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 geometric. If so, find the common ratio.

Accepted Answer

A)  - 4 
B)  2 
C)  $- \frac { 1 } { 4 }$ 
D)  4 
E)  not geometric 
A)  - 4 
B)  2 
C)  $- \frac { 1 } { 4 }$ 
D)  4 
E)  not geometric E

Question 3

Use the first and second differences of the first five terms of the given sequence to determine whether the sequence has a linear model, a quadratic model, or neither.
$$\begin{array} { l } 
a _ { 1 } = 0 \
a _ { n } = a _ { n - 1 } + 8 n
\end{array}$$

Accepted Answer

A) first differences: $1,2,3,4$ 
 second differences: 1, 1, 1
linear model 
B)  first differences: $8,24,56,120$ 
second differences: $16,32,64$
linear model 
C)  first differences: $8,16,32,64$
second differences: $8,16,32$
quadratic model 
D) first differences: $8,24,56,120$
 second differences: $16,32,64$
quadratic model 
E) first differences: $8,16,32,64$
 second differences: $8,16,32$
neither linear nor quadratic 
A) first differences: $1,2,3,4$ 
 second differences: 1, 1, 1
linear model 
B)  first differences: $8,24,56,120$ 
second differences: $16,32,64$
linear model 
C)  first differences: $8,16,32,64$
second differences: $8,16,32$
quadratic model 
D) first differences: $8,24,56,120$
 second differences: $16,32,64$
quadratic model 
E) first differences: $8,16,32,64$
 second differences: $8,16,32$
neither linear nor quadratic E

Question 4

Use the Binomial Theorem to expand the complex number. Simplify your result.
$$( 5 + 4 i ) ^ { 4 }$$

Accepted Answer

A)  $- 1519 + 720 i$ 
B)  $- 1519 - 720 i$ 
C)  $1519 + 720 i$ 
D)  $1519 - 720 i$ 
E)  625 
A)  $- 1519 + 720 i$ 
B)  $- 1519 - 720 i$ 
C)  $1519 + 720 i$ 
D)  $1519 - 720 i$ 
E)  625

Question 5

Determine whether the sequence is arithmetic. If so, find the common difference.
$$- 3 , - 7 , - 11 , - 15 , - 19$$

Accepted Answer

A)  1 
B)  - 4 
C)  - 3 
D)  4 
E)  not arithmetic 
A)  1 
B)  - 4 
C)  - 3 
D)  4 
E)  not arithmetic

Question 6

Use mathematical induction to prove the following inequality for all $n \geq 2$.
$$( 3 ) ^ { n + 2 } > \frac { 9 } { 32 } n ( 2 ) ^ { n + 5 }$$

Accepted Answer

i. Show true for @#LAT-DLM&
\[\begin{aligned}
( 3

Question 7

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

Accepted Answer

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

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

You are given the probability that an event will not happen. Find the probability that the event will happen.
$$P \left( E ^ { \prime } \right) = \frac { 5 } { 9 }$$

Accepted Answer

A)  $\frac { 5 } { 9 }$ 
B)  0 
C)  1 
D)  $\frac { 4 } { 9 }$ 
E)  $\frac { 2 } { 9 }$ 
A)  $\frac { 5 } { 9 }$ 
B)  0 
C)  1 
D)  $\frac { 4 } { 9 }$ 
E)  $\frac { 2 } { 9 }$

Question 10

Find the sum of the infinite geometric series.
$$\sum _ { n = 0 } ^ { \infty } 4 \left( - \frac { 1 } { 5 } \right) ^ { n }$$

Accepted Answer

A)  $- \frac { 10 } { 3 }$ 
B)  $\frac { 10 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $- \frac { 2 } { 3 }$ 
E)  undefined 
A)  $- \frac { 10 } { 3 }$ 
B)  $\frac { 10 } { 3 }$ 
C)  $\frac { 2 } { 3 }$ 
D)  $- \frac { 2 } { 3 }$ 
E)  undefined

Question 11

Use mathematical induction to prove the property for all positive integers n. $$\left[ a ^ { n } \right] ^ { 4 } = a ^ { 4 n }$$

Accepted Answer

1) @#LAT-DLM& :
\[\begin{array} { r } 
{ \left[ a

Question 12

Find the sum of the infinite series.
$$\sum _ { i = 1 } ^ { \infty } 3 \left( \frac { 1 } { 3 } \right) ^ { i }$$

