Question 1

Write out the first five terms of the sequence.
-$$a _ { n } = \frac { 3 } { n ^ { 2 } }$$

Accepted Answer

A)  $1 , \frac { 2 } { 4 } , \frac { 3 } { 9 } , \frac { 4 } { 16 } , \frac { 5 } { 25 }$ 
B)  $\frac { 1 } { 4 } , \frac { 2 } { 9 } , \frac { 3 } { 16 } , \frac { 4 } { 25 } , \frac { 5 } { 36 }$ 
C)  $1 , \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 }$ 
D)  $3 , \frac { 3 } { 4 } , \frac { 1 } { 3 } , \frac { 3 } { 16 } , \frac { 3 } { 25 }$ 
A)  $1 , \frac { 2 } { 4 } , \frac { 3 } { 9 } , \frac { 4 } { 16 } , \frac { 5 } { 25 }$ 
B)  $\frac { 1 } { 4 } , \frac { 2 } { 9 } , \frac { 3 } { 16 } , \frac { 4 } { 25 } , \frac { 5 } { 36 }$ 
C)  $1 , \frac { 1 } { 2 } , \frac { 1 } { 3 } , \frac { 1 } { 4 } , \frac { 1 } { 5 }$ 
D)  $3 , \frac { 3 } { 4 } , \frac { 1 } { 3 } , \frac { 3 } { 16 } , \frac { 3 } { 25 }$

Question 2

Solve the problem.
-Each of ten tickets is marked with a different number from 1 to 10 and put in a box. If you draw a ticket from the box, what is the probability that you will draw 1,6 , or 4 ?

Accepted Answer

A)  $\frac { 1 } { 10 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  1 
A)  $\frac { 1 } { 10 }$ 
B)  $\frac { 3 } { 10 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  1

Question 3

Find the sum of the first n terms of the following arithmetic sequence.
-Find the sum of the first 52 positive multiples of 7.

Accepted Answer

A)  $\frac { 19663 } { 2 }$ 
B)  9646 
C)  9464 
D)  9282 
A)  $\frac { 19663 } { 2 }$ 
B)  9646 
C)  9464 
D)  9282

Question 4

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$3 ^ { n } > 3 ^ { n - 1 }$$

Accepted Answer

Answers will vary. One possible proof fo

Question 5

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth.
-$$a _ { 2 } = 20.8 , a _ { 9 } = 130.52$$

Accepted Answer

A)  $\mathrm { a } _ { 1 } = 20.8 , \mathrm { r } = 1.3$ 
B)  $\mathrm { a } _ { 1 } = 16 , \mathrm { r } = 0.77$ 
C)  $\mathrm { a } _ { 1 } = 130.52 , \mathrm { r } = 0.77$ 
D)  $a _ { 1 } = 16 , r = 1.3$ 
A)  $\mathrm { a } _ { 1 } = 20.8 , \mathrm { r } = 1.3$ 
B)  $\mathrm { a } _ { 1 } = 16 , \mathrm { r } = 0.77$ 
C)  $\mathrm { a } _ { 1 } = 130.52 , \mathrm { r } = 0.77$ 
D)  $a _ { 1 } = 16 , r = 1.3$

Question 6

Use the formula for $$\mathrm { S } _ { \mathbf { n } }$$ to find the sum of the first five terms of the geometric sequence.
-$4 , - 16,64 , - 256 , \ldots$

Accepted Answer

A)  $- 820$ 
B)  $- 1364$ 
C)  1364 
D)  820 
A)  $- 820$ 
B)  $- 1364$ 
C)  1364 
D)  820

Question 7

Evaluate the sum. Round to two decimal places, if necessary.
-$$\sum _ { i = 3 } ^ { 6 } ( 3 i - 4 )$$

Accepted Answer

A)  42 
B)  38 
C)  33 
D)  23 
A)  42 
B)  38 
C)  33 
D)  23

Question 8

Find the probability.
-A bag contains 13 balls numbered 1 through 13 . What is the probability of selecting a ball that has an even number when one ball is drawn from the bag?

