Question 1

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 2

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 3

Find a formula for $a _ { n }$ for the arithmetic sequence.
$$a _ { 4 } = 10 , a _ { 8 } = 34$$

Accepted Answer

A)  $a _ { n } = - 8 + 6 n$ 
B)  $a _ { n } = 14 - 8 n$ 
C)  $a _ { n } = 6 - 8 n$ 
D)  $a _ { n } = - 8 ( 6 ) ^ { n }$ 
E)  $a _ { n } = - 14 + 6 n$ 
A)  $a _ { n } = - 8 + 6 n$ 
B)  $a _ { n } = 14 - 8 n$ 
C)  $a _ { n } = 6 - 8 n$ 
D)  $a _ { n } = - 8 ( 6 ) ^ { n }$ 
E)  $a _ { n } = - 14 + 6 n$

Question 4

Find the sum of the finite geometric sequence.
$$\sum _ { n = 1 } ^ { 5 } - \left( - \frac { 1 } { 6 } \right) ^ { n - 1 }$$

Accepted Answer

A)  $\frac { 6665 } { 1296 }$ 
B)  $- \frac { 1 } { 6 }$ 
C)  1111 
D)  $\frac { 185 } { 216 }$ 
E)  $- \frac { 1111 } { 1296 }$ 
A)  $\frac { 6665 } { 1296 }$ 
B)  $- \frac { 1 } { 6 }$ 
C)  1111 
D)  $\frac { 185 } { 216 }$ 
E)  $- \frac { 1111 } { 1296 }$

Question 5

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 6

Find a formula for $a _ { n }$ for the arithmetic sequence.
$$a _ { 3 } = 19 , a _ { 13 } = 99$$

Accepted Answer

A)  $a _ { n } = 3 + 8 n$ 
B)  $a _ { n } = 5 + 3 n$ 
C)  $a _ { n } = 8 + 3 n$ 
D)  $a _ { n } = 3 ( 8 ) ^ { n }$ 
E)  $a _ { n } = - 5 + 8 n$ 
A)  $a _ { n } = 3 + 8 n$ 
B)  $a _ { n } = 5 + 3 n$ 
C)  $a _ { n } = 8 + 3 n$ 
D)  $a _ { n } = 3 ( 8 ) ^ { n }$ 
E)  $a _ { n } = - 5 + 8 n$

Question 7

Write an expression for the apparent $n$th term of the sequence. (Assume that $n$ begins with 1.)
$- 4 , - 2,0,2,4$

Accepted Answer

A)  $a _ { n } = - 6 n + 2$ 
B)  $a _ { n } = ( - 1 ) ^ { n } ( 2 n - 6 )$ 
C)  $a _ { n } = 2 n$ 
D)  $a _ { n } = 2 ^ { n } - 6$ 
E)  $a _ { n } = 2 n - 6$ 
A)  $a _ { n } = - 6 n + 2$ 
B)  $a _ { n } = ( - 1 ) ^ { n } ( 2 n - 6 )$ 
C)  $a _ { n } = 2 n$ 
D)  $a _ { n } = 2 ^ { n } - 6$ 
E)  $a _ { n } = 2 n - 6$

Question 8

Use mathematical induction to prove the following for every positive integer $n$.
$1 + 6 + 36 + 216 + \ldots + 6 ^ { n - 1 } = \frac { 6 ^ { n } - 1 } { 5 }$

Accepted Answer

None

Question 9

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 10

Use mathematical induction to prove the following for every positive integer $n$.
$1 + 3 + 9 + 27 + \ldots + 3 ^ { n - 1 } = \frac { 3 ^ { n } - 1 } { 2 }$

Accepted Answer

None

Question 11

Find the indicated partial sum of the series.
$$\sum _ { i = 1 } ^ { \infty } 5 \left( - \frac { 1 } { 3 } \right) ^ { i }$$
fourth partial sum

