Question 1

Find the coefficient $a$ of the term in the expansion of the binomial.
$\begin{array}{ll}
\text { Binomial } & \text { Term } \
(2 x-5 y)^{7} & a x^{2} y^{5}
\end{array}$

Accepted Answer

A)  $a = 12$ 
B)  $a = - 262,500$ 
C)  $a = 14$ 
D)  $a = - 78,125$ 
E)  $\quad a = 2520$ 
A)  $a = 12$ 
B)  $a = - 262,500$ 
C)  $a = 14$ 
D)  $a = - 78,125$ 
E)  $\quad a = 2520$

Question 2

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

Accepted Answer

None

Question 3

Use mathematical induction to prove the following inequality for all $n \geq 2$.
$\frac { 1 } { \sqrt { 19 } } + \frac { 1 } { \sqrt { 38 } } + \frac { 1 } { \sqrt { 57 } } + \ldots + \frac { 1 } { \sqrt { 19 n } } > \frac { \sqrt { n } } { \sqrt { 19 } }$

Accepted Answer

None

Question 4

Determine the sample space for the experiment.
Three marbles are selected from marbles labeled A through D where the marbles are not replaced and the order of selection does not matter.

Accepted Answer

A)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } \}$ 
B)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { ACD } , \mathrm { BCD } \}$ 
C)  $S = \{ \mathrm { ABC } , \mathrm { BCD } , \mathrm { CDA } \}$ 
D)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } , \mathrm { CBA } , \mathrm { CDA } \}$ 
E)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } , \mathrm { CDB } \}$ 
A)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } \}$ 
B)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { ACD } , \mathrm { BCD } \}$ 
C)  $S = \{ \mathrm { ABC } , \mathrm { BCD } , \mathrm { CDA } \}$ 
D)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } , \mathrm { CBA } , \mathrm { CDA } \}$ 
E)  $S = \{ \mathrm { ABC } , \mathrm { ABD } , \mathrm { BCD } , \mathrm { CDB } \}$

Question 5

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

None

Question 6

Use mathematical induction to prove the following inequality for all $n \geq 2$.
$( 3 ) ^ { n + 3 } > \frac { 27 } { 2 } n ( 2 ) ^ { n + 1 }$

Accepted Answer

None

Question 7

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

None

Question 8

Find the coefficient $a$ of the term in the expansion of the binomial.
$\begin{array}{ll}
\text { Binomial } & \text { Term } \
(x-3 y)^{6} & a x^{3} y^{3}
\end{array}$

Accepted Answer

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

Question 9

Find the sum of the following infinite geometric series.
$$- 10 + 9 - \frac { 81 } { 10 } + \frac { 729 } { 100 } - \ldots$$

Accepted Answer

A)  $- \frac { 19 } { 100 }$ 
B)  $- \frac { 1 } { 100 }$ 
C)  $\frac { 19 } { 100 }$ 
D)  $- 10$ 
E)  $- \frac { 100 } { 19 }$ 
A)  $- \frac { 19 } { 100 }$ 
B)  $- \frac { 1 } { 100 }$ 
C)  $\frac { 19 } { 100 }$ 
D)  $- 10$ 
E)  $- \frac { 100 } { 19 }$

Question 10

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

Question 11

Expand the binomial by using Pascal's triangle to determine the coefficients.
$( 5 x - 4 y ) ^ { 6 }$

Accepted Answer

A)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 150000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 150000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
B)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 100000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 96000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
C)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 150000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 128000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
D)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 150000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 96000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
E)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 96000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 96000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
A)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 150000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 150000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
B)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 100000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 96000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
C)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 150000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 128000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
D)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 150000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 96000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$ 
E)  $15625 x ^ { 6 } - 75000 x ^ { 5 } y + 96000 x ^ { 4 } y ^ { 2 } - 160000 x ^ { 3 } y ^ { 3 } + 96000 x ^ { 2 } y ^ { 4 } - 30720 x y ^ { 5 } + 4096 y ^ { 6 }$

Question 12

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

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 13

Determine whether the sequence is geometric. If so, find the common ratio. $- 1,3,7,11 , \ldots$

