Question 1

Solve the problem.
-What are the odds in favor of drawing a 3 from these cards?

Accepted Answer

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

Question 2

Solve the problem.
-What are the odds in favor of spinning a D on this spinner?

Accepted Answer

A)  5 to 1 
B)  1 to 6 
C)  1 to 7 
D)  6 to 1 
A)  5 to 1 
B)  1 to 6 
C)  1 to 7 
D)  6 to 1

Question 3

Find the probability.
-A lottery game contains 22 balls numbered 1 through 22 . What is the probability of choosing a ball numbered 23?

Accepted Answer

A)  1 
B)  22 
C)  0 
D)  $\frac { 1 } { 22 }$ 
A)  1 
B)  22 
C)  0 
D)  $\frac { 1 } { 22 }$

Question 4

Find all natural number values for n for which the given statement is false.
-$n ! > 10 \mathrm { n }$

Accepted Answer

A)  $n = 1,2,3,4$, or 5 
B)  $n = 1,2$ or 3 
C)  $n = 1,2,3$, or 4 
D)  $n > 5$ 
A)  $n = 1,2,3,4$, or 5 
B)  $n = 1,2$ or 3 
C)  $n = 1,2,3$, or 4 
D)  $n > 5$

Question 5

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$3 \cdot 5 + 4 \cdot 6 + 5 \cdot 7 + \ldots + ( n + 3 ) ( n + 5 ) = \frac { n \left( 2 n ^ { 2 } + 27 n + 115 \right) } { 6 }$$

Accepted Answer

Answers will vary. One possible proof fo

Question 6

Find the common ratio r for the given infinite geometric sequence.
-A small start-up consulting company hopes to have 7 times the clientele it had the previous year for the next 2 years. If the company just meets its targets each of the 2 years and it had 7 clients to
Begin with, how many clients does it have at the end of 2 years?

Accepted Answer

A)  55 clients 
B)  56 clients 
C)  26 clients 
D)  84 clients 
A)  55 clients 
B)  56 clients 
C)  26 clients 
D)  84 clients

Question 7

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.
-The beginning population of a town was 36,000. If the population decreased by 250 people per year, how many people lived in the town 15 years later?

Accepted Answer

A)  3750 
B)  32,500 
C)  32,250 
D)  32,000 
A)  3750 
B)  32,500 
C)  32,250 
D)  32,000

Question 8

Write out the first five terms of the sequence.
-$$a _ { n } = \frac { n ^ { 2 } - 6 } { n ^ { 2 } + 6 }$$

Accepted Answer

A)  $- \frac { 5 } { 7 } , - \frac { 1 } { 5 } , \frac { 1 } { 5 } , \frac { 5 } { 11 } , \frac { 19 } { 31 }$ 
B)  1, 1, 1, 1, 1 
C)  $- \frac { 5 } { 7 } , 1 , \frac { 1 } { 5 } , 1 , \frac { 31 } { 19 }$ 
D)  $1 , - \frac { 1 } { 5 } , 1 , \frac { 5 } { 11 } , 1$ 
A)  $- \frac { 5 } { 7 } , - \frac { 1 } { 5 } , \frac { 1 } { 5 } , \frac { 5 } { 11 } , \frac { 19 } { 31 }$ 
B)  1, 1, 1, 1, 1 
C)  $- \frac { 5 } { 7 } , 1 , \frac { 1 } { 5 } , 1 , \frac { 31 } { 19 }$ 
D)  $1 , - \frac { 1 } { 5 } , 1 , \frac { 5 } { 11 } , 1$

Question 9

Solve the problem.
-What are the odds in favor of spinning an A on this spinner?

Accepted Answer

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

Question 10

Solve the problem.
-How many 4-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, if repetitions of digits are allowed?

Accepted Answer

A)  256 
B)  10,000 
C)  9999 
D)  24 
A)  256 
B)  10,000 
C)  9999 
D)  24

Question 11

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth.
-$$\sum _ { \mathrm { k } = 5 } ^ { 10 } ( 0.53 ) \mathrm { k }$$

Accepted Answer

A)  $0.085$ 
B)  $0.043$ 
C)  $0.087$ 
D)  $0.045$ 
A)  $0.085$ 
B)  $0.043$ 
C)  $0.087$ 
D)  $0.045$

Question 12

Evaluate the sum.
-$\sum _ { i = 1 } ^ { 1500 } i$

Accepted Answer

A)  1,127,251 
B)  1,126,500.5 
C)  1,124,250 
D)  1,125,750 
A)  1,127,251 
B)  1,126,500.5 
C)  1,124,250 
D)  1,125,750

Question 13

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question).
-$$a _ { 1 } = 6 , d = 1 , n = 6$$

Accepted Answer

A)  0, 6, 7, 8, 9, 10 
B)  6, 7, 8, 9, 10, 11 
C)  6, 6, 6, 6, 6, 6 
D)  6, 7, 8, 9, 10, 12 
A)  0, 6, 7, 8, 9, 10 
B)  6, 7, 8, 9, 10, 11 
C)  6, 6, 6, 6, 6, 6 
D)  6, 7, 8, 9, 10, 12

Question 14

Find a general term an for the geometric sequence.
-$$a _ { 1 } = - \frac { 1 } { 7 } , r = - 6$$

