Question 1

Decide whether the given sequence is finite or infinite.
-$$a _ { 1 } = 6 ; \text { for } n \geq 2 , a _ { n } = 9 \cdot a _ { n - 1 }$$

Accepted Answer

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

Question 2

Use a calculator to evaluate the expression.
-$${ } _ { 21 } \mathrm { C } _ { 4 }$$

Accepted Answer

A)  5985 
B)  23,940 
C)  35,910 
D)  143,640 
A)  5985 
B)  23,940 
C)  35,910 
D)  143,640

Question 3

Solve the problem.
-How many different three-digit numbers can be written using digits from the set { $\{ 5,6,7,8,9 \}$ without any repeating digits?

Accepted Answer

A)  20 
B)  120 
C)  10 
D)  60 
A)  20 
B)  120 
C)  10 
D)  60

Question 4

Find a general term an for the geometric sequence.
-$$\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 3$$

Accepted Answer

A)  $a _ { n } = 3 ( 3 ) ^ { n - 1 }$ 
B)  $a _ { n } = 3 ( 3 ) ( n - 1 )$ 
C)  $a _ { n } = 3 ^{n - 1} + 2$ 
D)  $a _ { n } = 3 + 6 ( n - 1 )$ 
A)  $a _ { n } = 3 ( 3 ) ^ { n - 1 }$ 
B)  $a _ { n } = 3 ( 3 ) ( n - 1 )$ 
C)  $a _ { n } = 3 ^{n - 1} + 2$ 
D)  $a _ { n } = 3 + 6 ( n - 1 )$

Question 5

Find the nth term of the geometric sequence.
-$$a _ { 1 } = 2 , r = 6 , n = 3$$

Accepted Answer

A)  $a _ { 3 } = \frac { 1 } { 18 }$ 
B)  $a _ { 3 } = 144$ 
C)  $a _ { 3 } = 36$ 
D)  $a _ { 3 } = 72$ 
A)  $a _ { 3 } = \frac { 1 } { 18 }$ 
B)  $a _ { 3 } = 144$ 
C)  $a _ { 3 } = 36$ 
D)  $a _ { 3 } = 72$

Question 6

Write the indicated term of the binomial expansion.
-$( 2 x + 5 ) ^ { 3 } ; \quad$ 3rd term

Accepted Answer

A)  $750 x$ 
B)  375 
C)  $150 x$ 
D)  $60 x ^ { 2 }$ 
A)  $750 x$ 
B)  375 
C)  $150 x$ 
D)  $60 x ^ { 2 }$

Question 7

Decide whether the given sequence is finite or infinite.
--5, -4, -3, -2

Accepted Answer

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

Question 8

Evaluate the sum using the given information.
-$x _ { 1 } = 0 , x _ { 2 } = 4$, and $x _ { 3 } = - 4$
$$\sum _ { i = 1 } ^ { 3 } \left( \frac { x _ { i } - 2 } { 2 x _ { i } + 4 } \right)$$

Accepted Answer

A)  $\frac { 7 } { 6 }$ 
B)  $- \frac { 11 } { 6 }$ 
C)  0 
D)  $\frac { 5 } { 6 }$ 
A)  $\frac { 7 } { 6 }$ 
B)  $- \frac { 11 } { 6 }$ 
C)  0 
D)  $\frac { 5 } { 6 }$

Question 9

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

Accepted Answer

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

Question 10

Solve the problem.
-How many different three-number "combinations" are possible on a combination lock having 32 numbers on its dial without repeating a number?

