Question 1

Find the sum of the following infinite geometric series.
$$- 10 + 9 - \frac { 81 } { 10 } + \frac { 729 } { 100 } - \ldots$$

Accepted Answer

A)  $- \frac { 19 } { 100 }$ 
B)  $- \frac { 1 } { 100 }$ 
C)  $\frac { 19 } { 100 }$ 
D)  $- 10$ 
E)  $- \frac { 100 } { 19 }$ 
A)  $- \frac { 19 } { 100 }$ 
B)  $- \frac { 1 } { 100 }$ 
C)  $\frac { 19 } { 100 }$ 
D)  $- 10$ 
E)  $- \frac { 100 } { 19 }$

Question 2

Determine whether the sequence is geometric. If so, find the common ratio.

Accepted Answer

A)  $- 3$ 
B)  $- 2$ 
C)  $- \frac { 1 } { 3 }$ 
D)  3 
E)  not geometric 
A)  $- 3$ 
B)  $- 2$ 
C)  $- \frac { 1 } { 3 }$ 
D)  3 
E)  not geometric

Question 3

Use mathematical induction to prove the following inequality for all $n \geq 2$.
$$\frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 4 } } + \frac { 1 } { \sqrt { 6 } } + \ldots + \frac { 1 } { \sqrt { 2 n } } > \frac { \sqrt { n } } { \sqrt { 2 } }$$

Accepted Answer

i. Show true for @#LAT-DLM&
\[\begin{aligned}
\fra

Question 4

How many 3 -digit numbers can be formed if the leading digit cannot be zero and repeats are not allowed?

Accepted Answer

A)  997 
B)  504 
C)  810 
D)  648 
E)  900 
A)  997 
B)  504 
C)  810 
D)  648 
E)  900

Question 5

Use mathematical induction to prove the following inequality for all $n \geq 2$.
$$\frac { 1 } { \sqrt { 19 } } + \frac { 1 } { \sqrt { 38 } } + \frac { 1 } { \sqrt { 57 } } + \ldots + \frac { 1 } { \sqrt { 19 n } } > \frac { \sqrt { n } } { \sqrt { 19 } }$$

Accepted Answer

i. Show true for @#LAT-DLM&
\[\begin{aligned}
\fra

Question 6

Write an expression for the apparent $n$th term of the sequence. (Assume that $n$ begins with 1.)
$- 5 , - 2,1,4,7$

Accepted Answer

A)  $a _ { n } = - 8 n + 3$ 
B)  $a _ { n } = ( - 1 ) ^ { n } ( 3 n - 8 )$ 
C)  $a _ { n } = 3 n$ 
D)  $a _ { n } = 3 ^ { n } - 8$ 
E)  $a _ { n } = 3 n - 8$ 
A)  $a _ { n } = - 8 n + 3$ 
B)  $a _ { n } = ( - 1 ) ^ { n } ( 3 n - 8 )$ 
C)  $a _ { n } = 3 n$ 
D)  $a _ { n } = 3 ^ { n } - 8$ 
E)  $a _ { n } = 3 n - 8$

Question 7

Find a quadratic model for the sequence with the indicated terms.
$$a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12$$

Accepted Answer

A)  $a _ { n } = n ^ { 2 } + 7$ 
B)  $a _ { n } = n ^ { 2 } - 4 n + 7$ 
C)  $a _ { n } = n ^ { 2 } - 7 n + 7$ 
D)  $a _ { n } = n ^ { 2 } - 7 n + 4$ 
E)  $\quad a _ { n } = n ^ { 2 } - n + 7$ 
A)  $a _ { n } = n ^ { 2 } + 7$ 
B)  $a _ { n } = n ^ { 2 } - 4 n + 7$ 
C)  $a _ { n } = n ^ { 2 } - 7 n + 7$ 
D)  $a _ { n } = n ^ { 2 } - 7 n + 4$ 
E)  $\quad a _ { n } = n ^ { 2 } - n + 7$

Question 8

Find the indicated $n$th term of the geometric sequence. 4th term: $4 , - 12,36 , \ldots$

