { 13 } { 2 }$ 
C)  $\frac { 5 } { 13 }$ 
D)  10 
A)  $\frac { 2 } { 13 }$ 
B)  $\frac { 13 } { 2 }$ 
C)  $\frac { 5 } { 13 }$ 
D)  10

Question 15

Solve the problem.
-What are the odds in favor of drawing an even number from these cards?

142454

Accepted Answer

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

Question 16

Solve the problem.
-A bag contains 6 apples and 4 oranges. If you select 5 pieces of fruit without looking, how many ways can you get 5 oranges?

Accepted Answer

A)  24 
B)  10 
C)  0 
D)  6 
A)  24 
B)  10 
C)  0 
D)  6

Question 17

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$2 ^ { n } > n$$

Accepted Answer

Answers will vary. One possible proof fo

Question 18

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
-$$\mathrm { a } _ { 1 } = 3 - \sqrt { 7 } , \mathrm { a } _ { 2 } = 3 , \mathrm { n } = 4$$

Accepted Answer

A)  $0,1 - \sqrt { 7 } , 4 - 4 \sqrt { 7 } , 5 - 5 \sqrt { 7 }$ 
B)  $0,3 - \sqrt { 7 } , 3,3 + \sqrt { 7 }$ 
C)  $3 - \sqrt { 7 } , 3,3 + \sqrt { 7 } , 3 + 2 \sqrt { 7 }$ 
D)  $3 - \sqrt { 7 } , 4 - \sqrt { 7 } , 5 - \sqrt { 7 } , 6 - \sqrt { 7 }$ 
A)  $0,1 - \sqrt { 7 } , 4 - 4 \sqrt { 7 } , 5 - 5 \sqrt { 7 }$ 
B)  $0,3 - \sqrt { 7 } , 3,3 + \sqrt { 7 }$ 
C)  $3 - \sqrt { 7 } , 3,3 + \sqrt { 7 } , 3 + 2 \sqrt { 7 }$ 
D)  $3 - \sqrt { 7 } , 4 - \sqrt { 7 } , 5 - \sqrt { 7 } , 6 - \sqrt { 7 }$

Question 19

Use mathematical induction to prove that the statement is true for every positive integer n.
-If $0 < a < 1$, then $a ^ { n } < 1$.
(Assume that a is a constant.)

Accepted Answer

Answers will vary. One possible proof fo

Question 20

Solve the problem.
-Suppose there are 3 roads connecting town A to town B and 7 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

Accepted Answer

A)  21 
B)  9 
C)  49 
D)  10 
A)  21 
B)  9 
C)  49 
D)  10

Solve the problem. -An ordinary die is tossed. What are the odds in favor of the die showing a 4 ?

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $2,6,10,14,18 , \ldots$

Evaluate the sum. - $\sum _ { i = 3 } ^ { 14 } ( i + 7 )$

Write the indicated term of the binomial expansion. - $( 3 x - 2 y ) ^ { 12 } ; \quad 11$ th term

Find a general term an for the geometric sequence. - $a _ { 1 } = \frac { 1 } { 5 } , r = - 5$

Write the indicated term of the binomial expansion. - $( 4 x + 7 ) ^ { 3 } ; \quad 3 r d$ term

Write the binomial expansion of the expression. - $\left( \frac { 1 } { x } - \sqrt { 13 } y \right) ^ { 3 }$

Find a formula for the nth term of the arithmetic sequence shown in the graph. -

Use mathematical induction to prove that the statement is true for every positive integer n. - $2 \cdot 4 + 3 \cdot 5 + 4 \cdot 6 + \ldots + ( n + 1 ) ( n + 3 ) = \frac { n \left( 2 n ^ { 2 } + 15 n + 31 \right) } { 6 }$

List the elements in the sample space of the experiment. -A group of 13 people are assigned numbers 1 through 13 . List the sample space of the event choosing a person with a number 5 or less.

Evaluate the sum. - $\sum _ { \mathrm { i } = 1 } ^ { 8 } ( \mathrm { i } + 4 )$

List the elements in the sample space of the experiment. -A box contains 3 blue cards numbered 1 through 3 , and 4 green cards numbered 1 through 4 . List the sample space of picking a blue card followed by a green card.

Solve the problem. -A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing an ace or 9?

Solve the problem. -What are the odds in favor of drawing an even number from these cards? 142454

Solve the problem. -A bag contains 6 apples and 4 oranges. If you select 5 pieces of fruit without looking, how many ways can you get 5 oranges?