Accepted Answer

A)  $\frac { 13 } { 6 }$ 
B)  6 
C)  $\frac { 6 } { 13 }$ 
D)  $\frac { 2 } { 13 }$ 
A)  $\frac { 13 } { 6 }$ 
B)  6 
C)  $\frac { 6 } { 13 }$ 
D)  $\frac { 2 } { 13 }$

Question 9

Provide an appropriate response.
-Let $f ( x ) = 8 \left( 6 ^ { x } \right)$, and consider the sequence $f ( 1 ) , f ( 2 ) , f ( 3 ) , \ldots$ Is this a geometric sequence, and if so, what is the common ratio?

Accepted Answer

A)  Yes; $x$ 
B)  Yes; 8 
C)  No 
D)  Yes; 6 
A)  Yes; $x$ 
B)  Yes; 8 
C)  No 
D)  Yes; 6

Question 10

Write out the first five terms of the sequence.
-$$a _ { n } = n - 4$$

Accepted Answer

A)  $- 1,0,1,2,3$ 
B)  $- 4 , - 3 , - 2 , - 1,0$ 
C)  $- 16 , - 12 , - 8 , - 4,0$ 
D)  $- 3 , - 2 , - 1,0,1$ 
A)  $- 1,0,1,2,3$ 
B)  $- 4 , - 3 , - 2 , - 1,0$ 
C)  $- 16 , - 12 , - 8 , - 4,0$ 
D)  $- 3 , - 2 , - 1,0,1$

Question 11

Write the binomial expansion of the expression.
-$$\left( - 2 + \frac { y } { 3 } \right) ^ { 5 }$$

Accepted Answer

A)  $- 32 + \frac { 80 } { 3 } y - \frac { 80 } { 9 } y ^ { 2 } + \frac { 40 } { 27 } y ^ { 3 } - \frac { 10 } { 81 } y ^ { 4 } + \frac { 1 } { 243 } y ^ { 5 }$ 
B)  $- 32 - \frac { 10000 } { 3 } y - \frac { 8000 } { 9 } y ^ { 2 } - \frac { 400 } { 27 } y ^ { 3 } - \frac { 10 } { 81 } y ^ { 4 } - \frac { 1 } { 243 } y ^ { 5 }$ 
C)  $- 32 - \frac { 80 } { 3 } y + \frac { 80 } { 9 } y ^ { 2 } - \frac { 40 } { 27 } y ^ { 3 } + \frac { 10 } { 81 } y ^ { 4 } - \frac { 1 } { 243 } y ^ { 5 }$ 
D)  $32 + \frac { 80 } { 3 } y + \frac { 80 } { 9 } y ^ { 2 } + \frac { 40 } { 27 } y ^ { 3 } + \frac { 10 } { 81 } y ^ { 4 } + \frac { 1 } { 243 } y ^ { 5 }$ 
A)  $- 32 + \frac { 80 } { 3 } y - \frac { 80 } { 9 } y ^ { 2 } + \frac { 40 } { 27 } y ^ { 3 } - \frac { 10 } { 81 } y ^ { 4 } + \frac { 1 } { 243 } y ^ { 5 }$ 
B)  $- 32 - \frac { 10000 } { 3 } y - \frac { 8000 } { 9 } y ^ { 2 } - \frac { 400 } { 27 } y ^ { 3 } - \frac { 10 } { 81 } y ^ { 4 } - \frac { 1 } { 243 } y ^ { 5 }$ 
C)  $- 32 - \frac { 80 } { 3 } y + \frac { 80 } { 9 } y ^ { 2 } - \frac { 40 } { 27 } y ^ { 3 } + \frac { 10 } { 81 } y ^ { 4 } - \frac { 1 } { 243 } y ^ { 5 }$ 
D)  $32 + \frac { 80 } { 3 } y + \frac { 80 } { 9 } y ^ { 2 } + \frac { 40 } { 27 } y ^ { 3 } + \frac { 10 } { 81 } y ^ { 4 } + \frac { 1 } { 243 } y ^ { 5 }$

Question 12

Find the probability.
-A 6-sided die is rolled. What is the probability of rolling a number less than $6 ?$