Accepted Answer

A)  $- 5$ 
B)  $- 108$ 
C)  324 
D)  $- 192$ 
E)  $- 768$ 
A)  $- 5$ 
B)  $- 108$ 
C)  324 
D)  $- 192$ 
E)  $- 768$

Question 9

W Write the $n$th term of the geometric sequence as a function of $n$.
$$a _ { 1 } = 4 , a _ { k + 1 } = 2 a _ { k }$$

Accepted Answer

A)  $a _ { n } = 4 ( 2 ) ^ { n - 1 }$ 
B)  $a _ { n } = 2 ( 4 ) ^ { n - 1 }$ 
C)  $a _ { n } = 4 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
D)  $a _ { n } = 2 + 2 n$ 
E)  $a _ { n } = 4 ( 2 ) ^ { n }$ 
A)  $a _ { n } = 4 ( 2 ) ^ { n - 1 }$ 
B)  $a _ { n } = 2 ( 4 ) ^ { n - 1 }$ 
C)  $a _ { n } = 4 \left( \frac { 1 } { 2 } \right) ^ { n - 1 }$ 
D)  $a _ { n } = 2 + 2 n$ 
E)  $a _ { n } = 4 ( 2 ) ^ { n }$

Question 10

Determine the sample space for the experiment.
Four coins are flipped and the number of heads observed is recorded.

Accepted Answer

A)  $S = \{ 1,2,3,4 \}$ 
B)  $S = \{ 4 \}$ 
C)  $S = \{ 0,1,2,3,4 \}$ 
D)  $S = \{ \mathrm { HT } , \mathrm { TH } \}$ 
E)  $S = \{ \mathrm { HHHH } \}$ 
A)  $S = \{ 1,2,3,4 \}$ 
B)  $S = \{ 4 \}$ 
C)  $S = \{ 0,1,2,3,4 \}$ 
D)  $S = \{ \mathrm { HT } , \mathrm { TH } \}$ 
E)  $S = \{ \mathrm { HHHH } \}$

Question 11

Write the first five terms of the geometric sequence. $$a _ { 1 } = 4 , r = - \frac { 1 } { 5 }$$

Accepted Answer

A)  $4 , - 1 , - 6 , - 11 , - 16$ 
B)  $4 , - 20,100 , - 500,2500$ 
C)  $- \frac { 4 } { 5 } , \frac { 4 } { 25 } , - \frac { 4 } { 125 } , \frac { 4 } { 625 } , - \frac { 4 } { 3125 }$ 
D)  $1,4 , - \frac { 4 } { 5 } , \frac { 4 } { 25 } , - \frac { 4 } { 125 }$ 
E)  $4 , - \frac { 4 } { 5 } , \frac { 4 } { 25 } , - \frac { 4 } { 125 } , \frac { 4 } { 625 }$ 
A)  $4 , - 1 , - 6 , - 11 , - 16$ 
B)  $4 , - 20,100 , - 500,2500$ 
C)  $- \frac { 4 } { 5 } , \frac { 4 } { 25 } , - \frac { 4 } { 125 } , \frac { 4 } { 625 } , - \frac { 4 } { 3125 }$ 
D)  $1,4 , - \frac { 4 } { 5 } , \frac { 4 } { 25 } , - \frac { 4 } { 125 }$ 
E)  $4 , - \frac { 4 } { 5 } , \frac { 4 } { 25 } , - \frac { 4 } { 125 } , \frac { 4 } { 625 }$

Question 12

Solve for $n$.
$$42 \cdot { } _ { n - 1 } P _ { 5 } = { } _ { n + 1 } P _ { 6 }$$

Accepted Answer

A)  $n = 35$ 
B)  $n = 7,34$ 
C)  $n = 6,35$ 
D)  $n = 6$ 
E)  no solution 
A)  $n = 35$ 
B)  $n = 7,34$ 
C)  $n = 6,35$ 
D)  $n = 6$ 
E)  no solution

Question 13

Determine the sample space for the experiment. Three marbles are selected from marbles labeled A through D where the marbles
Are not replaced and the order of selection does not matter.