Accepted Answer

A)  $a _ { n } = - \frac { 1 } { 7 } \cdot ( - 6 ) ^ { n - 1 }$ 
B)  $a _ { n } = \left( - \frac { 1 } { 7 } \right) ^ { n - 1 } + \frac { 1 } { 6 }$ 
C)  $a _ { n } = - \frac { 1 } { 7 } - \frac { 1 } { 6 } ( n - 1 )$ 
D)  $a _ { n } = - \frac { 1 } { 7 } + \frac { 1 } { 6 } ( n - 1 )$ 
A)  $a _ { n } = - \frac { 1 } { 7 } \cdot ( - 6 ) ^ { n - 1 }$ 
B)  $a _ { n } = \left( - \frac { 1 } { 7 } \right) ^ { n - 1 } + \frac { 1 } { 6 }$ 
C)  $a _ { n } = - \frac { 1 } { 7 } - \frac { 1 } { 6 } ( n - 1 )$ 
D)  $a _ { n } = - \frac { 1 } { 7 } + \frac { 1 } { 6 } ( n - 1 )$

Question 15

Evaluate the sum using the given information.
-$x _ { 1 } = 4 , x _ { 2 } = 0 , x _ { 3 } = 3 , x _ { 4 } = - 4$, and $x _ { 5 } = 2$
$$\sum _ { i = 1 } ^ { 5 } \left( - 2 x _ { i } + 3 \right)$$

Accepted Answer

A)  -7 
B)  5 
C)  -15 
D)  -13 
A)  -7 
B)  5 
C)  -15 
D)  -13

Question 16

Use mathematical induction to prove that the statement is true for every positive integer n.
-The series of sketches below starts with an equilateral triangle having sides of length 1 
(one). In the following steps, equilateral triangles are constructed by joining the midpoints
of the sides of the preceding triangle. Develop a formula for the area of the nth new
triangle. Use math induction to prove your answer.

Accepted Answer

The answer of Use mathematical induction to prove that the...

Question 17

Solve the problem.
-

Accepted Answer

A)  $\frac { 5 } { 3 }$ 
B)  1 
C)  3 
D)  $\frac { 2 } { 5 }$ 
A)  $\frac { 5 } { 3 }$ 
B)  1 
C)  3 
D)  $\frac { 2 } { 5 }$

Question 18

Evaluate the sum. Round to two decimal places, if necessary.
-$$\sum _ { i = 2 } ^ { 5 } \frac { - 5 } { i }$$

Accepted Answer

A)  -4.69 
B)  1.42 
C)  -6.42 
D)  172.5 
A)  -4.69 
B)  1.42 
C)  -6.42 
D)  172.5

Question 19

Use the summation properties to evaluate the series. The following rules may be needed: $$\sum _ { i = 1 } ^ { n } i = \frac { n ( n + 1 ) } { 2 } ; \quad \sum _ { i = 1 } ^ { n } i ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } ; \quad \sum _ { i = 1 } ^ { n } i ^ { 3 } = \frac { n ^ { 2 } ( n + 1 ) ^ { 2 } } { 4 } .$$
-$$\sum _ { i = 1 } ^ { 5 } \left( - 4 i ^ { 2 } - i ^ { 3 } \right)$$

Accepted Answer

A)  - 5340 
B)  - 445 
C)  - 1780 
D)  5 
A)  - 5340 
B)  - 445 
C)  - 1780 
D)  5

Question 20

Evaluate the expression.
-$$\left( \begin{array} { l } 
4 \
1
\end{array} \right)$$

Accepted Answer

A)  12 
B)  24 
C)  4 
D)  2 
A)  12 
B)  24 
C)  4 
D)  2

Solve the problem. -What are the odds in favor of drawing a 3 from these cards?

Solve the problem. -What are the odds in favor of spinning a D on this spinner?

Find the probability. -A lottery game contains 22 balls numbered 1 through 22 . What is the probability of choosing a ball numbered 23?

Find all natural number values for n for which the given statement is false. - $n ! > 10 \mathrm { n }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $3 \cdot 5 + 4 \cdot 6 + 5 \cdot 7 + \ldots + ( n + 3 ) ( n + 5 ) = \frac { n \left( 2 n ^ { 2 } + 27 n + 115 \right) } { 6 }$

Write out the first five terms of the sequence. - $a _ { n } = \frac { n ^ { 2 } - 6 } { n ^ { 2 } + 6 }$

Solve the problem. -What are the odds in favor of spinning an A on this spinner?

Solve the problem. -How many 4-digit numbers can be formed using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, if repetitions of digits are allowed?

Use a graphing calculator to evaluate the sum. Round to the nearest thousandth. - $\sum _ { \mathrm { k } = 5 } ^ { 10 } ( 0.53 ) \mathrm { k }$

Evaluate the sum. - $\sum _ { i = 1 } ^ { 1500 } i$

Write the first n terms of the given arithmetic sequence (the value of n is indicated in the question). - $a _ { 1 } = 6 , d = 1 , n = 6$

Find a general term an for the geometric sequence. - $a _ { 1 } = - \frac { 1 } { 7 } , r = - 6$

Evaluate the sum using the given information. - $x _ { 1 } = 4 , x _ { 2 } = 0 , x _ { 3 } = 3 , x _ { 4 } = - 4$ , and $x _ { 5 } = 2$ $\sum _ { i = 1 } ^ { 5 } \left( - 2 x _ { i } + 3 \right)$

Solve the problem. -

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { i = 2 } ^ { 5 } \frac { - 5 } { i }$

Evaluate the expression. - $\left( \begin{array} { l } 4 \\1\end{array} \right)$

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