Question 1

Evaluate using a graphing utility: ${ } _ { 20 } P _ { 5 }$

Accepted Answer

A)  15,504 
B)  116,280 
C)  $1,860,480$ 
D)  100 
E)  undefined 
A)  15,504 
B)  116,280 
C)  $1,860,480$ 
D)  100 
E)  undefined C

Question 2

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that $n$ begins with 1.)
$$a _ { n } = - 6 + 4 n$$

Accepted Answer

A)  $- 4$ 
B)  6 
C)  $- 6$ 
D)  4 
E)  not arithmetic 
A)  $- 4$ 
B)  6 
C)  $- 6$ 
D)  4 
E)  not arithmetic D

Question 3

Find the sum using the formulas for the sums of powers of integers.
$\sum _ { n = 1 } ^ { 12 } n ^ { 3 }$
A) 4356
B) 12,168
C) 1728
D) 650
E) 6084 
Use mathematical induction to prove the property for all positive integers $n$.
$\left[ a ^ { n } \right] ^ { 4 } = a ^ { 4 n }$

Accepted Answer

E 1) $n = 1$ :
$\begin{array} { r } 
{ \left[ a ^ { 1 } \right] ^ { 4 } \stackrel { ? } { = } a ^ { 4 \cdot 1 } } \
{ [ a ] ^ { 4 } \stackrel { ? } { = } a ^ { 4 } } \
a ^ { 4 } = a ^ { 4 }
\end{array}$
The statement is true for $n = 1$.
2) Assume $\left[ a ^ { k } \right] ^ { 4 } = a ^ { 4 k }$. Then,
$\begin{array} { c } 
( a ) ^ { 4 } \left[ a ^ { k } \right] ^ { 4 } = ( a ) ^ { 4 } \left( a ^ { 4 k } \right) \
{ \left[ a \cdot a ^ { k } \right] ^ { 4 } = a ^ { 4 + 4 k } } \
{ \left[ a ^ { k + 1 } \right] ^ { 4 } = a ^ { 4 ( k + 1 ) } }
\end{array}$
By mathematical induction, the property is true for all positive values of $n$.

Question 4

Find the sum of the finite geometric sequence. Round to the nearest thousandth.
$$\sum _ { i = 0 } ^ { 6 } 200 ( 1.07 ) ^ { i }$$

Accepted Answer

A)  $1730.804$ 
B)  $1530.804$ 
C)  $1153.308$ 
D)  $1430.658$ 
E)  $1851.961$ 
A)  $1730.804$ 
B)  $1530.804$ 
C)  $1153.308$ 
D)  $1430.658$ 
E)  $1851.961$

Question 5

Find the sum of the following infinite geometric series.
$$- 14 + 13 - \frac { 169 } { 14 } + \frac { 2197 } { 196 } - \ldots$$

Accepted Answer

A)  $- \frac { 27 } { 196 }$ 
B)  $- \frac { 1 } { 196 }$ 
C)  $\frac { 27 } { 196 }$ 
D)  $- 14$ 
E)  $- \frac { 196 } { 27 }$ 
A)  $- \frac { 27 } { 196 }$ 
B)  $- \frac { 1 } { 196 }$ 
C)  $\frac { 27 } { 196 }$ 
D)  $- 14$ 
E)  $- \frac { 196 } { 27 }$

Question 6

Use mathematical induction to prove that 160 is a factor of $2 ^ { 4 n + 3 } + 32$ for all positive $n$.

Accepted Answer

None

Question 7

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

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 8

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing four green, six yellow, and five red marbles such that both marbles are yellow.

Accepted Answer

A)  $\frac { 1 } { 7 }$ 
B)  $\frac { 4 } { 25 }$ 
C)  $\frac { 6 } { 35 }$ 
D)  $\frac { 1 } { 3 }$ 
E)  $\frac { 2 } { 3 }$ 
A)  $\frac { 1 } { 7 }$ 
B)  $\frac { 4 } { 25 }$ 
C)  $\frac { 6 } { 35 }$ 
D)  $\frac { 1 } { 3 }$ 
E)  $\frac { 2 } { 3 }$

Question 9

Evaluate: ${ } _ { 8 } P _ { 5 }$

Accepted Answer

A)  56 
B)  336 
C)  6720 
D)  40 
E)  undefined 
A)  56 
B)  336 
C)  6720 
D)  40 
E)  undefined