Accepted Answer

A)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } \}$ 
B)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { ACD } , \mathrm { BCD } \}$ 
C)  $S = \{ \mathrm { ABC } , \mathrm { BCD } , \mathrm { CDA } \}$ 
D)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } , \mathrm { CBA } , \mathrm { CDA } \}$ 
E)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } , \mathrm { CDB } \}$ 
A)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } \}$ 
B)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { ACD } , \mathrm { BCD } \}$ 
C)  $S = \{ \mathrm { ABC } , \mathrm { BCD } , \mathrm { CDA } \}$ 
D)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } , \mathrm { CBA } , \mathrm { CDA } \}$ 
E)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } , \mathrm { CDB } \}$

Question 14

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that $n$ begins with 1.)
$$a _ { n } = - \left( \frac { 1 } { 6 } \right) ^ { n }$$

Accepted Answer

A)  $- 1$ 
B)  6 
C)  $\frac { 1 } { 6 }$ 
D)  1 
E)  not arithmetic 
A)  $- 1$ 
B)  6 
C)  $\frac { 1 } { 6 }$ 
D)  1 
E)  not arithmetic

Question 15

Find the sum of the following infinite geometric series. $$- 15 + 14 - \frac { 196 } { 15 } + \frac { 2744 } { 225 } - \ldots$$

Accepted Answer

A)  $- \frac { 29 } { 225 }$ 
B)  $- \frac { 1 } { 225 }$ 
C)  $\frac { 29 } { 225 }$ 
D)  $- 15$ 
E)  $- \frac { 225 } { 29 }$ 
A)  $- \frac { 29 } { 225 }$ 
B)  $- \frac { 1 } { 225 }$ 
C)  $\frac { 29 } { 225 }$ 
D)  $- 15$ 
E)  $- \frac { 225 } { 29 }$

Question 16

Find the number of distinguishable permutations of the group of letters. $\mathrm { E } , \mathrm { S } , \mathrm { T } , \mathrm { I } , \mathrm { M } , \mathrm { A } , \mathrm { T } , \mathrm { E }$

Accepted Answer

A)  10,080 
B)  8 
C)  20,160 
D)  40,320 
E)  3360 
A)  10,080 
B)  8 
C)  20,160 
D)  40,320 
E)  3360

Question 17

Use mathematical induction to prove the following inequality for all $n \geq 2$.
$$\frac { 1 } { \sqrt { 23 } } + \frac { 1 } { \sqrt { 46 } } + \frac { 1 } { \sqrt { 69 } } + \ldots + \frac { 1 } { \sqrt { 23 n } } > \frac { \sqrt { n } } { \sqrt { 23 } }$$

Accepted Answer

i. Show true for @#LAT-DLM&
\[\begin{aligned}
\fra

Question 18

Find the indicated term of the sequence.
$$\begin{array} { l } 
a _ { n } = ( - 1 ) ^ { n } ( 5 n - 1 ) \
a _ { 29 } = \square
\end{array}$$

Accepted Answer

A)  139 
B)  - 144 
C)  - 146 
D)  4 
E)  - 140 
A)  139 
B)  - 144 
C)  - 146 
D)  4 
E)  - 140

Question 19

Find $P _ { k + 1 }$ for the given $P _ { k }$.
$$P _ { k } = 7 + 13 + 19 + \ldots + [ 6 ( k - 1 ) + 1 ] + [ 6 k + 1 ]$$

Accepted Answer

A)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 k + 7 ] + [ 6 ( k + 1 ) + 1 ]$ 
B)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 k + 1 ] + [ 6 ( k + 1 ) + 7 ]$ 
C)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 k + 7 ] + [ 6 ( k + 1 ) + 7 ]$ 
D)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 k + 1 ] + [ 6 ( k + 1 ) + 1 ]$ 
E)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 ( k - 1 ) + 1 ] + [ 6 k + 1 ]$ 
A)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 k + 7 ] + [ 6 ( k + 1 ) + 1 ]$ 
B)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 k + 1 ] + [ 6 ( k + 1 ) + 7 ]$ 
C)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 k + 7 ] + [ 6 ( k + 1 ) + 7 ]$ 
D)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 k + 1 ] + [ 6 ( k + 1 ) + 1 ]$ 
E)  $P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 ( k - 1 ) + 1 ] + [ 6 k + 1 ]$

