Question 1

Find the specified $n$th term in the expansion of the binomial. (Write the expansion in descending powers of $x$.)
$$( 2 x - 3 y ) ^ { 8 } , n = 5$$

Accepted Answer

A)  $448 x ^ { 3 } y ^ { 5 }$ 
B)  $90,720 x ^ { 4 } y ^ { 4 }$ 
C)  $56 x ^ { 3 } y ^ { 5 }$ 
D)  $6561 y ^ { 8 }$ 
E)  $6720 x ^ { 3 } y ^ { 5 }$ 
A)  $448 x ^ { 3 } y ^ { 5 }$ 
B)  $90,720 x ^ { 4 } y ^ { 4 }$ 
C)  $56 x ^ { 3 } y ^ { 5 }$ 
D)  $6561 y ^ { 8 }$ 
E)  $6720 x ^ { 3 } y ^ { 5 }$

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

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 4

Write the first five terms of the sequence. (Assume that $n$ begins with 1.)
$$a _ { n } = ( - 1 ) ^ { n } ( n - 1 ) ( n - 2 )$$

Accepted Answer

A)  $4,0,0 , - 2,6$ 
B)  $0,0 , - 2,6 , - 12$ 
C)  $0,0 , - 2 , - 6 , - 12$ 
D)  $0,2 , - 6,12 , - 20$ 
E)  $0 , - 1,2 , - 3,4$ 
A)  $4,0,0 , - 2,6$ 
B)  $0,0 , - 2,6 , - 12$ 
C)  $0,0 , - 2 , - 6 , - 12$ 
D)  $0,2 , - 6,12 , - 20$ 
E)  $0 , - 1,2 , - 3,4$

Question 5

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 6

Write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of n. (Assume that n begins with 1.) $a _ { 1 } = 21 , a _ { k + 1 } = a _ { k } - 5$

Accepted Answer

A)  $a _ { n } = 26 - 5 n$ 
B)  $a _ { n } = 21 - 5 n$ 
C)  $a _ { n } = 31 - 5 n$ 
D)  $a _ { n } = 26$ 
E)  $\quad a _ { n } = 21 ( n - 1 )$ 
A)  $a _ { n } = 26 - 5 n$ 
B)  $a _ { n } = 21 - 5 n$ 
C)  $a _ { n } = 31 - 5 n$ 
D)  $a _ { n } = 26$ 
E)  $\quad a _ { n } = 21 ( n - 1 )$

Question 7

Expand the following expression in the difference quotient and simplify. $$\frac { f ( x + h ) - f ( x ) } { h } , h \neq 0 . \quad f ( x ) = \frac { 5 } { 2 x }$$

Accepted Answer

A)  $\frac { 5 } { 2 \left( x ^ { 2 } + h \right) }$ 
B)  $- \frac { 5 } { 2 ( x + h ) ^ { 2 } }$ 
C)  $- \frac { 5 } { 2 x ( x + h ) }$ 
D)  $\frac { 5 } { 2 ( x + h ) }$ 
E)  $- \frac { h } { x ( x + h ) }$ 
A)  $\frac { 5 } { 2 \left( x ^ { 2 } + h \right) }$ 
B)  $- \frac { 5 } { 2 ( x + h ) ^ { 2 } }$ 
C)  $- \frac { 5 } { 2 x ( x + h ) }$ 
D)  $\frac { 5 } { 2 ( x + h ) }$ 
E)  $- \frac { h } { x ( x + h ) }$

Question 8

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 9

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 10

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)  $729 x ^ { 6 } + 2916 x ^ { 5 } y + 2160 x ^ { 4 } y ^ { 2 } + 4320 x ^ { 3 } y ^ { 3 } + 2160 x ^ { 2 } y ^ { 4 } + 576 x y ^ { 5 } + 64 y ^ { 6 }$

Question 11

Find the rational number representation of the repeating decimal. $0 . \overline { 157 }$