Accepted Answer

A)  undefined 
B)  $\frac { 3 } { 4 }$ 
C)  3 
D)  $\frac { 9 } { 4 }$ 
E)  $\frac { 3 } { 2 }$ 
A)  undefined 
B)  $\frac { 3 } { 4 }$ 
C)  3 
D)  $\frac { 9 } { 4 }$ 
E)  $\frac { 3 } { 2 }$

Question 13

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 14

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

Accepted Answer

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

Question 15

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 16

Find the sum using the formulas for the sums of powers of integers.
$$\sum _ { n = 1 } ^ { 9 } n ^ { 3 }$$

Accepted Answer

A)  1296 
B)  4050 
C)  729 
D)  285 
E)  2025 
A)  1296 
B)  4050 
C)  729 
D)  285 
E)  2025

Question 17

Find $P _ { k + 1 }$ for the given $P _ { k }$.
$$P _ { k } = \frac { 4 } { k ( k + 1 ) }$$

Accepted Answer

A)  $ P _ { k + 1 } = \frac { 4 } { k ( k + 1 ) } + 1$ 
B)  $P _ { k + 1 } = \frac { 4 } { k ( k + 1 ) } + \frac { 4 } { ( k + 1 ) ( k + 2 ) }$ 
C)  $P _ { k + 1 } = \frac { 4 } { ( k + 1 ) ( k + 2 ) }$ 
D)  $P _ { k + 1 } = \frac { 4 } { k ( k + 2 ) }$ 
E)  $P _ { k + 1 } = \frac { 16 } { ( k + 1 ) ( k + 2 ) }$ 
A)  $ P _ { k + 1 } = \frac { 4 } { k ( k + 1 ) } + 1$ 
B)  $P _ { k + 1 } = \frac { 4 } { k ( k + 1 ) } + \frac { 4 } { ( k + 1 ) ( k + 2 ) }$ 
C)  $P _ { k + 1 } = \frac { 4 } { ( k + 1 ) ( k + 2 ) }$ 
D)  $P _ { k + 1 } = \frac { 4 } { k ( k + 2 ) }$ 
E)  $P _ { k + 1 } = \frac { 16 } { ( k + 1 ) ( k + 2 ) }$

Question 18

Write the first five terms of the arithmetic sequence. $$a _ { 1 } = 5 , d = 7$$

Accepted Answer

A)  5, 12, 19, 26, 33 
B)  5, -2, -9, -16, -23 
C)  12, 19, 26, 33, 40 
D)  5, 35, 245, 1715, 12005 
E)  5, 12, 17, 22, 27 
A)  5, 12, 19, 26, 33 
B)  5, -2, -9, -16, -23 
C)  12, 19, 26, 33, 40 
D)  5, 35, 245, 1715, 12005 
E)  5, 12, 17, 22, 27

Question 19

Find the number of distinguishable permutations of the group of letters.
$$\mathrm { G } , \mathrm { A } , \mathrm { U } , \mathrm { S } , \mathrm { S }$$

Accepted Answer

A)  60 
B)  120 
C)  30 
D)  5 
E)  15 
A)  60 
B)  120 
C)  30 
D)  5 
E)  15

Question 20

Evaluate using a graphing utility: ${ } _ { 15 } P _ { 6 }$

Accepted Answer

A)  5005 
B)  360,360 
C)  3,603,600 
D)  90 
E)  undefined 
A)  5005 
B)  360,360 
C)  3,603,600 
D)  90 
E)  undefined

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

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

Use the first and second differences of the first five terms of the given sequence to determine whether the sequence has a linear model, a quadratic model, or neither. =0 =+8n

Use the Binomial Theorem to expand the complex number. Simplify your result. $( 5 + 4 i ) ^ { 4 }$

Determine whether the sequence is arithmetic. If so, find the common difference. $- 3 , - 7 , - 11 , - 15 , - 19$

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $( 3 ) ^ { n + 2 } > \frac { 9 } { 32 } n ( 2 ) ^ { n + 5 }$

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.