Accepted Answer

A)  57,536 
B)  14,880 
C)  29,760 
D)  28,768 
A)  57,536 
B)  14,880 
C)  29,760 
D)  28,768

Question 11

Find all natural number values for n for which the given statement is false.
-$$2 ^ { n } > 2 n + 2$$

Accepted Answer

A)  $n = 1,2$, or 3 
B)  $n = 1$ or 2 
C)  $\mathrm { n } = 1,2$, or 4 
D)  $n > 3$ 
A)  $n = 1,2$, or 3 
B)  $n = 1$ or 2 
C)  $\mathrm { n } = 1,2$, or 4 
D)  $n > 3$

Question 12

Find a general term an for the geometric sequence.
-$$\mathrm { a } _ { 1 } = \frac { 1 } { 3 } , \mathrm { r } = 6$$

Accepted Answer

A)  $a _ { n } = \frac { 1 } { 3 } + \frac { 1 } { 6 } ( n - 1 )$ 
B)  $a _ { n } = \left( \frac { 1 } { 3 } \right) ^ { n - 1 } - \frac { 5 } { 18 }$ 
C)  $a _ { n } = \frac { 1 } { 3 } \cdot 6 ^ { n - 1 }$ 
D)  $a _ { n } = \frac { 1 } { 3 } - \frac { 5 } { 18 } ( n - 1 )$ 
A)  $a _ { n } = \frac { 1 } { 3 } + \frac { 1 } { 6 } ( n - 1 )$ 
B)  $a _ { n } = \left( \frac { 1 } { 3 } \right) ^ { n - 1 } - \frac { 5 } { 18 }$ 
C)  $a _ { n } = \frac { 1 } { 3 } \cdot 6 ^ { n - 1 }$ 
D)  $a _ { n } = \frac { 1 } { 3 } - \frac { 5 } { 18 } ( n - 1 )$

Question 13

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
-3, 9, 15, 21, 27, . . .

Accepted Answer

A)  $a _ { n } = 3 n - 6 ; a _ { 6 } = 12$ 
B)  $a _ { n } = 6 n - 1 ; a _ { 6 } = 35$ 
C)  $a _ { n } = 3 ( 2 n - 1 ) ; a _ { 6 } = 33$ 
D)  $a _ { n } = 3 ( 6 ) ^ { n - 1 } ; a _ { 6 } = 18$ 
A)  $a _ { n } = 3 n - 6 ; a _ { 6 } = 12$ 
B)  $a _ { n } = 6 n - 1 ; a _ { 6 } = 35$ 
C)  $a _ { n } = 3 ( 2 n - 1 ) ; a _ { 6 } = 33$ 
D)  $a _ { n } = 3 ( 6 ) ^ { n - 1 } ; a _ { 6 } = 18$

Question 14

Provide an appropriate response.
-Consider the arrangements of sixteen students in a line. Is this a combination, a permutation, or neither?

Accepted Answer

A)  Neither 
B)  Combination 
C)  Permutation 
A)  Neither 
B)  Combination 
C)  Permutation

Question 15

Solve the problem.
-A bag contains 5 apples and 3 oranges. If you select 4 pieces of fruit without looking, how many ways can you get exactly 3 apples?

Accepted Answer

A)  20 
B)  60 
C)  180 
D)  30 
A)  20 
B)  60 
C)  180 
D)  30

Question 16

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

Accepted Answer

Answers will vary. One possible proof fo

Question 17

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.
-$( 1.03 ) ^ { - 3 }$

Accepted Answer

A)  $1.090$ 
B)  $1.093$ 
C)  $0.918$ 
D)  $0.915$ 
A)  $1.090$ 
B)  $1.093$ 
C)  $0.918$ 
D)  $0.915$

Question 18

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
-The first term is $- 1 + \sqrt { 15 }$, and the common difference is $3 ; n = 3$

Accepted Answer

A)  $- 1 + \sqrt { 15 } , 3 + \sqrt { 15 } , 6 + \sqrt { 15 }$ 
B)  $- 1 - \sqrt { 15 } , 2 + \sqrt { 15 } , 5 + \sqrt { 15 }$ 
C)  $1 + \sqrt { 15 } , 2 + \sqrt { 15 } , 5 + \sqrt { 15 }$ 
D)  $- 1 + \sqrt { 15 } , 2 + \sqrt { 15 } , 5 + \sqrt { 15 }$ 
A)  $- 1 + \sqrt { 15 } , 3 + \sqrt { 15 } , 6 + \sqrt { 15 }$ 
B)  $- 1 - \sqrt { 15 } , 2 + \sqrt { 15 } , 5 + \sqrt { 15 }$ 
C)  $1 + \sqrt { 15 } , 2 + \sqrt { 15 } , 5 + \sqrt { 15 }$ 
D)  $- 1 + \sqrt { 15 } , 2 + \sqrt { 15 } , 5 + \sqrt { 15 }$