Accepted Answer

A)  $\frac { 157 } { 9999 }$ 
B)  $\frac { 15.7 } { 999 }$ 
C)  $\frac { 157 } { 9 }$ 
D)  $\frac { 157 } { 999 }$ 
E)  $\frac { 157 } { 99 }$ 
A)  $\frac { 157 } { 9999 }$ 
B)  $\frac { 15.7 } { 999 }$ 
C)  $\frac { 157 } { 9 }$ 
D)  $\frac { 157 } { 999 }$ 
E)  $\frac { 157 } { 99 }$

Question 12

Find the sum using the formulas for the sums of powers of integers.
$$\sum _ { n = 1 } ^ { 12 } n ^ { 3 }$$

Accepted Answer

A)  4356 
B)  12,168 
C)  1728 
D)  650 
E)  6084 
A)  4356 
B)  12,168 
C)  1728 
D)  650 
E)  6084

Question 13

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 14

Write the first five terms of the arithmetic sequence.
$$a _ { 5 } = - 29 , a _ { 10 } = - 59$$

Accepted Answer

A)  $- 5,1,7,13,19$ 
B)  $- 11 , - 17 , - 23 , - 29 , - 35$ 
C)  $- 5 , - 11 , - 17 , - 23 , - 29$ 
D)  $- 5,30 , - 180,1080 , - 6480$ 
E)  $- 5 , - 11 , - 16 , - 21 , - 26$ 
A)  $- 5,1,7,13,19$ 
B)  $- 11 , - 17 , - 23 , - 29 , - 35$ 
C)  $- 5 , - 11 , - 17 , - 23 , - 29$ 
D)  $- 5,30 , - 180,1080 , - 6480$ 
E)  $- 5 , - 11 , - 16 , - 21 , - 26$

Question 15

Find the coefficient $a$ of the term in the expansion of the binomial.

Accepted Answer

A)  $a = 9$ 
B)  $a = - 540$ 
C)  $a = 18$ 
D)  $a = 729$ 
E)  $ a = 120$ 
A)  $a = 9$ 
B)  $a = - 540$ 
C)  $a = 18$ 
D)  $a = 729$ 
E)  $ a = 120$

Question 16

Determine the sample space for the experiment.
Two marbles are selected from marbles labeled A through $\mathrm { E }$ where the marbles are not replaced and the order of selection does not matter.

Accepted Answer

A)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { CD } \}$ 
B)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { CD } , \mathrm { CE } , \mathrm { DE } \}$ 
C)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { DE } \}$ 
D)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { CD } , \mathrm { CE } , \mathrm { DE } , \mathrm { DA } , \mathrm { DB } \}$ 
E)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { CA } , \mathrm { CB } , \mathrm { CD } , \mathrm { CE } , \mathrm { DE } \}$ 
A)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { CD } \}$ 
B)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { CD } , \mathrm { CE } , \mathrm { DE } \}$ 
C)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { DE } \}$ 
D)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { CD } , \mathrm { CE } , \mathrm { DE } , \mathrm { DA } , \mathrm { DB } \}$ 
E)  $S = \{ \mathrm { AB } , \mathrm { AC } , \mathrm { AD } , \mathrm { AE } , \mathrm { BC } , \mathrm { BD } , \mathrm { BE } , \mathrm { CA } , \mathrm { CB } , \mathrm { CD } , \mathrm { CE } , \mathrm { DE } \}$

Question 17

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 18

Use mathematical induction to prove the following inequality for all $n \geq 1$.
$$15 ^ { n } \geq 14 n$$

Accepted Answer

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

Question 19

Write the first five terms of the sequence. (Assume that n begins with 0.) $$a _ { n } = \frac { 3 ^ { n } } { ( n + 1 ) ! }$$