Accepted Answer

A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 5 } { 7 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  $\frac { 5 } { 6 }$ 
A)  $\frac { 1 } { 3 }$ 
B)  $\frac { 5 } { 7 }$ 
C)  $\frac { 1 } { 6 }$ 
D)  $\frac { 5 } { 6 }$

Question 13

Write the binomial expansion of the expression.
-$( 3 x - 2 ) ^ { 5 }$

Accepted Answer

A)  $243 x ^ { 5 } - 162 x ^ { 4 } + 108 x ^ { 3 } - 72 x ^ { 2 } + 48 x - 32$ 
B)  $243 x ^ { 5 } - 810 x ^ { 4 } + 1080 x ^ { 3 } - 720 x ^ { 2 } + 240 x - 32$ 
C)  $243 x ^ { 5 } + 240 x ^ { 4 } - 720 x ^ { 3 } - 720 x ^ { 2 } + 240 x - 32$ 
D)  $\left( 9 x ^ { 2 } - 12 x + 4 \right) ^ { 5 }$ 
A)  $243 x ^ { 5 } - 162 x ^ { 4 } + 108 x ^ { 3 } - 72 x ^ { 2 } + 48 x - 32$ 
B)  $243 x ^ { 5 } - 810 x ^ { 4 } + 1080 x ^ { 3 } - 720 x ^ { 2 } + 240 x - 32$ 
C)  $243 x ^ { 5 } + 240 x ^ { 4 } - 720 x ^ { 3 } - 720 x ^ { 2 } + 240 x - 32$ 
D)  $\left( 9 x ^ { 2 } - 12 x + 4 \right) ^ { 5 }$

Question 14

Solve the problem.
-How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O?

Accepted Answer

A)  16,900 
B)  33,800 
C)  456,976 
D)  35,152 
A)  16,900 
B)  33,800 
C)  456,976 
D)  35,152

Question 15

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or
diverges.
-$$\mathrm { a } _ { \mathrm { n } } = \frac { 1 } { 36 } \mathrm { e } ^ { \mathrm { n } }$$

Accepted Answer

A)  Diverges 
B)  Converges 
A)  Diverges 
B)  Converges

Question 16

Use mathematical induction to prove that the statement is true for every positive integer n.
-$0.12 ^ { n } < 1$

Accepted Answer

Answers will vary. One possible proof fo

Question 17

Evaluate the sum. Round to two decimal places, if necessary.
-$$\sum _ { \mathrm { k } = 1 } ^ { 4 } ( - 1 ) ^ { \mathrm { k } } ( \mathrm { k } + 4 )$$

Accepted Answer

A)  -26 
B)  2 
C)  26 
D)  18 
A)  -26 
B)  2 
C)  26 
D)  18

Question 18

Find a general term an for the geometric sequence.
-$$a _ { 1 } = \frac { 1 } { 5 } , r = 9$$

Accepted Answer

A)  $a _ { n } = \frac { 1 } { 5 } \cdot 9 ^{n - 1}$ 
B)  $a _ { n } = \frac { 1 } { 5 } + \frac { 8 } { 5 } ( n - 1 )$ 
C)  $a _ { n } = \left( \frac { 1 } { 5 } \right) n - 1 + \frac { 8 } { 5 }$ 
D)  $a _ { n } = \frac { 1 } { 5 } + 9 ( n - 1 )$ 
A)  $a _ { n } = \frac { 1 } { 5 } \cdot 9 ^{n - 1}$ 
B)  $a _ { n } = \frac { 1 } { 5 } + \frac { 8 } { 5 } ( n - 1 )$ 
C)  $a _ { n } = \left( \frac { 1 } { 5 } \right) n - 1 + \frac { 8 } { 5 }$ 
D)  $a _ { n } = \frac { 1 } { 5 } + 9 ( n - 1 )$

Question 19

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$\left( 2 ^ { 2 } \right) ^ { n } = 2 ^ { 2 n }$$

Accepted Answer

Answers will vary. One possible proof fo

Question 20

Solve the problem.
-If it has been determined that the probability of an earthquake occurring on a certain day in a certain area is $0.04$, what are the odds against an earthquake?

