Question 1

Use the formula for $$\mathrm { S } _ { \mathbf { n } }$$ to find the sum of the first five terms of the geometric sequence.
-1, 3, 9, 27, . . .

Accepted Answer

A)  122 
B)  121 
C)  131 
D)  124 
A)  122 
B)  121 
C)  131 
D)  124

Question 2

Evaluate the sum. Round to two decimal places, if necessary.
-$$\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k \cdot - 3 k }$$

Accepted Answer

A)  -6 
B)  -30 
C)  81 
D)  30 
A)  -6 
B)  -30 
C)  81 
D)  30

Question 3

Find the common ratio r for the given infinite geometric sequence.
-A woman borrowed $10,000 at 6% interest compounded annually. If the loan was paid off in one lump sum at the end of 4 years, how much did she pay?

Accepted Answer

A)  $10,600.00 
B)  $12,624.77 
C)  $2624.77 
D)  $12,400.00 
A)  $10,600.00 
B)  $12,624.77 
C)  $2624.77 
D)  $12,400.00

Question 4

Evaluate the sum using the given information.
-$$\begin{array} { l } 
\mathrm { x } _ { 1 } = - 4 , \mathrm { x } _ { 2 } = 4 , \mathrm { x } _ { 3 } = - 1 , \mathrm { x } _ { 4 } = - 2 , \text { and } \Delta \mathrm { x } = 0.7 ; \mathrm { f } ( \mathrm { x } ) = \frac { \mathrm { x } } { \mathrm { x } - 5 } \
\sum _ { \mathrm { i } = 1 } ^ { 4 } \mathrm { f } \left( \mathrm { x } _ { \mathrm { i } } \right) \Delta \mathrm { x } \text { (Round to the nearest tenth, if necessary.) }
\end{array}$$

Accepted Answer

A)  -1 
B)  -2.4 
C)  -2.2 
D)  3.4 
A)  -1 
B)  -2.4 
C)  -2.2 
D)  3.4

Question 5

Provide an appropriate response.
-What is the common ratio in the geometric sequence $\left. a _ { n } = \log \left[ 6 ^ { \left( 5 ^ { n - 1 } \right.} \right) \right]$ ?

Accepted Answer

A)  $\log 6$ 
B)  6 
C)  $\log 5$ 
D)  5 
A)  $\log 6$ 
B)  6 
C)  $\log 5$ 
D)  5

Question 6

Solve the problem.
-How many ways can a committee of 4 be selected from a club with 12 members?

Accepted Answer

A)  495 
B)  248 
C)  11,880 
D)  24 
A)  495 
B)  248 
C)  11,880 
D)  24

Question 7

Evaluate the sum. Round to two decimal places, if necessary.
-$$\sum _ { \mathrm { k } = 2 } ^ { 5 } ( - 1 ) ^ { \mathrm { k } + 1 } ( \mathrm { k } + 3 ) ^ { 2 }$$

Accepted Answer

A)  26 
B)  174 
C)  -174 
D)  30,558 
A)  26 
B)  174 
C)  -174 
D)  30,558

Question 8

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

Accepted Answer

Answers will vary. One possible proof fo

Question 9

Solve the problem.
-Suppose that a family has 5 children and that the probability of having a girl is $\frac { 1 } { 2 }$. What is the probability of having at least three boys?

Accepted Answer

A)  $0.4688$ 
B)  $0.1563$ 
C)  $0.5000$ 
D)  $0.3125$ 
A)  $0.4688$ 
B)  $0.1563$ 
C)  $0.5000$ 
D)  $0.3125$

Question 10

Solve the problem.
-A signal is made by placing 3 flags, one above the other, on a flag pole. If there are 8 different flags available, how many possible signals can be flown?

Accepted Answer

A)  336 
B)  512 
C)  56 
D)  24 
A)  336 
B)  512 
C)  56 
D)  24

Question 11

Use mathematical induction to prove that the statement is true for every positive integer n.
-$$4 + 4 \cdot \frac { 1 } { 8 } + 4 \cdot \left( \frac { 1 } { 8 } \right) ^ { 2 } + \ldots + 4 \cdot \left( \frac { 1 } { 8 } \right) ^ { n - 1 } = \frac { 4 \left( 1 - \left( \frac { 1 } { 8 } \right) ^ { n } \right) } { 1 - \frac { 1 } { 8 } }$$

Accepted Answer

Answers will vary. One possible proof fo

Question 12

Find the nth term of the geometric sequence.
-$$a _ { 1 } = \frac { 1 } { 4 } , r = \frac { 1 } { 2 } , n = 6$$

Accepted Answer

A)  $a _ { 6 } = \frac { 1 } { 32 }$ 
B)  $a _ { 6 } = \frac { 1 } { 2048 }$ 
C)  $\mathrm { a } _ { 6 } = \frac { 1 } { 128 }$ 
D)  $\mathrm { a } _ { 6 } = \frac { 1 } { 256 }$ 
A)  $a _ { 6 } = \frac { 1 } { 32 }$ 
B)  $a _ { 6 } = \frac { 1 } { 2048 }$ 
C)  $\mathrm { a } _ { 6 } = \frac { 1 } { 128 }$ 
D)  $\mathrm { a } _ { 6 } = \frac { 1 } { 256 }$

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.
-During the first day of an epidemic 7000 people get sick, during the second day 14,000 people get sick, and so on with the number of people getting sick increasing by 7000 each day. The epidemic
Peaks after 30 days with 7000 fewer people getting sick on the 31st day than on the 30th day, and so
Forth. What is the total number of people who get sick?