Question 10

Find the sum of the infinite series.
$$\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 4 } \right) ^ { i }$$

Accepted Answer

A)  undefined 
B)  $\frac { 2 } { 5 }$ 
C)  2 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 2 } { 3 }$ 
A)  undefined 
B)  $\frac { 2 } { 5 }$ 
C)  2 
D)  $\frac { 4 } { 3 }$ 
E)  $\frac { 2 } { 3 }$

Question 11

Use mathematical induction to prove the following for every positive integer $n$.
$\sum _ { i = 1 } ^ { n } \frac { 8 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 8 n } { 2 n + 1 }$

Accepted Answer

None

Question 12

Find the indicated $n$th term of the geometric sequence.
6th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 8 } = \frac { 4 } { 2187 }$

Accepted Answer

A)  $\frac { 4 } { 729 }$ 
B)  $\frac { 3 } { 1024 }$ 
C)  $\frac { 4 } { 2187 }$ 
D)  $\frac { 4 } { 243 }$ 
E)  $\frac { 4 } { 81 }$ 
A)  $\frac { 4 } { 729 }$ 
B)  $\frac { 3 } { 1024 }$ 
C)  $\frac { 4 } { 2187 }$ 
D)  $\frac { 4 } { 243 }$ 
E)  $\frac { 4 } { 81 }$

Question 13

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 14

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)  $\quad P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 ( k - 1 ) + 1 ] + [ 6 k + 1 ]$ 
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)  $\quad P _ { k + 1 } = 7 + 13 + 19 + \ldots + [ 6 ( k - 1 ) + 1 ] + [ 6 k + 1 ]$

Question 15

Given the sequence $4 + \frac { 7 } { 8 } , 4 + \frac { 8 } { 9 } , 4 + \frac { 9 } { 10 } , 4 + \frac { 10 } { 11 } , 4 + \frac { 11 } { 12 } , \ldots$, write an expression for the apparent $n$th term assuming $n$ begins with 1 .

Accepted Answer

A)  $\frac { 6 n + 34 } { n + 7 }$ 
B)  $\frac { 5 n + 35 } { n + 7 }$ 
C)  $\frac { 5 n + 34 } { n + 7 }$ 
D)  $\frac { 5 n + 34 } { n + 8 }$ 
E)  $\frac { 6 n + 35 } { n + 8 }$ 
A)  $\frac { 6 n + 34 } { n + 7 }$ 
B)  $\frac { 5 n + 35 } { n + 7 }$ 
C)  $\frac { 5 n + 34 } { n + 7 }$ 
D)  $\frac { 5 n + 34 } { n + 8 }$ 
E)  $\frac { 6 n + 35 } { n + 8 }$

Question 16

Determine whether the sequence is geometric. If so, find the common ratio. $- 2,6 , - 18,54 , \ldots$

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 17

Use the Binomial Theorem to expand the following complex number. Write your answer in standard form.
$\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$

Accepted Answer

A)  $\frac { 36 } { 343 } - \frac { 20 \sqrt { 7 } } { 343 } i$ 
B)  $\frac { 36 } { 343 } + \frac { 20 \sqrt { 7 } } { 49 } i$ 
C)  $- \frac { 36 } { 49 } - \frac { 20 \sqrt { 7 } } { 343 } i$ 
D)  $- \frac { 27 } { 343 } + \frac { \sqrt { 7 } } { 49 } i$ 
E)  $- \frac { 27 } { 343 } - \frac { \sqrt { 7 } } { 49 } i$ 
A)  $\frac { 36 } { 343 } - \frac { 20 \sqrt { 7 } } { 343 } i$ 
B)  $\frac { 36 } { 343 } + \frac { 20 \sqrt { 7 } } { 49 } i$ 
C)  $- \frac { 36 } { 49 } - \frac { 20 \sqrt { 7 } } { 343 } i$ 
D)  $- \frac { 27 } { 343 } + \frac { \sqrt { 7 } } { 49 } i$ 
E)  $- \frac { 27 } { 343 } - \frac { \sqrt { 7 } } { 49 } i$

Question 18

Expand the following binomial by using Pascal's Triangle.
$$( 2 x - 4 ) ^ { 5 }$$