Accepted Answer

A)  $1 , \frac { 3 } { 2 } , \frac { 3 } { 2 } , \frac { 9 } { 8 } , \frac { 27 } { 40 }$ 
B)  $\frac { 3 } { 2 } , \frac { 9 } { 8 } , \frac { 27 } { 40 } , \frac { 27 } { 80 } , \frac { 81 } { 560 }$ 
C)  $0 , \frac { 3 } { 2 } , \frac { 9 } { 8 } , \frac { 27 } { 40 } , \frac { 27 } { 80 }$ 
D)  $3 , \frac { 3 } { 8 } , \frac { 9 } { 40 } , \frac { 9 } { 80 } , \frac { 27 } { 560 }$ 
E)  $3 , \frac { 3 } { 2 } , \frac { 3 } { 8 } , \frac { 3 } { 40 } , \frac { 1 } { 80 }$ 
A)  $1 , \frac { 3 } { 2 } , \frac { 3 } { 2 } , \frac { 9 } { 8 } , \frac { 27 } { 40 }$ 
B)  $\frac { 3 } { 2 } , \frac { 9 } { 8 } , \frac { 27 } { 40 } , \frac { 27 } { 80 } , \frac { 81 } { 560 }$ 
C)  $0 , \frac { 3 } { 2 } , \frac { 9 } { 8 } , \frac { 27 } { 40 } , \frac { 27 } { 80 }$ 
D)  $3 , \frac { 3 } { 8 } , \frac { 9 } { 40 } , \frac { 9 } { 80 } , \frac { 27 } { 560 }$ 
E)  $3 , \frac { 3 } { 2 } , \frac { 3 } { 8 } , \frac { 3 } { 40 } , \frac { 1 } { 80 }$

Question 20

Write the first five terms of the geometric sequence.
$$a _ { 1 } = - 5 , r = - \frac { 1 } { 9 }$$

Accepted Answer

A)  $- 5 , - 14 , - 23 , - 32 , - 41$ 
B)  $- 5,45 , - 405,3645 , - 32,805$ 
C)  $\frac { 5 } { 9 } , - \frac { 5 } { 81 } , \frac { 5 } { 729 } , - \frac { 5 } { 6561 } , \frac { 5 } { 59,049 }$ 
D)  $1 , - 5 , \frac { 5 } { 9 } , - \frac { 5 } { 81 } , \frac { 5 } { 729 }$ 
E)  $- 5 , \frac { 5 } { 9 } , - \frac { 5 } { 81 } , \frac { 5 } { 729 } , - \frac { 5 } { 6561 }$ 
A)  $- 5 , - 14 , - 23 , - 32 , - 41$ 
B)  $- 5,45 , - 405,3645 , - 32,805$ 
C)  $\frac { 5 } { 9 } , - \frac { 5 } { 81 } , \frac { 5 } { 729 } , - \frac { 5 } { 6561 } , \frac { 5 } { 59,049 }$ 
D)  $1 , - 5 , \frac { 5 } { 9 } , - \frac { 5 } { 81 } , \frac { 5 } { 729 }$ 
E)  $- 5 , \frac { 5 } { 9 } , - \frac { 5 } { 81 } , \frac { 5 } { 729 } , - \frac { 5 } { 6561 }$

Find the specified $n$ th term in the expansion of the binomial. (Write the expansion in descending powers of $x$ .) $( 2 x - 3 y ) ^ { 8 } , n = 5$

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

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

Write the first five terms of the sequence. (Assume that $n$ begins with 1.) $a _ { n } = ( - 1 ) ^ { n } ( n - 1 ) ( n - 2 )$

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

Write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of n. (Assume that n begins with 1.) $a _ { 1 } = 21 , a _ { k + 1 } = a _ { k } - 5$

Expand the following expression in the difference quotient and simplify. $\frac { f ( x + h ) - f ( x ) } { h } , h \neq 0 . \quad f ( x ) = \frac { 5 } { 2 x }$

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

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

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

Find the rational number representation of the repeating decimal. $0 . \overline { 157 }$

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 12 } n ^ { 3 }$

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

Write the first five terms of the arithmetic sequence. $a _ { 5 } = - 29 , a _ { 10 } = - 59$

Find the coefficient $a$ of the term in the expansion of the binomial.

Determine the sample space for the experiment. Two marbles are selected from marbles labeled A through $\mathrm { E }$ where the marbles are not replaced and the order of selection does not matter.

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 .