Accepted Answer

A)  6,510,000 
B)  6,300,000 
C)  6,405,000 
D)  6,090,000 
A)  6,510,000 
B)  6,300,000 
C)  6,405,000 
D)  6,090,000

Question 14

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

Accepted Answer

A)  $15,11,7,3 , - 1$ 
B)  $11,7,3 , - 1 , - 5$ 
C)  $11,7,2 , - 1 , - 5$ 
D)  $7,3 , - 1 , - 5 , - 9$ 
A)  $15,11,7,3 , - 1$ 
B)  $11,7,3 , - 1 , - 5$ 
C)  $11,7,2 , - 1 , - 5$ 
D)  $7,3 , - 1 , - 5 , - 9$

Question 15

Solve the problem.
-License plates are made using 3 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

Accepted Answer

A)  175,760 
B)  11,881,376 
C)  1,757,600 
D)  100,000 
A)  175,760 
B)  11,881,376 
C)  1,757,600 
D)  100,000

Question 16

Evaluate the sum. Round to two decimal places, if necessary.
-$$\sum _ { k = 3 } ^ { 6 } \left( k ^ { 2 } - 5 \right)$$

Accepted Answer

A)  93 
B)  66 
C)  86 
D)  35 
A)  93 
B)  66 
C)  86 
D)  35

Question 17

Graph the function corresponding to the sequence defined. Use the graph to decide whether the sequence converges or
diverges.
-Suppose that an insect population density, in thousands, during year $n$ can be modeled by the recursively defined sequence: $a _ { 1 } = 1 ; a _ { n } = 2.67 a _ { n - 1 } - 0.17 a \underset { n - 1 } { 2 } \quad$ for $n > 1$.
Use technology to graph the sequence for $n = 1,2,3 , \ldots , 20$. Describe what happens to the population density function.

Accepted Answer

A)  The insect population stabilizes near $6.62$ thousand. 
B)  The insect population stabilizes near $9.82$ thousand. 
C)  The insect population increases every year. 
D)  The insect population stabilizes near $10.75$ thousand. 
A)  The insect population stabilizes near $6.62$ thousand. 
B)  The insect population stabilizes near $9.82$ thousand. 
C)  The insect population increases every year. 
D)  The insect population stabilizes near $10.75$ thousand.

Question 18

Evaluate the sum.
-$\sum _ { j = 5 } ^ { 12 } ( 5 j - 4 )$

Accepted Answer

A)  248 
B)  283 
C)  308 
D)  223 
A)  248 
B)  283 
C)  308 
D)  223

Question 19

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 length of the sides of the
new triangle.

Accepted Answer

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

Question 20

Find the common ratio r for the given infinite geometric sequence.
-$$\frac { 4 } { 3 } , \frac { 16 } { 3 } , \frac { 64 } { 3 } , \frac { 256 } { 3 } , \frac { 1024 } { 3 } , \ldots$$

Accepted Answer

A)  $\frac { 1 } { 4 }$ 
B)  10 
C)  12 
D)  4 
A)  $\frac { 1 } { 4 }$ 
B)  10 
C)  12 
D)  4

Use the formula for $\mathrm { S } _ { \mathbf { n } }$ to find the sum of the first five terms of the geometric sequence. -1, 3, 9, 27, . . .

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k \cdot - 3 k }$

Find the common ratio r for the given infinite geometric sequence. -A woman borrowed $10,000 at 6% interest compounded annually. If the loan was paid off in one lump sum at the end of 4 years, how much did she pay?

Evaluate the sum using the given information. - =-4,=4,=-1,=-2, and \Delta=0.7;()= \Delta (Round to the nearest tenth, if necessary.)

Provide an appropriate response. -What is the common ratio in the geometric sequence $\left. a _ { n } = \log \left[ 6 ^ { \left( 5 ^ { n - 1 } \right.} \right) \right]$ ?

Solve the problem. -How many ways can a committee of 4 be selected from a club with 12 members?

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { \mathrm { k } = 2 } ^ { 5 } ( - 1 ) ^ { \mathrm { k } + 1 } ( \mathrm { k } + 3 ) ^ { 2 }$

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

Solve the problem. -Suppose that a family has 5 children and that the probability of having a girl is $\frac { 1 } { 2 }$ . What is the probability of having at least three boys?

Solve the problem. -A signal is made by placing 3 flags, one above the other, on a flag pole. If there are 8 different flags available, how many possible signals can be flown?

Find the nth term of the geometric sequence. - $a _ { 1 } = \frac { 1 } { 4 } , r = \frac { 1 } { 2 } , n = 6$

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

Solve the problem. -License plates are made using 3 letters followed by 2 digits. How many plates can be made if repetition of letters and digits is allowed?

Evaluate the sum. Round to two decimal places, if necessary. - $\sum _ { k = 3 } ^ { 6 } \left( k ^ { 2 } - 5 \right)$

Evaluate the sum. - $\sum _ { j = 5 } ^ { 12 } ( 5 j - 4 )$

Find the common ratio r for the given infinite geometric sequence. - $\frac { 4 } { 3 } , \frac { 16 } { 3 } , \frac { 64 } { 3 } , \frac { 256 } { 3 } , \frac { 1024 } { 3 } , \ldots$

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

Use the formula for Sn\mathrm { S } _ { \mathbf { n } }Sn​ to find the sum of the first five terms of the geometric sequence. -1, 3, 9, 27, . . .

Evaluate the sum. Round to two decimal places, if necessary. - ∑k=14(−1)k⋅−3k\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k \cdot - 3 k }∑k=14​(−1)k⋅−3k