Question 1

Use the Binomial Theorem to expand and simplify the expression.
$$( w - 2 ) ^ { 5 }$$

Accepted Answer

A)  $w ^ { 5 } - 10 w ^ { 4 } + 40 w ^ { 3 } - 80 w ^ { 2 } + 80 w$ 
B)  $w ^ { 4 } - 8 w ^ { 3 } + 24 w ^ { 2 } - 32 w + 16$ 
C)  $w ^ { 5 } - 10 w ^ { 4 } + 60 w ^ { 3 } - 120 w ^ { 2 } + 80 w - 32$ 
D)  $w ^ { 5 } - 8 w ^ { 4 } + 36 w ^ { 3 } - 72 w ^ { 2 } + 64 w - 32$ 
E)  $w ^ { 5 } - 10 w ^ { 4 } + 40 w ^ { 3 } - 80 w ^ { 2 } + 80 w - 32$ 
A)  $w ^ { 5 } - 10 w ^ { 4 } + 40 w ^ { 3 } - 80 w ^ { 2 } + 80 w$ 
B)  $w ^ { 4 } - 8 w ^ { 3 } + 24 w ^ { 2 } - 32 w + 16$ 
C)  $w ^ { 5 } - 10 w ^ { 4 } + 60 w ^ { 3 } - 120 w ^ { 2 } + 80 w - 32$ 
D)  $w ^ { 5 } - 8 w ^ { 4 } + 36 w ^ { 3 } - 72 w ^ { 2 } + 64 w - 32$ 
E)  $w ^ { 5 } - 10 w ^ { 4 } + 40 w ^ { 3 } - 80 w ^ { 2 } + 80 w - 32$

Question 2

Use mathematical induction to prove the following for every positive integer $n$.
$$\sum _ { i = 1 } ^ { n } 96 i ^ { 5 } = 8 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$$

Accepted Answer

i. Show true for @#LAT-DLM&
\[\begin{aligned}
96 \

Question 3

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 4

Determine whether the sequence is geometric. If so, find the common ratio. $- 1 , - 2 , - 4 , - 8 , \ldots$

Accepted Answer

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

Question 5

Given the sequence $4 + \frac { 12 } { 13 } , 4 + \frac { 19 } { 20 } , 4 + \frac { 26 } { 27 } , 4 + \frac { 33 } { 34 } , 4 + \frac { 40 } { 41 } , \ldots$, write an expression for the apparent $n$th term assuming $n$ begins with 1 .

Accepted Answer

A)  $\frac { 36 n + 29 } { 7 n + 6 }$ 
B)  $\frac { 35 n + 30 } { 7 n + 6 }$ 
C)  $\frac { 35 n + 29 } { 7 n + 6 }$ 
D)  $\frac { 35 n + 29 } { 7 n + 7 }$ 
E)  $\frac { 36 n + 30 } { 7 n + 7 }$ 
A)  $\frac { 36 n + 29 } { 7 n + 6 }$ 
B)  $\frac { 35 n + 30 } { 7 n + 6 }$ 
C)  $\frac { 35 n + 29 } { 7 n + 6 }$ 
D)  $\frac { 35 n + 29 } { 7 n + 7 }$ 
E)  $\frac { 36 n + 30 } { 7 n + 7 }$

Question 6

$$\text { Use mathematical induction to prove that } 80 \text { is a factor of } 2 ^ { 8 n + 2 } + 16 \text { for all positive } n \text {. }$$

Accepted Answer

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

Question 7

Find the sum of the finite geometric sequence.
$$\sum _ { n = 1 } ^ { 6 } 2 \left( - \frac { 2 } { 3 } \right) ^ { n - 1 }$$

Accepted Answer

A)  $- \frac { 463 } { 243 }$ 
B)  $- \frac { 1024 } { 3 }$ 
C)  4256 
D)  $\frac { 55 } { 81 }$ 
E)  $\frac { 266 } { 243 }$ 
A)  $- \frac { 463 } { 243 }$ 
B)  $- \frac { 1024 } { 3 }$ 
C)  4256 
D)  $\frac { 55 } { 81 }$ 
E)  $\frac { 266 } { 243 }$