Use mathematical induction to prove the following inequality for all $n \geq 1$ . $15 ^ { n } \geq 14 n$

Write the first five terms of the sequence. (Assume that n begins with 0.) $a _ { n } = \frac { 3 ^ { n } } { ( n + 1 ) ! }$

Write the first five terms of the geometric sequence. $a _ { 1 } = - 5 , r = - \frac { 1 } { 9 }$

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 specified nnn th term in the expansion of the binomial. (Write the expansion in descending powers of xxx .) (2x−3y)8,n=5( 2 x - 3 y ) ^ { 8 } , n = 5(2x−3y)8,n=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)

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,…

Write the first five terms of the sequence. (Assume that nnn begins with 1.) an=(−1)n(n−1)(n−2)a _ { n } = ( - 1 ) ^ { n } ( n - 1 ) ( n - 2 )an​=(−1)n(n−1)(n−2)

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

Write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of n. (Assume that n begins with 1.) a1=21,ak+1=ak−5a _ { 1 } = 21 , a _ { k + 1 } = a _ { k } - 5a1​=21,ak+1​=ak​−5

Expand the following expression in the difference quotient and simplify. f(x+h)−f(x)h,h≠0.f(x)=52x\frac { f ( x + h ) - f ( x ) } { h } , h \neq 0 . \quad f ( x ) = \frac { 5 } { 2 x }hf(x+h)−f(x)​,h=0.f(x)=2x5​

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

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,…

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

Find the rational number representation of the repeating decimal. 0.157‾0 . \overline { 157 }0.157

Find the sum using the formulas for the sums of powers of integers. ∑n=112n3\sum _ { n = 1 } ^ { 12 } n ^ { 3 }∑n=112​n3

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

Write the first five terms of the arithmetic sequence. a5=−29,a10=−59a _ { 5 } = - 29 , a _ { 10 } = - 59a5​=−29,a10​=−59

Find the coefficient aaa of the term in the expansion of the binomial.

Determine the sample space for the experiment. Two marbles are selected from marbles labeled A through E\mathrm { E }E where the marbles are not replaced and the order of selection does not matter.

Use mathematical induction to prove the following inequality for all n≥1n \geq 1n≥1 . 15n≥14n15 ^ { n } \geq 14 n15n≥14n

Write the first five terms of the sequence. (Assume that n begins with 0.) an=3n(n+1)!a _ { n } = \frac { 3 ^ { n } } { ( n + 1 ) ! }an​=(n+1)!3n​

Write the first five terms of the geometric sequence. a1=−5,r=−19a _ { 1 } = - 5 , r = - \frac { 1 } { 9 }a1​=−5,r=−91​

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 specified $n$ th term in the expansion of the binomial. (Write the expansion in descending powers of $x$ .) $( 2 x - 3 y ) ^ { 8 } , n = 5$

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

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

Write the first five terms of the sequence. (Assume that $n$ begins with 1.) $a _ { n } = ( - 1 ) ^ { n } ( n - 1 ) ( n - 2 )$

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

Write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of n. (Assume that n begins with 1.) $a _ { 1 } = 21 , a _ { k + 1 } = a _ { k } - 5$

Expand the following expression in the difference quotient and simplify. $\frac { f ( x + h ) - f ( x ) } { h } , h \neq 0 . \quad f ( x ) = \frac { 5 } { 2 x }$

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

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

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

Find the rational number representation of the repeating decimal. $0 . \overline { 157 }$

Find the sum using the formulas for the sums of powers of integers. $\sum _ { n = 1 } ^ { 12 } n ^ { 3 }$

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

Write the first five terms of the arithmetic sequence. $a _ { 5 } = - 29 , a _ { 10 } = - 59$

Find the coefficient $a$ of the term in the expansion of the binomial.

Determine the sample space for the experiment. Two marbles are selected from marbles labeled A through $\mathrm { E }$ where the marbles are not replaced and the order of selection does not matter.

Use mathematical induction to prove the following inequality for all $n \geq 1$ . $15 ^ { n } \geq 14 n$

Write the first five terms of the sequence. (Assume that n begins with 0.) $a _ { n } = \frac { 3 ^ { n } } { ( n + 1 ) ! }$

Write the first five terms of the geometric sequence. $a _ { 1 } = - 5 , r = - \frac { 1 } { 9 }$