Question 19

Find the common ratio r for the given infinite geometric sequence.
-A man borrowed $3000 at 1.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

Accepted Answer

A)  $3270.00 
B)  $3280.33 
C)  $3045.00 
D)  $280.33 
A)  $3270.00 
B)  $3280.33 
C)  $3045.00 
D)  $280.33

Question 20

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

Accepted Answer

A)  $a _ { n } = 4 - n$ 
B)  $a _ { n } = n + 2$ 
C)  $a _ { n } = n - 4$ 
D)  $a _ { n } = n + 3$ 
A)  $a _ { n } = 4 - n$ 
B)  $a _ { n } = n + 2$ 
C)  $a _ { n } = n - 4$ 
D)  $a _ { n } = n + 3$

Decide whether the given sequence is finite or infinite. - $a _ { 1 } = 6 ; \text { for } n \geq 2 , a _ { n } = 9 \cdot a _ { n - 1 }$

Use a calculator to evaluate the expression. - ${ } _ { 21 } \mathrm { C } _ { 4 }$

Solve the problem. -How many different three-digit numbers can be written using digits from the set { $\{ 5,6,7,8,9 \}$ without any repeating digits?

Find a general term an for the geometric sequence. - $\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 3$

Find the nth term of the geometric sequence. - $a _ { 1 } = 2 , r = 6 , n = 3$

Write the indicated term of the binomial expansion. - $( 2 x + 5 ) ^ { 3 } ; \quad$ 3rd term

Decide whether the given sequence is finite or infinite. --5, -4, -3, -2

Evaluate the sum using the given information. - $x _ { 1 } = 0 , x _ { 2 } = 4$ , and $x _ { 3 } = - 4$ $\sum _ { i = 1 } ^ { 3 } \left( \frac { x _ { i } - 2 } { 2 x _ { i } + 4 } \right)$

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

Solve the problem. -How many different three-number "combinations" are possible on a combination lock having 32 numbers on its dial without repeating a number?

Find all natural number values for n for which the given statement is false. - $2 ^ { n } > 2 n + 2$

Find a general term an for the geometric sequence. - $\mathrm { a } _ { 1 } = \frac { 1 } { 3 } , \mathrm { r } = 6$

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). -3, 9, 15, 21, 27, . . .

Provide an appropriate response. -Consider the arrangements of sixteen students in a line. Is this a combination, a permutation, or neither?

Solve the problem. -A bag contains 5 apples and 3 oranges. If you select 4 pieces of fruit without looking, how many ways can you get exactly 3 apples?

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

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). -The first term is $- 1 + \sqrt { 15 }$ , and the common difference is $3 ; n = 3$

Find the common ratio r for the given infinite geometric sequence. -A man borrowed $3000 at 1.5% interest compounded annually. If he paid off the loan in full at the end of 6 years, how much did he pay?

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

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

Decide whether the given sequence is finite or infinite. - a1=6; for n≥2,an=9⋅an−1a _ { 1 } = 6 ; \text { for } n \geq 2 , a _ { n } = 9 \cdot a _ { n - 1 }a1​=6; for n≥2,an​=9⋅an−1​

Use a calculator to evaluate the expression. - 21C4{ } _ { 21 } \mathrm { C } _ { 4 }21​C4​

Solve the problem. -How many different three-digit numbers can be written using digits from the set { {5,6,7,8,9}\{ 5,6,7,8,9 \}{5,6,7,8,9} without any repeating digits?

Find a general term an for the geometric sequence. - a1=3,r=3\mathrm { a } _ { 1 } = 3 , \mathrm { r } = 3a1​=3,r=3

Find the nth term of the geometric sequence. - a1=2,r=6,n=3a _ { 1 } = 2 , r = 6 , n = 3a1​=2,r=6,n=3

Write the indicated term of the binomial expansion. - (2x+5)3;( 2 x + 5 ) ^ { 3 } ; \quad(2x+5)3; 3rd term