Accepted Answer

A)  1 to 25 
B)  25 to 1 
C)  23 to 1 
D)  24 to 1 
A)  1 to 25 
B)  25 to 1 
C)  23 to 1 
D)  24 to 1

Write out the first five terms of the sequence. - $a _ { n } = \frac { 3 } { n ^ { 2 } }$

Solve the problem. -Each of ten tickets is marked with a different number from 1 to 10 and put in a box. If you draw a ticket from the box, what is the probability that you will draw 1,6 , or 4 ?

Find the sum of the first n terms of the following arithmetic sequence. -Find the sum of the first 52 positive multiples of 7.

Use mathematical induction to prove that the statement is true for every positive integer n. - $3 ^ { n } > 3 ^ { n - 1 }$

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - $a _ { 2 } = 20.8 , a _ { 9 } = 130.52$

Use the formula for $\mathrm { S } _ { \mathbf { n } }$ to find the sum of the first five terms of the geometric sequence. - $4 , - 16,64 , - 256 , \ldots$

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { i = 3 } ^ { 6 } ( 3 i - 4 )$

Find the probability. -A bag contains 13 balls numbered 1 through 13 . What is the probability of selecting a ball that has an even number when one ball is drawn from the bag?

Provide an appropriate response. -Let $f ( x ) = 8 \left( 6 ^ { x } \right)$ , and consider the sequence $f ( 1 ) , f ( 2 ) , f ( 3 ) , \ldots$ Is this a geometric sequence, and if so, what is the common ratio?

Write out the first five terms of the sequence. - $a _ { n } = n - 4$

Write the binomial expansion of the expression. - $\left( - 2 + \frac { y } { 3 } \right) ^ { 5 }$

Find the probability. -A 6-sided die is rolled. What is the probability of rolling a number less than $6 ?$

Write the binomial expansion of the expression. - $( 3 x - 2 ) ^ { 5 }$

Solve the problem. -How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O?

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $\mathrm { a } _ { \mathrm { n } } = \frac { 1 } { 36 } \mathrm { e } ^ { \mathrm { n } }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $0.12 ^ { n } < 1$

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { \mathrm { k } = 1 } ^ { 4 } ( - 1 ) ^ { \mathrm { k } } ( \mathrm { k } + 4 )$

Find a general term an for the geometric sequence. - $a _ { 1 } = \frac { 1 } { 5 } , r = 9$

Use mathematical induction to prove that the statement is true for every positive integer n. - $\left( 2 ^ { 2 } \right) ^ { n } = 2 ^ { 2 n }$

Solve the problem. -If it has been determined that the probability of an earthquake occurring on a certain day in a certain area is $0.04$ , what are the odds against an earthquake?

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

Write out the first five terms of the sequence. - an=3n2a _ { n } = \frac { 3 } { n ^ { 2 } }an​=n23​

Solve the problem. -Each of ten tickets is marked with a different number from 1 to 10 and put in a box. If you draw a ticket from the box, what is the probability that you will draw 1,6 , or 4 ?

Find the sum of the first n terms of the following arithmetic sequence. -Find the sum of the first 52 positive multiples of 7.

Use mathematical induction to prove that the statement is true for every positive integer n. - 3n>3n−13 ^ { n } > 3 ^ { n - 1 }3n>3n−1

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - a2=20.8,a9=130.52a _ { 2 } = 20.8 , a _ { 9 } = 130.52a2​=20.8,a9​=130.52

Use the formula for Sn\mathrm { S } _ { \mathbf { n } }Sn​ to find the sum of the first five terms of the geometric sequence. - 4,−16,64,−256,…4 , - 16,64 , - 256 , \ldots4,−16,64,−256,…

Evaluate the sum. Round to two decimal places, if necessary. - ∑i=36(3i−4)\sum _ { i = 3 } ^ { 6 } ( 3 i - 4 )∑i=36​(3i−4)

Find the probability. -A bag contains 13 balls numbered 1 through 13 . What is the probability of selecting a ball that has an even number when one ball is drawn from the bag?