Question 8

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)  $\quad a _ { n } = 3 n - 8$ 
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)  $\quad a _ { n } = 3 n - 8$

Question 9

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

Question 10

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

Question 11

Find the sum.
$$\sum _ { i = 1 } ^ { 4 } ( - i - 4 )$$

Accepted Answer

A)  $- 18$ 
B)  $- 30$ 
C)  $- 26$ 
D)  $- 8$ 
E)  $- 10$ 
A)  $- 18$ 
B)  $- 30$ 
C)  $- 26$ 
D)  $- 8$ 
E)  $- 10$

Question 12

Determine whether the sequence is arithmetic. If so, find the common difference. $7,8,9,10,11$

Accepted Answer

A)  6 
B)  1 
C)  7 
D)  $- 1$ 
E)  not arithmetic 
A)  6 
B)  1 
C)  7 
D)  $- 1$ 
E)  not arithmetic

Question 13

Find the rational number representation of the repeating decimal. $0 . \overline { 167 }$

Accepted Answer

A)  $\frac { 167 } { 9999 }$ 
B)  $\frac { 16.7 } { 999 }$ 
C)  $\frac { 167 } { 9 }$ 
D)  $\frac { 167 } { 999 }$ 
E)  $\frac { 167 } { 99 }$ 
A)  $\frac { 167 } { 9999 }$ 
B)  $\frac { 16.7 } { 999 }$ 
C)  $\frac { 167 } { 9 }$ 
D)  $\frac { 167 } { 999 }$ 
E)  $\frac { 167 } { 99 }$

Question 14

Determine whether the sequence is geometric. If so, find the common ratio. $- 1,2,5,8 , \ldots$

Accepted Answer

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

Question 15

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)  $\quad 32 x ^ { 5 } - 64 x ^ { 4 } + 128 x ^ { 3 } - 256 x ^ { 2 } + 512 x - 1024$ 
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)  $\quad 32 x ^ { 5 } - 64 x ^ { 4 } + 128 x ^ { 3 } - 256 x ^ { 2 } + 512 x - 1024$

Question 16

Find a formula for $a _ { n }$ for the arithmetic sequence.
$$a _ { 4 } = - 13 , a _ { 13 } = - 31$$

Accepted Answer

A)  $a _ { n } = - 7 - 2 n$ 
B)  $a _ { n } = 5 - 7 n$ 
C)  $a _ { n } = - 2 - 7 n$ 
D)  $a _ { n } = - 7 ( - 2 ) ^ { n }$ 
E)  $a _ { n } = - 5 - 2 n$ 
A)  $a _ { n } = - 7 - 2 n$ 
B)  $a _ { n } = 5 - 7 n$ 
C)  $a _ { n } = - 2 - 7 n$ 
D)  $a _ { n } = - 7 ( - 2 ) ^ { n }$ 
E)  $a _ { n } = - 5 - 2 n$

Question 17

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 18

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

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 19

Determine the number of ways a computer can randomly generate a prime integer between 10 and 20 .

Accepted Answer

A)  4 
B) 3 
C) 2 
D) 5 
E) 11 
A)  4 
B) 3 
C) 2 
D) 5 
E) 11

Question 20

Expand the binomial by using Pascal's triangle to determine the coefficients.
$$( 3 x + 2 y ) ^ { 6 }$$

Accepted Answer

A)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 4860 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 4860 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
B)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 3240 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2160 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
C)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 4860 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2880 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
D)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 4860 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2160 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
E)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 2160 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2160 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
A)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 4860 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 4860 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
B)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 3240 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2160 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
C)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 4860 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2880 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
D)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 4860 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2160 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$ 
E)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 2160 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2160 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$

Use the Binomial Theorem to expand and simplify the expression. $( w - 2 ) ^ { 5 }$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } 96 i ^ { 5 } = 8 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$

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

Determine whether the sequence is geometric. If so, find the common ratio. $- 1 , - 2 , - 4 , - 8 , \ldots$

Given the sequence $4 + \frac { 12 } { 13 } , 4 + \frac { 19 } { 20 } , 4 + \frac { 26 } { 27 } , 4 + \frac { 33 } { 34 } , 4 + \frac { 40 } { 41 } , \ldots$ , write an expression for the apparent $n$ th term assuming $n$ begins with 1 .