You are given the probability that an event will not happen. Find the probability that the event will happen. $P \left( E ^ { \prime } \right) = \frac { 5 } { 9 }$

Find the sum of the infinite geometric series. $\sum _ { n = 0 } ^ { \infty } 4 \left( - \frac { 1 } { 5 } \right) ^ { n }$

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 infinite series. $\sum _ { i = 1 } ^ { \infty } 3 \left( \frac { 1 } { 3 } \right) ^ { i }$

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

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

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

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 9 } n ^ { 3 }$

Find $P _ { k + 1 }$ for the given $P _ { k }$ . $P _ { k } = \frac { 4 } { k ( k + 1 ) }$

Write the first five terms of the arithmetic sequence. $a _ { 1 } = 5 , d = 7$

Find the number of distinguishable permutations of the group of letters. $\mathrm { G } , \mathrm { A } , \mathrm { U } , \mathrm { S } , \mathrm { S }$

Evaluate using a graphing utility: ${ } _ { 15 } P _ { 6 }$

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

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

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

Use the first and second differences of the first five terms of the given sequence to determine whether the sequence has a linear model, a quadratic model, or neither. =0 =+8n

Use the Binomial Theorem to expand the complex number. Simplify your result. (5+4i)4( 5 + 4 i ) ^ { 4 }(5+4i)4

Determine whether the sequence is arithmetic. If so, find the common difference. −3,−7,−11,−15,−19- 3 , - 7 , - 11 , - 15 , - 19−3,−7,−11,−15,−19

Use mathematical induction to prove the following inequality for all n≥2n \geq 2n≥2 . (3)n+2>932n(2)n+5( 3 ) ^ { n + 2 } > \frac { 9 } { 32 } n ( 2 ) ^ { n + 5 }(3)n+2>329​n(2)n+5

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.

You are given the probability that an event will not happen. Find the probability that the event will happen. P(E′)=59P \left( E ^ { \prime } \right) = \frac { 5 } { 9 }P(E′)=95​

Find the sum of the infinite geometric series. ∑n=0∞4(−15)n\sum _ { n = 0 } ^ { \infty } 4 \left( - \frac { 1 } { 5 } \right) ^ { n }∑n=0∞​4(−51​)n

Use mathematical induction to prove the property for all positive integers n. [an]4=a4n\left[ a ^ { n } \right] ^ { 4 } = a ^ { 4 n }[an]4=a4n

Find the sum of the infinite series. ∑i=1∞3(13)i\sum _ { i = 1 } ^ { \infty } 3 \left( \frac { 1 } { 3 } \right) ^ { i }∑i=1∞​3(31​)i

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 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

Find the sum using the formulas for the sums of powers of integers. ∑n=19n3\sum _ { n = 1 } ^ { 9 } n ^ { 3 }∑n=19​n3

Find Pk+1P _ { k + 1 }Pk+1​ for the given PkP _ { k }Pk​ . Pk=4k(k+1)P _ { k } = \frac { 4 } { k ( k + 1 ) }Pk​=k(k+1)4​

Write the first five terms of the arithmetic sequence. a1=5,d=7a _ { 1 } = 5 , d = 7a1​=5,d=7

Find the number of distinguishable permutations of the group of letters. G,A,U,S,S\mathrm { G } , \mathrm { A } , \mathrm { U } , \mathrm { S } , \mathrm { S }G,A,U,S,S

Evaluate using a graphing utility: 15P6{ } _ { 15 } P _ { 6 }15​P6​

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

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

Use the Binomial Theorem to expand the complex number. Simplify your result. $( 5 + 4 i ) ^ { 4 }$

Determine whether the sequence is arithmetic. If so, find the common difference. $- 3 , - 7 , - 11 , - 15 , - 19$

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $( 3 ) ^ { n + 2 } > \frac { 9 } { 32 } n ( 2 ) ^ { n + 5 }$

You are given the probability that an event will not happen. Find the probability that the event will happen. $P \left( E ^ { \prime } \right) = \frac { 5 } { 9 }$

Find the sum of the infinite geometric series. $\sum _ { n = 0 } ^ { \infty } 4 \left( - \frac { 1 } { 5 } \right) ^ { n }$

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 infinite series. $\sum _ { i = 1 } ^ { \infty } 3 \left( \frac { 1 } { 3 } \right) ^ { i }$

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

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

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 9 } n ^ { 3 }$

Find $P _ { k + 1 }$ for the given $P _ { k }$ . $P _ { k } = \frac { 4 } { k ( k + 1 ) }$

Write the first five terms of the arithmetic sequence. $a _ { 1 } = 5 , d = 7$

Find the number of distinguishable permutations of the group of letters. $\mathrm { G } , \mathrm { A } , \mathrm { U } , \mathrm { S } , \mathrm { S }$

Evaluate using a graphing utility: ${ } _ { 15 } P _ { 6 }$