Accepted Answer

A)  $\frac { 5 } { 81 }$ 
B)  $\frac { 305 } { 81 }$ 
C)  $- \frac { 100 } { 81 }$ 
D)  $\frac { 35 } { 81 }$ 
E)  $- \frac { 205 } { 162 }$ 
A)  $\frac { 5 } { 81 }$ 
B)  $\frac { 305 } { 81 }$ 
C)  $- \frac { 100 } { 81 }$ 
D)  $\frac { 35 } { 81 }$ 
E)  $- \frac { 205 } { 162 }$

Question 12

Find a formula for the $n t h$ term of the following geometric sequence, then find the 4 th term of the sequence.
$9,36,144 , \ldots$

Accepted Answer

A)  $a _ { n } = 9 ( 4 ) ^ { n } ; a _ { 4 } = 576$ 
B)  $a _ { n } = 9 ( 4 ) ^ { n } ; a _ { 4 } = 2304$ 
C)  $a _ { n } = 9 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 2304$ 
D)  $a _ { n } = 9 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 576$ 
E)  $a _ { n } = 9 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 144$ 
A)  $a _ { n } = 9 ( 4 ) ^ { n } ; a _ { 4 } = 576$ 
B)  $a _ { n } = 9 ( 4 ) ^ { n } ; a _ { 4 } = 2304$ 
C)  $a _ { n } = 9 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 2304$ 
D)  $a _ { n } = 9 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 576$ 
E)  $a _ { n } = 9 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 144$

Question 13

Simplify the factorial expression.
$\frac { 11 ! } { 9 ! }$

Accepted Answer

A)  990 
B)  110 
C)  $\frac { 11 } { 9 }$ 
D)  11 
E)  1320 
A)  990 
B)  110 
C)  $\frac { 11 } { 9 }$ 
D)  11 
E)  1320

Question 14

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 15

Find the probability for the experiment of selecting one card from a standard deck of 52 playing cards such that the card is not a red face card.

Accepted Answer

A)  $\frac { 3 } { 26 }$ 
B)  $\frac { 10 } { 13 }$ 
C)  $\frac { 11 } { 13 }$ 
D)  $\frac { 23 } { 26 }$ 
E)  $\frac { 49 } { 52 }$ 
A)  $\frac { 3 } { 26 }$ 
B)  $\frac { 10 } { 13 }$ 
C)  $\frac { 11 } { 13 }$ 
D)  $\frac { 23 } { 26 }$ 
E)  $\frac { 49 } { 52 }$

Question 16

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 17

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 18

Use mathematical induction to prove the following for every positive integer $n$.
$\sum _ { i = 1 } ^ { n } \frac { 17 } { i ( i + 1 ) ( i + 2 ) } = \frac { 17 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }$

Accepted Answer

None

Question 19

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 20

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

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

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

Find a formula for $a _ { n }$ for the arithmetic sequence. $a _ { 4 } = 10 , a _ { 8 } = 34$

Find the sum of the finite geometric sequence. $\sum _ { n = 1 } ^ { 5 } - \left( - \frac { 1 } { 6 } \right) ^ { n - 1 }$

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

Find a formula for $a _ { n }$ for the arithmetic sequence. $a _ { 3 } = 19 , a _ { 13 } = 99$

Write an expression for the apparent $n$ th term of the sequence. (Assume that $n$ begins with 1.) $- 4 , - 2,0,2,4$

Use mathematical induction to prove the following for every positive integer $n$ . $1 + 6 + 36 + 216 + \ldots + 6 ^ { n - 1 } = \frac { 6 ^ { n } - 1 } { 5 }$

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

Use mathematical induction to prove the following for every positive integer $n$ . $1 + 3 + 9 + 27 + \ldots + 3 ^ { n - 1 } = \frac { 3 ^ { n } - 1 } { 2 }$

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 5 \left( - \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum

Find a formula for the $n t h$ term of the following geometric sequence, then find the 4 th term of the sequence. $9,36,144 , \ldots$

Simplify the factorial expression. $\frac { 11 ! } { 9 ! }$

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

Find the probability for the experiment of selecting one card from a standard deck of 52 playing cards such that the card is not a red face card.