Accepted Answer

A)  $32 x ^ { 5 } - 1024$ 
B)  $- 1024 x ^ { 5 } + 1280 x ^ { 3 } + 32$ 
C)  $32 x ^ { 5 } - 320 x ^ { 4 } + 1280 x ^ { 3 } - 2560 x ^ { 2 } + 2560 x - 1024$ 
D)  $32 x ^ { 5 } + 2560 x ^ { 4 } - 2560 x ^ { 3 } + 1280 x ^ { 2 } - 320 x - 1024$ 
E)  $32 x ^ { 5 } - 64 x ^ { 4 } + 128 x ^ { 3 } - 256 x ^ { 2 } + 512 x - 1024$ 
A)  $32 x ^ { 5 } - 1024$ 
B)  $- 1024 x ^ { 5 } + 1280 x ^ { 3 } + 32$ 
C)  $32 x ^ { 5 } - 320 x ^ { 4 } + 1280 x ^ { 3 } - 2560 x ^ { 2 } + 2560 x - 1024$ 
D)  $32 x ^ { 5 } + 2560 x ^ { 4 } - 2560 x ^ { 3 } + 1280 x ^ { 2 } - 320 x - 1024$ 
E)  $32 x ^ { 5 } - 64 x ^ { 4 } + 128 x ^ { 3 } - 256 x ^ { 2 } + 512 x - 1024$

Question 19

Find the indicated term of the sequence.
$a _ { n } = ( - 1 ) ^ { n } ( 5 n - 1 )$

Accepted Answer

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

Question 20

Use mathematical induction to prove the following equality. $\quad \ln \left( 3 ^ { n } x _ { 1 } x _ { 2 } \ldots x _ { n } \right) = \ln \left( 3 x _ { 1 } \right) + \ln \left( 3 x _ { 2 } \right) + \ldots + \ln \left( 3 x _ { n } \right)$, where $x _ { 1 } > 0 , x _ { 2 } > 0 , \ldots , x _ { n } > 0$ $x _ { 1 } > 0 , x _ { 2 } > 0 , \ldots , x _ { n } > 0$

Accepted Answer

None

Evaluate using a graphing utility: ${ } _ { 20 } P _ { 5 }$

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that $n$ begins with 1.) $a _ { n } = - 6 + 4 n$

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 12 } n ^ { 3 }$ A) 4356 B) 12,168 C) 1728 D) 650 E) 6084 Use mathematical induction to prove the property for all positive integers $n$ . $\left[ a ^ { n } \right] ^ { 4 } = a ^ { 4 n }$

Find the sum of the finite geometric sequence. Round to the nearest thousandth. $\sum _ { i = 0 } ^ { 6 } 200 ( 1.07 ) ^ { i }$

Find the sum of the following infinite geometric series. $- 14 + 13 - \frac { 169 } { 14 } + \frac { 2197 } { 196 } - \ldots$

Use mathematical induction to prove that 160 is a factor of $2 ^ { 4 n + 3 } + 32$ for all positive $n$ .

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

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing four green, six yellow, and five red marbles such that both marbles are yellow.

Evaluate: ${ } _ { 8 } P _ { 5 }$

Find the sum of the infinite series. $\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 4 } \right) ^ { i }$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 8 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 8 n } { 2 n + 1 }$

Find the indicated $n$ th term of the geometric sequence. 6th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 8 } = \frac { 4 } { 2187 }$

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

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

Given the sequence $4 + \frac { 7 } { 8 } , 4 + \frac { 8 } { 9 } , 4 + \frac { 9 } { 10 } , 4 + \frac { 10 } { 11 } , 4 + \frac { 11 } { 12 } , \ldots$ , write an expression for the apparent $n$ th term assuming $n$ begins with 1 .