Question 20

Use mathematical induction to prove the following equality.
$$\ln \left( 2 ^ { n } x _ { 1 } x _ { 2 } \ldots x _ { n } \right) = \ln \left( 2 x _ { 1 } \right) + \ln \left( 2 x _ { 2 } \right) + \ldots + \ln \left( 2 x _ { n } \right) \text {, where } x _ { 1 } > 0 , x _ { 2 } > 0 , \ldots , x _ { n } > 0$$

Accepted Answer

i. Show true for @#LAT-DLM&
\[\begin{array} { l }

Find the sum of the following infinite geometric series. $- 10 + 9 - \frac { 81 } { 10 } + \frac { 729 } { 100 } - \ldots$

Determine whether the sequence is geometric. If so, find the common ratio.

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $\frac { 1 } { \sqrt { 2 } } + \frac { 1 } { \sqrt { 4 } } + \frac { 1 } { \sqrt { 6 } } + \ldots + \frac { 1 } { \sqrt { 2 n } } > \frac { \sqrt { n } } { \sqrt { 2 } }$

How many 3 -digit numbers can be formed if the leading digit cannot be zero and repeats are not allowed?

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $\frac { 1 } { \sqrt { 19 } } + \frac { 1 } { \sqrt { 38 } } + \frac { 1 } { \sqrt { 57 } } + \ldots + \frac { 1 } { \sqrt { 19 n } } > \frac { \sqrt { n } } { \sqrt { 19 } }$

Write an expression for the apparent $n$ th term of the sequence. (Assume that $n$ begins with 1.) $- 5 , - 2,1,4,7$

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12$

Find the indicated $n$ th term of the geometric sequence. 4th term: $4 , - 12,36 , \ldots$

W Write the $n$ th term of the geometric sequence as a function of $n$ . $a _ { 1 } = 4 , a _ { k + 1 } = 2 a _ { k }$

Determine the sample space for the experiment. Four coins are flipped and the number of heads observed is recorded.

Write the first five terms of the geometric sequence. $a _ { 1 } = 4 , r = - \frac { 1 } { 5 }$

Solve for $n$ . $42 \cdot { } _ { n - 1 } P _ { 5 } = { } _ { n + 1 } P _ { 6 }$

Determine the sample space for the experiment. Three marbles are selected from marbles labeled A through D where the marbles Are not replaced and the order of selection does not matter.

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that $n$ begins with 1.) $a _ { n } = - \left( \frac { 1 } { 6 } \right) ^ { n }$

Find the sum of the following infinite geometric series. $- 15 + 14 - \frac { 196 } { 15 } + \frac { 2744 } { 225 } - \ldots$

Find the number of distinguishable permutations of the group of letters. $\mathrm { E } , \mathrm { S } , \mathrm { T } , \mathrm { I } , \mathrm { M } , \mathrm { A } , \mathrm { T } , \mathrm { E }$

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $\frac { 1 } { \sqrt { 23 } } + \frac { 1 } { \sqrt { 46 } } + \frac { 1 } { \sqrt { 69 } } + \ldots + \frac { 1 } { \sqrt { 23 n } } > \frac { \sqrt { n } } { \sqrt { 23 } }$

Find the indicated term of the sequence. =(-1(5n-1) =\square

Find $P _ { k + 1 }$ for the given $P _ { k }$ . $P _ { k } = 7 + 13 + 19 + \ldots + [ 6 ( k - 1 ) + 1 ] + [ 6 k + 1 ]$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Exam 8: Sequences, Series, and Probability

Find the sum of the following infinite geometric series. −10+9−8110+729100−…- 10 + 9 - \frac { 81 } { 10 } + \frac { 729 } { 100 } - \ldots−10+9−1081​+100729​−…

Determine whether the sequence is geometric. If so, find the common ratio.

How many 3 -digit numbers can be formed if the leading digit cannot be zero and repeats are not allowed?