$\text { Use mathematical induction to prove that } 80 \text { is a factor of } 2 ^ { 8 n + 2 } + 16 \text { for all positive } n \text {. }$

Find the sum of the finite geometric sequence. $\sum _ { n = 1 } ^ { 6 } 2 \left( - \frac { 2 } { 3 } \right) ^ { n - 1 }$

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

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. $\sum _ { i = 1 } ^ { 4 } ( - i - 4 )$

Determine whether the sequence is arithmetic. If so, find the common difference. $7,8,9,10,11$

Find the rational number representation of the repeating decimal. $0 . \overline { 167 }$

Determine whether the sequence is geometric. If so, find the common ratio. $- 1,2,5,8 , \ldots$

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

Find a formula for $a _ { n }$ for the arithmetic sequence. $a _ { 4 } = - 13 , a _ { 13 } = - 31$

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

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

Determine the number of ways a computer can randomly generate a prime integer between 10 and 20 .

Expand the binomial by using Pascal's triangle to determine the coefficients. $( 3 x + 2 y ) ^ { 6 }$

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

Calculus Practice Problems

Filters

Exam 8: Sequences, Series, and Probability

Use the Binomial Theorem to expand and simplify the expression. (w−2)5( w - 2 ) ^ { 5 }(w−2)5

Use mathematical induction to prove the following for every positive integer nnn . ∑i=1n96i5=8n2(n+1)2(2n2+2n−1)\sum _ { i = 1 } ^ { n } 96 i ^ { 5 } = 8 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)∑i=1n​96i5=8n2(n+1)2(2n2+2n−1)

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

Determine whether the sequence is geometric. If so, find the common ratio. −1,−2,−4,−8,…- 1 , - 2 , - 4 , - 8 , \ldots−1,−2,−4,−8,…

Find the sum of the finite geometric sequence. ∑n=162(−23)n−1\sum _ { n = 1 } ^ { 6 } 2 \left( - \frac { 2 } { 3 } \right) ^ { n - 1 }∑n=16​2(−32​)n−1

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

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. ∑i=14(−i−4)\sum _ { i = 1 } ^ { 4 } ( - i - 4 )∑i=14​(−i−4)

Determine whether the sequence is arithmetic. If so, find the common difference. 7,8,9,10,117,8,9,10,117,8,9,10,11

Find the rational number representation of the repeating decimal. 0.167‾0 . \overline { 167 }0.167

Determine whether the sequence is geometric. If so, find the common ratio. −1,2,5,8,…- 1,2,5,8 , \ldots−1,2,5,8,…

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

Find a formula for ana _ { n }an​ for the arithmetic sequence. a4=−13,a13=−31a _ { 4 } = - 13 , a _ { 13 } = - 31a4​=−13,a13​=−31

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

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

Determine the number of ways a computer can randomly generate a prime integer between 10 and 20 .

Expand the binomial by using Pascal's triangle to determine the coefficients. (3x+2y)6( 3 x + 2 y ) ^ { 6 }(3x+2y)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

Calculus Practice Problems

Filters

Use the Binomial Theorem to expand and simplify the expression. $( w - 2 ) ^ { 5 }$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } 96 i ^ { 5 } = 8 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$

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

Determine whether the sequence is geometric. If so, find the common ratio. $- 1 , - 2 , - 4 , - 8 , \ldots$

Find the sum of the finite geometric sequence. $\sum _ { n = 1 } ^ { 6 } 2 \left( - \frac { 2 } { 3 } \right) ^ { n - 1 }$

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

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. $\sum _ { i = 1 } ^ { 4 } ( - i - 4 )$

Determine whether the sequence is arithmetic. If so, find the common difference. $7,8,9,10,11$

Find the rational number representation of the repeating decimal. $0 . \overline { 167 }$

Determine whether the sequence is geometric. If so, find the common ratio. $- 1,2,5,8 , \ldots$

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

Find a formula for $a _ { n }$ for the arithmetic sequence. $a _ { 4 } = - 13 , a _ { 13 } = - 31$

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

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

Expand the binomial by using Pascal's triangle to determine the coefficients. $( 3 x + 2 y ) ^ { 6 }$