Question 1

Write out the sum. Do not evaluate.
-$$\sum _ { k = 1 } ^ { 5 } \frac { k + 1 } { k + 2 }$$

Accepted Answer

A)  $\frac { 1 } { 3 } + \frac { 2 } { 4 } + \frac { 3 } { 5 } + \frac { 4 } { 6 }$ 
B)  $\frac { 6 } { 7 }$ 
C)  $\frac { 2 } { 3 } + \frac { 3 } { 4 } + \frac { 4 } { 5 } + \frac { 5 } { 6 } + \frac { 6 } { 7 }$ 
D)  $\frac { 1 } { 2 } + \frac { 2 } { 3 } + \frac { 3 } { 4 } + \frac { 4 } { 5 } + \frac { 5 } { 6 }$ 
A)  $\frac { 1 } { 3 } + \frac { 2 } { 4 } + \frac { 3 } { 5 } + \frac { 4 } { 6 }$ 
B)  $\frac { 6 } { 7 }$ 
C)  $\frac { 2 } { 3 } + \frac { 3 } { 4 } + \frac { 4 } { 5 } + \frac { 5 } { 6 } + \frac { 6 } { 7 }$ 
D)  $\frac { 1 } { 2 } + \frac { 2 } { 3 } + \frac { 3 } { 4 } + \frac { 4 } { 5 } + \frac { 5 } { 6 }$

Question 2

Find the indicated term using the given information.
-$\mathrm { a } _ { 18 } = 87 , \mathrm { a } _ { 20 } = 99 ; \mathrm { a } _ { 6 }$

Accepted Answer

A)  $- 15$ 
B)  21 
C)  15 
D)  6 
A)  $- 15$ 
B)  21 
C)  15 
D)  6

Question 3

Determine whether the sequence is arithmetic.
-$$- 1,1,3,5,7 , \ldots$$

Accepted Answer

A)  Arithmetic 
B)  Not arithmetic 
A)  Arithmetic 
B)  Not arithmetic

Question 4

Choose the one alternative that best completes the statement or answers the question.
Evaluate the expression.
-$\left( \begin{array} { c } 10 \ 5 \end{array} \right)$

Accepted Answer

A)  252 
B)  30,240 
C)  126 
D)  504 
A)  252 
B)  30,240 
C)  126 
D)  504

Question 5

Find the sum of the sequence.
-$$\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k } \cdot - 8 k$$

Accepted Answer

A)  $- 16$ 
B)  $- 80$ 
C)  80 
D)  4,096 
A)  $- 16$ 
B)  $- 80$ 
C)  80 
D)  4,096

Question 6

Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$3 + 8 + 13 + \ldots + ( 5 n - 2 ) = \frac { n } { 2 } ( 5 n + 1 )$

Accepted Answer

First we show that the statement is true

Question 7

Write out the first five terms of the sequence.
-$$\{ 4 n - 2 \}$$

Accepted Answer

A)  $6,10,14,18,22$ 
B)  $2,3,4,5,6$ 
C)  $- 2 , - 6 , - 10 , - 14 , - 18$ 
D)  $2,6,10,14,18$ 
A)  $6,10,14,18,22$ 
B)  $2,3,4,5,6$ 
C)  $- 2 , - 6 , - 10 , - 14 , - 18$ 
D)  $2,6,10,14,18$

Question 8

Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$7 + 14 + 21 + \ldots + 7 n = \frac { 7 n ( n + 1 ) } { 2 }$$

Accepted Answer

First, we show the statement is true whe

Question 9

An arithmetic sequence is given. Find the common difference and write out the first four terms.
-$\{ 3 n + 2 \}$

Accepted Answer

A)  $\mathrm { d } = 2 ; 3,5,8,11$ 
B)  $d = 3 ; 5,8,11,14$ 
C)  $\mathrm { d } = 3 ; 3,5,8,11$ 
D)  $\mathrm { d } = 2 ; 5,8,11,14$ 
A)  $\mathrm { d } = 2 ; 3,5,8,11$ 
B)  $d = 3 ; 5,8,11,14$ 
C)  $\mathrm { d } = 3 ; 3,5,8,11$ 
D)  $\mathrm { d } = 2 ; 5,8,11,14$

Question 10

Choose the one alternative that best completes the statement or answers the question.
-$$a _ { 1 } = z ; a _ { n } = a _ { n - 1 } + U$$

Accepted Answer

A)  $U , U + z , U + 2 z , U + 3 z$ 
B)  $z , z + U , z + 2 U , z + 3 U$ 
C)  $z , U , 2 U , 3 U$ 
D)  $z , z - U , z - 2 U , z - 3 U$ 
A)  $U , U + z , U + 2 z , U + 3 z$ 
B)  $z , z + U , z + 2 U , z + 3 U$ 
C)  $z , U , 2 U , 3 U$ 
D)  $z , z - U , z - 2 U , z - 3 U$