Write an expression for the apparent nnn th term of the sequence. (Assume that nnn begins with 1.) −5,−2,1,4,7- 5 , - 2,1,4,7−5,−2,1,4,7

Find a quadratic model for the sequence with the indicated terms. a0=7,a2=3,a5=12a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12a0​=7,a2​=3,a5​=12

Find the indicated nnn th term of the geometric sequence. 4th term: 4,−12,36,…4 , - 12,36 , \ldots4,−12,36,…

W Write the nnn th term of the geometric sequence as a function of nnn . a1=4,ak+1=2aka _ { 1 } = 4 , a _ { k + 1 } = 2 a _ { k }a1​=4,ak+1​=2ak​

Determine the sample space for the experiment. Four coins are flipped and the number of heads observed is recorded.

Write the first five terms of the geometric sequence. a1=4,r=−15a _ { 1 } = 4 , r = - \frac { 1 } { 5 }a1​=4,r=−51​

Solve for nnn . 42⋅n−1P5=n+1P642 \cdot { } _ { n - 1 } P _ { 5 } = { } _ { n + 1 } P _ { 6 }42⋅n−1​P5​=n+1​P6​

Determine the sample space for the experiment. Three marbles are selected from marbles labeled A through D where the marbles Are not replaced and the order of selection does not matter.

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that nnn begins with 1.) an=−(16)na _ { n } = - \left( \frac { 1 } { 6 } \right) ^ { n }an​=−(61​)n

Find the sum of the following infinite geometric series. −15+14−19615+2744225−…- 15 + 14 - \frac { 196 } { 15 } + \frac { 2744 } { 225 } - \ldots−15+14−15196​+2252744​−…

Find the number of distinguishable permutations of the group of letters. E,S,T,I,M,A,T,E\mathrm { E } , \mathrm { S } , \mathrm { T } , \mathrm { I } , \mathrm { M } , \mathrm { A } , \mathrm { T } , \mathrm { E }E,S,T,I,M,A,T,E

Find the indicated term of the sequence. =(-1(5n-1) =\square

Find Pk+1P _ { k + 1 }Pk+1​ for the given PkP _ { k }Pk​ . Pk=7+13+19+…+[6(k−1)+1]+[6k+1]P _ { k } = 7 + 13 + 19 + \ldots + [ 6 ( k - 1 ) + 1 ] + [ 6 k + 1 ]Pk​=7+13+19+…+[6(k−1)+1]+[6k+1]

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

Topics in Analytic Geometry

Analytic Geometry in Three Dimensions

Limits and an Introduction to Calculus

Review of Graphs, Equations, and Inequalities

Filters

Find the sum of the following infinite geometric series. $- 10 + 9 - \frac { 81 } { 10 } + \frac { 729 } { 100 } - \ldots$

Write an expression for the apparent $n$ th term of the sequence. (Assume that $n$ begins with 1.) $- 5 , - 2,1,4,7$

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12$

Find the indicated $n$ th term of the geometric sequence. 4th term: $4 , - 12,36 , \ldots$

W Write the $n$ th term of the geometric sequence as a function of $n$ . $a _ { 1 } = 4 , a _ { k + 1 } = 2 a _ { k }$

Write the first five terms of the geometric sequence. $a _ { 1 } = 4 , r = - \frac { 1 } { 5 }$

Solve for $n$ . $42 \cdot { } _ { n - 1 } P _ { 5 } = { } _ { n + 1 } P _ { 6 }$

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that $n$ begins with 1.) $a _ { n } = - \left( \frac { 1 } { 6 } \right) ^ { n }$

Find the sum of the following infinite geometric series. $- 15 + 14 - \frac { 196 } { 15 } + \frac { 2744 } { 225 } - \ldots$

Find the number of distinguishable permutations of the group of letters. $\mathrm { E } , \mathrm { S } , \mathrm { T } , \mathrm { I } , \mathrm { M } , \mathrm { A } , \mathrm { T } , \mathrm { E }$

Find $P _ { k + 1 }$ for the given $P _ { k }$ . $P _ { k } = 7 + 13 + 19 + \ldots + [ 6 ( k - 1 ) + 1 ] + [ 6 k + 1 ]$