Provide an appropriate response. -Let f(x)=8(6x)f ( x ) = 8 \left( 6 ^ { x } \right)f(x)=8(6x) , and consider the sequence f(1),f(2),f(3),…f ( 1 ) , f ( 2 ) , f ( 3 ) , \ldotsf(1),f(2),f(3),… Is this a geometric sequence, and if so, what is the common ratio?

Write out the first five terms of the sequence. - an=n−4a _ { n } = n - 4an​=n−4

Write the binomial expansion of the expression. - (−2+y3)5\left( - 2 + \frac { y } { 3 } \right) ^ { 5 }(−2+3y​)5

Find the probability. -A 6-sided die is rolled. What is the probability of rolling a number less than 6?6 ?6?

Write the binomial expansion of the expression. - (3x−2)5( 3 x - 2 ) ^ { 5 }(3x−2)5

Solve the problem. -How many different 4-letter radio-station call letters can be made if the first letter must be K or W, repeats are allowed, but the call letters cannot end in an O?

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - an=136en\mathrm { a } _ { \mathrm { n } } = \frac { 1 } { 36 } \mathrm { e } ^ { \mathrm { n } }an​=361​en

Use mathematical induction to prove that the statement is true for every positive integer n. - 0.12n<10.12 ^ { n } < 10.12n<1

Evaluate the sum. Round to two decimal places, if necessary. - ∑k=14(−1)k(k+4)\sum _ { \mathrm { k } = 1 } ^ { 4 } ( - 1 ) ^ { \mathrm { k } } ( \mathrm { k } + 4 )∑k=14​(−1)k(k+4)

Find a general term an for the geometric sequence. - a1=15,r=9a _ { 1 } = \frac { 1 } { 5 } , r = 9a1​=51​,r=9

Use mathematical induction to prove that the statement is true for every positive integer n. - (22)n=22n\left( 2 ^ { 2 } \right) ^ { n } = 2 ^ { 2 n }(22)n=22n

Solve the problem. -If it has been determined that the probability of an earthquake occurring on a certain day in a certain area is 0.040.040.04 , what are the odds against an earthquake?

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 out the first five terms of the sequence. - $a _ { n } = \frac { 3 } { n ^ { 2 } }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $3 ^ { n } > 3 ^ { n - 1 }$

Find the first term and the common ratio for the geometric sequence. Round approximations to the nearest hundredth. - $a _ { 2 } = 20.8 , a _ { 9 } = 130.52$

Use the formula for $\mathrm { S } _ { \mathbf { n } }$ to find the sum of the first five terms of the geometric sequence. - $4 , - 16,64 , - 256 , \ldots$

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { i = 3 } ^ { 6 } ( 3 i - 4 )$

Provide an appropriate response. -Let $f ( x ) = 8 \left( 6 ^ { x } \right)$ , and consider the sequence $f ( 1 ) , f ( 2 ) , f ( 3 ) , \ldots$ Is this a geometric sequence, and if so, what is the common ratio?

Write out the first five terms of the sequence. - $a _ { n } = n - 4$

Write the binomial expansion of the expression. - $\left( - 2 + \frac { y } { 3 } \right) ^ { 5 }$

Find the probability. -A 6-sided die is rolled. What is the probability of rolling a number less than $6 ?$

Write the binomial expansion of the expression. - $( 3 x - 2 ) ^ { 5 }$

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or diverges. - $\mathrm { a } _ { \mathrm { n } } = \frac { 1 } { 36 } \mathrm { e } ^ { \mathrm { n } }$

Use mathematical induction to prove that the statement is true for every positive integer n. - $0.12 ^ { n } < 1$

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { \mathrm { k } = 1 } ^ { 4 } ( - 1 ) ^ { \mathrm { k } } ( \mathrm { k } + 4 )$

Find a general term an for the geometric sequence. - $a _ { 1 } = \frac { 1 } { 5 } , r = 9$

Use mathematical induction to prove that the statement is true for every positive integer n. - $\left( 2 ^ { 2 } \right) ^ { n } = 2 ^ { 2 n }$

Solve the problem. -If it has been determined that the probability of an earthquake occurring on a certain day in a certain area is $0.04$ , what are the odds against an earthquake?