Use mathematical induction to prove that the statement is true for every positive integer n. - $2 ^ { n } > n$

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $\mathrm { a } _ { 1 } = 3 - \sqrt { 7 } , \mathrm { a } _ { 2 } = 3 , \mathrm { n } = 4$

Use mathematical induction to prove that the statement is true for every positive integer n. -If $0 < a < 1$ , then $a ^ { n } < 1$ . (Assume that a is a constant.)

Solve the problem. -Suppose there are 3 roads connecting town A to town B and 7 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

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

Solve the problem. -An ordinary die is tossed. What are the odds in favor of the die showing a 4 ?

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - 2,6,10,14,18,…2,6,10,14,18 , \ldots2,6,10,14,18,…

Evaluate the sum. - ∑i=314(i+7)\sum _ { i = 3 } ^ { 14 } ( i + 7 )∑i=314​(i+7)

Write the indicated term of the binomial expansion. - (3x−2y)12;11( 3 x - 2 y ) ^ { 12 } ; \quad 11(3x−2y)12;11 th term

Find a general term an for the geometric sequence. - a1=15,r=−5a _ { 1 } = \frac { 1 } { 5 } , r = - 5a1​=51​,r=−5

Write the indicated term of the binomial expansion. - (4x+7)3;3rd( 4 x + 7 ) ^ { 3 } ; \quad 3 r d(4x+7)3;3rd term

Write the binomial expansion of the expression. - (1x−13y)3\left( \frac { 1 } { x } - \sqrt { 13 } y \right) ^ { 3 }(x1​−13​y)3

Find a formula for the nth term of the arithmetic sequence shown in the graph. -

List the elements in the sample space of the experiment. -A group of 13 people are assigned numbers 1 through 13 . List the sample space of the event choosing a person with a number 5 or less.

Evaluate the sum. - ∑i=18(i+4)\sum _ { \mathrm { i } = 1 } ^ { 8 } ( \mathrm { i } + 4 )∑i=18​(i+4)

List the elements in the sample space of the experiment. -A box contains 3 blue cards numbered 1 through 3 , and 4 green cards numbered 1 through 4 . List the sample space of picking a blue card followed by a green card.

Solve the problem. -A card is drawn from a well-shuffled deck of 52 cards. What is the probability of drawing an ace or 9?

Solve the problem. -What are the odds in favor of drawing an even number from these cards? 142454

Solve the problem. -A bag contains 6 apples and 4 oranges. If you select 5 pieces of fruit without looking, how many ways can you get 5 oranges?

Use mathematical induction to prove that the statement is true for every positive integer n. - 2n>n2 ^ { n } > n2n>n

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - a1=3−7,a2=3,n=4\mathrm { a } _ { 1 } = 3 - \sqrt { 7 } , \mathrm { a } _ { 2 } = 3 , \mathrm { n } = 4a1​=3−7​,a2​=3,n=4

Use mathematical induction to prove that the statement is true for every positive integer n. -If 0<a<10 < a < 10<a<1 , then an<1a ^ { n } < 1an<1 . (Assume that a is a constant.)

Solve the problem. -Suppose there are 3 roads connecting town A to town B and 7 roads connecting town B to town C. In how many ways can a person travel from A to C via B?

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

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $2,6,10,14,18 , \ldots$

Evaluate the sum. - $\sum _ { i = 3 } ^ { 14 } ( i + 7 )$

Write the indicated term of the binomial expansion. - $( 3 x - 2 y ) ^ { 12 } ; \quad 11$ th term

Find a general term an for the geometric sequence. - $a _ { 1 } = \frac { 1 } { 5 } , r = - 5$

Write the indicated term of the binomial expansion. - $( 4 x + 7 ) ^ { 3 } ; \quad 3 r d$ term

Write the binomial expansion of the expression. - $\left( \frac { 1 } { x } - \sqrt { 13 } y \right) ^ { 3 }$

Evaluate the sum. - $\sum _ { \mathrm { i } = 1 } ^ { 8 } ( \mathrm { i } + 4 )$

Use mathematical induction to prove that the statement is true for every positive integer n. - $2 ^ { n } > n$

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $\mathrm { a } _ { 1 } = 3 - \sqrt { 7 } , \mathrm { a } _ { 2 } = 3 , \mathrm { n } = 4$

Use mathematical induction to prove that the statement is true for every positive integer n. -If $0 < a < 1$ , then $a ^ { n } < 1$ . (Assume that a is a constant.)