Question 11

Find the sum of the arithmetic sequence.
-$$( - 6 ) + ( - 1 ) + 4 + 9 + \ldots + 39$$

Accepted Answer

A)  165 
B)  330 
C)  170 
D)  160 
A)  165 
B)  330 
C)  170 
D)  160

Question 12

Evaluate the factorial expression.
-$$\frac { ( n + 9 ) ! } { n + 9 }$$

Accepted Answer

A)  $n + 9 !$ 
B)  1 
C)  $( n + 8 ) !$ 
D)  $9 !$ 
A)  $n + 9 !$ 
B)  1 
C)  $( n + 8 ) !$ 
D)  $9 !$

Question 13

The sequence is defined recursively. Write the first four terms.
-$a _ { 1 } = 2 , a _ { 2 } = 5$ and $a _ { n } = a _ { n - 2 } - 3 a _ { n } - 1$ for $n \geq 3$

Accepted Answer

A)  $2,5 , - 1 , - 16$ 
B)  $2,5 , - 13,44$ 
C)  $2,5,1,2$ 
D)  $2,5,17 , - 46$ 
A)  $2,5 , - 1 , - 16$ 
B)  $2,5 , - 13,44$ 
C)  $2,5,1,2$ 
D)  $2,5,17 , - 46$

Question 14

Express the repeating decimal as a fraction in lowest terms.
-$$0 . \overline { 4 } = \frac { 4 } { 10 } + \frac { 4 } { 100 } + \frac { 4 } { 1,000 } + \frac { 4 } { 10,000 } \ldots$$

Accepted Answer

A)  $\frac { 111 } { 250 }$ 
B)  $\frac { 4 } { 99 }$ 
C)  $\frac { 11 } { 25 }$ 
D)  $\frac { 4 } { 9 }$ 
A)  $\frac { 111 } { 250 }$ 
B)  $\frac { 4 } { 99 }$ 
C)  $\frac { 11 } { 25 }$ 
D)  $\frac { 4 } { 9 }$

Question 15

Write out the first five terms of the sequence.
-$$\{ 4 ( 4 n - 2 ) \}$$

Accepted Answer

A)  $- 8,8,24,40,56$ 
B)  $2,6,10,14,18$ 
C)  $8,16,24,32,40$ 
D)  $8,24,40,56,72$ 
A)  $- 8,8,24,40,56$ 
B)  $2,6,10,14,18$ 
C)  $8,16,24,32,40$ 
D)  $8,24,40,56,72$

Question 16

Determine whether the given sequence is arithmetic, geometric, or neither. If arithmetic, find the common difference. If
geometric, find the common ratio.
-$\{ 4 n - 2 \}$

Accepted Answer

A)  Arithmetic, $d = - 2$ 
B)  Geometric, $r = 4$ 
C)  Neither 
D)  Arithmetic, $\mathrm { d } = 4$ 
A)  Arithmetic, $d = - 2$ 
B)  Geometric, $r = 4$ 
C)  Neither 
D)  Arithmetic, $\mathrm { d } = 4$

Question 17

Write the word or phrase that best completes each statement or answers the question.
Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n.
-$$\left( 1 - \frac { 1 } { 2 } \right) \left( 1 - \frac { 1 } { 3 } \right) \ldots \left( 1 - \frac { 1 } { n + 1 } \right) = \frac { 1 } { n + 1 }$$

Accepted Answer

First we show that the statement is true

Question 18

Find the sum of the arithmetic sequence.
-2 + 4 + 6 + ... + 1,836

Accepted Answer

A)  841,806 
B)  842,724 
C)  844,561 
D)  843,642 
A)  841,806 
B)  842,724 
C)  844,561 
D)  843,642

Question 19

Write the word or phrase that best completes each statement or answers the question.
-Initially, a pendulum swings through an arc of 3 feet. On each successive swing, the length of the arc is 0.8 of the
previous length. After 10 swings, what total length will the pendulum have swung (to the nearest tenth of a
foot)?

Accepted Answer

approximat

Question 20

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary.
-$$4 + 12 + 36 + 108 + 324 + \ldots + 4 \cdot 310$$

Accepted Answer

A)  354,292 
B)  354,294 
C)  354,329 
D)  354,272 
A)  354,292 
B)  354,294 
C)  354,329 
D)  354,272

Write out the sum. Do not evaluate. - $\sum _ { k = 1 } ^ { 5 } \frac { k + 1 } { k + 2 }$

Find the indicated term using the given information. - $\mathrm { a } _ { 18 } = 87 , \mathrm { a } _ { 20 } = 99 ; \mathrm { a } _ { 6 }$

Determine whether the sequence is arithmetic. - $- 1,1,3,5,7 , \ldots$

Choose the one alternative that best completes the statement or answers the question. Evaluate the expression. - $\left( \begin{array} { c } 10 \\ 5 \end{array} \right)$

Find the sum of the sequence. - $\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k } \cdot - 8 k$

Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $3 + 8 + 13 + \ldots + ( 5 n - 2 ) = \frac { n } { 2 } ( 5 n + 1 )$

Write out the first five terms of the sequence. - $\{ 4 n - 2 \}$

Write the word or phrase that best completes each statement or answers the question. Use the Principle of Mathematical Induction to show that the statement is true for all natural numbers n. - $7 + 14 + 21 + \ldots + 7 n = \frac { 7 n ( n + 1 ) } { 2 }$

An arithmetic sequence is given. Find the common difference and write out the first four terms. - $\{ 3 n + 2 \}$

Choose the one alternative that best completes the statement or answers the question. - $a _ { 1 } = z ; a _ { n } = a _ { n - 1 } + U$

Find the sum of the arithmetic sequence. - $( - 6 ) + ( - 1 ) + 4 + 9 + \ldots + 39$

Evaluate the factorial expression. - $\frac { ( n + 9 ) ! } { n + 9 }$

The sequence is defined recursively. Write the first four terms. - $a _ { 1 } = 2 , a _ { 2 } = 5$ and $a _ { n } = a _ { n - 2 } - 3 a _ { n } - 1$ for $n \geq 3$

Express the repeating decimal as a fraction in lowest terms. - $0 . \overline { 4 } = \frac { 4 } { 10 } + \frac { 4 } { 100 } + \frac { 4 } { 1,000 } + \frac { 4 } { 10,000 } \ldots$

Write out the first five terms of the sequence. - $\{ 4 ( 4 n - 2 ) \}$

Determine whether the given sequence is arithmetic, geometric, or neither. If arithmetic, find the common difference. If geometric, find the common ratio. - $\{ 4 n - 2 \}$

Find the sum of the arithmetic sequence. -2 + 4 + 6 + ... + 1,836

Use a graphing utility to find the sum of the geometric sequence. Round answer to two decimal places, if necessary. - $4 + 12 + 36 + 108 + 324 + \ldots + 4 \cdot 310$

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Polar Coordinates; Vectors

Analytic Geometry

Systems of Equations and Inequalities

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 11: Sequences; Induction; the Binomial Theorem

Write out the sum. Do not evaluate. - ∑k=15k+1k+2\sum _ { k = 1 } ^ { 5 } \frac { k + 1 } { k + 2 }∑k=15​k+2k+1​

Find the indicated term using the given information. - a18=87,a20=99;a6\mathrm { a } _ { 18 } = 87 , \mathrm { a } _ { 20 } = 99 ; \mathrm { a } _ { 6 }a18​=87,a20​=99;a6​

Determine whether the sequence is arithmetic. - −1,1,3,5,7,…- 1,1,3,5,7 , \ldots−1,1,3,5,7,…

Choose the one alternative that best completes the statement or answers the question. Evaluate the expression. - (105)\left( \begin{array} { c } 10 \\ 5 \end{array} \right)(105​)

Find the sum of the sequence. - ∑k=14(−1)k⋅−8k\sum _ { k = 1 } ^ { 4 } ( - 1 ) ^ { k } \cdot - 8 k∑k=14​(−1)k⋅−8k

Write out the first five terms of the sequence. - {4n−2}\{ 4 n - 2 \}{4n−2}

An arithmetic sequence is given. Find the common difference and write out the first four terms. - {3n+2}\{ 3 n + 2 \}{3n+2}

Choose the one alternative that best completes the statement or answers the question. - a1=z;an=an−1+Ua _ { 1 } = z ; a _ { n } = a _ { n - 1 } + Ua1​=z;an​=an−1​+U

Find the sum of the arithmetic sequence. - (−6)+(−1)+4+9+…+39( - 6 ) + ( - 1 ) + 4 + 9 + \ldots + 39(−6)+(−1)+4+9+…+39

Evaluate the factorial expression. - (n+9)!n+9\frac { ( n + 9 ) ! } { n + 9 }n+9(n+9)!​

The sequence is defined recursively. Write the first four terms. - a1=2,a2=5a _ { 1 } = 2 , a _ { 2 } = 5a1​=2,a2​=5 and an=an−2−3an−1a _ { n } = a _ { n - 2 } - 3 a _ { n } - 1an​=an−2​−3an​−1 for n≥3n \geq 3n≥3

Express the repeating decimal as a fraction in lowest terms. - 0.4‾=410+4100+41,000+410,000…0 . \overline { 4 } = \frac { 4 } { 10 } + \frac { 4 } { 100 } + \frac { 4 } { 1,000 } + \frac { 4 } { 10,000 } \ldots0.4=104​+1004​+1,0004​+10,0004​…

Write out the first five terms of the sequence. - {4(4n−2)}\{ 4 ( 4 n - 2 ) \}{4(4n−2)}

Determine whether the given sequence is arithmetic, geometric, or neither. If arithmetic, find the common difference. If geometric, find the common ratio. - {4n−2}\{ 4 n - 2 \}{4n−2}