Determine whether the sequence is geometric. If so, find the common ratio. $- 2,6 , - 18,54 , \ldots$

Use the Binomial Theorem to expand the following complex number. Write your answer in standard form. $\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$

Expand the following binomial by using Pascal's Triangle. $( 2 x - 4 ) ^ { 5 }$

Find the indicated term of the sequence. $a _ { n } = ( - 1 ) ^ { n } ( 5 n - 1 )$ $Find the indicated term of the sequence. a _ { n } = ( - 1 ) ^ { n } ( 5 n - 1 )$

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

Filters

Exam 8: Sequences, Series, and Probability

Evaluate using a graphing utility: 20P5{ } _ { 20 } P _ { 5 }20​P5​

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that nnn begins with 1.) an=−6+4na _ { n } = - 6 + 4 nan​=−6+4n

Find the sum of the finite geometric sequence. Round to the nearest thousandth. ∑i=06200(1.07)i\sum _ { i = 0 } ^ { 6 } 200 ( 1.07 ) ^ { i }∑i=06​200(1.07)i

Find the sum of the following infinite geometric series. −14+13−16914+2197196−…- 14 + 13 - \frac { 169 } { 14 } + \frac { 2197 } { 196 } - \ldots−14+13−14169​+1962197​−…

Use mathematical induction to prove that 160 is a factor of 24n+3+322 ^ { 4 n + 3 } + 3224n+3+32 for all positive nnn .

Find the number of distinguishable permutations of the group of letters. E,S,T,I,M,A,T,EE , S , T , I , M , A , T , EE,S,T,I,M,A,T,E

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing four green, six yellow, and five red marbles such that both marbles are yellow.

Evaluate: 8P5{ } _ { 8 } P _ { 5 }8​P5​

Find the sum of the infinite series. ∑i=1∞2(14)i\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 4 } \right) ^ { i }∑i=1∞​2(41​)i

Use mathematical induction to prove the following for every positive integer nnn . ∑i=1n8(2i−1)(2i+1)=8n2n+1\sum _ { i = 1 } ^ { n } \frac { 8 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 8 n } { 2 n + 1 }∑i=1n​(2i−1)(2i+1)8​=2n+18n​

Find the indicated nnn th term of the geometric sequence. 6th term: a5=481,a8=42187a _ { 5 } = \frac { 4 } { 81 } , a _ { 8 } = \frac { 4 } { 2187 }a5​=814​,a8​=21874​

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 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]

Determine whether the sequence is geometric. If so, find the common ratio. −2,6,−18,54,…- 2,6 , - 18,54 , \ldots−2,6,−18,54,…

Use the Binomial Theorem to expand the following complex number. Write your answer in standard form. (−37−77i)3\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }(−73​−77​​i)3

Expand the following binomial by using Pascal's Triangle. (2x−4)5( 2 x - 4 ) ^ { 5 }(2x−4)5

Find the indicated term of the sequence. an=(−1)n(5n−1)a _ { n } = ( - 1 ) ^ { n } ( 5 n - 1 )an​=(−1)n(5n−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

Filters

Evaluate using a graphing utility: ${ } _ { 20 } P _ { 5 }$

Determine whether the sequence is arithmetic. If so, find the common difference. (Assume that $n$ begins with 1.) $a _ { n } = - 6 + 4 n$

Find the sum of the finite geometric sequence. Round to the nearest thousandth. $\sum _ { i = 0 } ^ { 6 } 200 ( 1.07 ) ^ { i }$

Find the sum of the following infinite geometric series. $- 14 + 13 - \frac { 169 } { 14 } + \frac { 2197 } { 196 } - \ldots$

Use mathematical induction to prove that 160 is a factor of $2 ^ { 4 n + 3 } + 32$ for all positive $n$ .

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

Evaluate: ${ } _ { 8 } P _ { 5 }$

Find the sum of the infinite series. $\sum _ { i = 1 } ^ { \infty } 2 \left( \frac { 1 } { 4 } \right) ^ { i }$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 8 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 8 n } { 2 n + 1 }$

Find the indicated $n$ th term of the geometric sequence. 6th term: $a _ { 5 } = \frac { 4 } { 81 } , a _ { 8 } = \frac { 4 } { 2187 }$

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

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

Determine whether the sequence is geometric. If so, find the common ratio. $- 2,6 , - 18,54 , \ldots$

Use the Binomial Theorem to expand the following complex number. Write your answer in standard form. $\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$

Expand the following binomial by using Pascal's Triangle. $( 2 x - 4 ) ^ { 5 }$

Find the indicated term of the sequence. $a _ { n } = ( - 1 ) ^ { n } ( 5 n - 1 )$ $Find the indicated term of the sequence. a _ { n } = ( - 1 ) ^ { n } ( 5 n - 1 )$