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

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

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 17 } { i ( i + 1 ) ( i + 2 ) } = \frac { 17 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }$

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

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

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

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​

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

Find a formula for ana _ { n }an​ for the arithmetic sequence. a4=10,a8=34a _ { 4 } = 10 , a _ { 8 } = 34a4​=10,a8​=34

Find the sum of the finite geometric sequence. ∑n=15−(−16)n−1\sum _ { n = 1 } ^ { 5 } - \left( - \frac { 1 } { 6 } \right) ^ { n - 1 }∑n=15​−(−61​)n−1

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,…

Find a formula for ana _ { n }an​ for the arithmetic sequence. a3=19,a13=99a _ { 3 } = 19 , a _ { 13 } = 99a3​=19,a13​=99

Write an expression for the apparent nnn th term of the sequence. (Assume that nnn begins with 1.) −4,−2,0,2,4- 4 , - 2,0,2,4−4,−2,0,2,4

Use mathematical induction to prove the following for every positive integer nnn . 1+6+36+216+…+6n−1=6n−151 + 6 + 36 + 216 + \ldots + 6 ^ { n - 1 } = \frac { 6 ^ { n } - 1 } { 5 }1+6+36+216+…+6n−1=56n−1​

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

Use mathematical induction to prove the following for every positive integer nnn . 1+3+9+27+…+3n−1=3n−121 + 3 + 9 + 27 + \ldots + 3 ^ { n - 1 } = \frac { 3 ^ { n } - 1 } { 2 }1+3+9+27+…+3n−1=23n−1​

Find the indicated partial sum of the series. ∑i=1∞5(−13)i\sum _ { i = 1 } ^ { \infty } 5 \left( - \frac { 1 } { 3 } \right) ^ { i }∑i=1∞​5(−31​)i fourth partial sum

Find a formula for the nthn t hnth term of the following geometric sequence, then find the 4 th term of the sequence. 9,36,144,…9,36,144 , \ldots9,36,144,…

Simplify the factorial expression. 11!9!\frac { 11 ! } { 9 ! }9!11!​

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 the probability for the experiment of selecting one card from a standard deck of 52 playing cards such that the card is not a red face card.

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

Use mathematical induction to prove the following for every positive integer nnn . ∑i=1n17i(i+1)(i+2)=17n(n+3)4(n+1)(n+2)\sum _ { i = 1 } ^ { n } \frac { 17 } { i ( i + 1 ) ( i + 2 ) } = \frac { 17 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }∑i=1n​i(i+1)(i+2)17​=4(n+1)(n+2)17n(n+3)​

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

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

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

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

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

Find a formula for $a _ { n }$ for the arithmetic sequence. $a _ { 4 } = 10 , a _ { 8 } = 34$

Find the sum of the finite geometric sequence. $\sum _ { n = 1 } ^ { 5 } - \left( - \frac { 1 } { 6 } \right) ^ { n - 1 }$

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

Find a formula for $a _ { n }$ for the arithmetic sequence. $a _ { 3 } = 19 , a _ { 13 } = 99$

Write an expression for the apparent $n$ th term of the sequence. (Assume that $n$ begins with 1.) $- 4 , - 2,0,2,4$

Use mathematical induction to prove the following for every positive integer $n$ . $1 + 6 + 36 + 216 + \ldots + 6 ^ { n - 1 } = \frac { 6 ^ { n } - 1 } { 5 }$

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

Use mathematical induction to prove the following for every positive integer $n$ . $1 + 3 + 9 + 27 + \ldots + 3 ^ { n - 1 } = \frac { 3 ^ { n } - 1 } { 2 }$

Find the indicated partial sum of the series. $\sum _ { i = 1 } ^ { \infty } 5 \left( - \frac { 1 } { 3 } \right) ^ { i }$ fourth partial sum

Find a formula for the $n t h$ term of the following geometric sequence, then find the 4 th term of the sequence. $9,36,144 , \ldots$

Simplify the factorial expression. $\frac { 11 ! } { 9 ! }$

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

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

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 17 } { i ( i + 1 ) ( i + 2 ) } = \frac { 17 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }$

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

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