Accepted Answer

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

Question 14

Use mathematical induction to prove the following for every positive integer $n$.
$\sum _ { i = 1 } ^ { n } \frac { 9 } { i ( i + 1 ) ( i + 2 ) } = \frac { 9 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }$

Accepted Answer

None

Question 15

Write the first five terms of the sequence defined recursively. Use the pattern to write the $n$th 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 16

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 17

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 18

Find a quadratic model for the sequence with the indicated terms.
$$a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12$$

Accepted Answer

A)  $a _ { n } = n ^ { 2 } + 7$ 
B)  $a _ { n } = n ^ { 2 } - 4 n + 7$ 
C)  $a _ { n } = n ^ { 2 } - 7 n + 7$ 
D)  $a _ { n } = n ^ { 2 } - 7 n + 4$ 
E)  $a _ { n } = n ^ { 2 } - n + 7$ 
A)  $a _ { n } = n ^ { 2 } + 7$ 
B)  $a _ { n } = n ^ { 2 } - 4 n + 7$ 
C)  $a _ { n } = n ^ { 2 } - 7 n + 7$ 
D)  $a _ { n } = n ^ { 2 } - 7 n + 4$ 
E)  $a _ { n } = n ^ { 2 } - n + 7$

Question 19

Use mathematical induction to prove the formula for every positive integer $n$. Show all your work.
$$7 + 10 + 13 + 16 + \ldots + ( 3 n + 4 ) = \frac { n } { 2 } ( 3 n + 11 )$$

Accepted Answer

1) When @#LAT-DLM&. The formula is valid for @#LAT-DLM&.
2)

Question 20

Find the number of distinguishable permutations of the group of letters. G, A, U, S, S

Accepted Answer

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

Find the coefficient $a$ of the term in the expansion of the binomial. Binomial Term (2x-5y a

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

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $\frac { 1 } { \sqrt { 19 } } + \frac { 1 } { \sqrt { 38 } } + \frac { 1 } { \sqrt { 57 } } + \ldots + \frac { 1 } { \sqrt { 19 n } } > \frac { \sqrt { n } } { \sqrt { 19 } }$

Determine the sample space for the experiment. Three marbles are selected from marbles labeled A through D where the marbles are not replaced and the order of selection does not matter.

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

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $( 3 ) ^ { n + 3 } > \frac { 27 } { 2 } n ( 2 ) ^ { n + 1 }$

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 coefficient $a$ of the term in the expansion of the binomial. Binomial Term (x-3y a

Find the sum of the following infinite geometric series. $- 10 + 9 - \frac { 81 } { 10 } + \frac { 729 } { 100 } - \ldots$

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

Expand the binomial by using Pascal's triangle to determine the coefficients. $( 5 x - 4 y ) ^ { 6 }$

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

Determine whether the sequence is geometric. If so, find the common ratio. $- 1,3,7,11 , \ldots$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 9 } { i ( i + 1 ) ( i + 2 ) } = \frac { 9 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }$

Write the first five terms of the sequence defined recursively. Use the pattern to write the $n$ th 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 binomial by using Pascal's triangle to determine the coefficients. $( 3 x + 2 y ) ^ { 6 }$

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

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12$

Use mathematical induction to prove the formula for every positive integer $n$ . Show all your work. $7 + 10 + 13 + 16 + \ldots + ( 3 n + 4 ) = \frac { n } { 2 } ( 3 n + 11 )$

Find the number of distinguishable permutations of the group of letters. G, A, U, S, S

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

Find the coefficient aaa of the term in the expansion of the binomial. Binomial Term (2x-5y a

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

Determine the sample space for the experiment. Three marbles are selected from marbles labeled A through D 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≥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

Use mathematical induction to prove the following inequality for all n≥2n \geq 2n≥2 . (3)n+3>272n(2)n+1( 3 ) ^ { n + 3 } > \frac { 27 } { 2 } n ( 2 ) ^ { n + 1 }(3)n+3>227​n(2)n+1

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 coefficient aaa of the term in the expansion of the binomial. Binomial Term (x-3y a

Find the sum of the following infinite geometric series. −10+9−8110+729100−…- 10 + 9 - \frac { 81 } { 10 } + \frac { 729 } { 100 } - \ldots−10+9−1081​+100729​−…

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

Expand the binomial by using Pascal's triangle to determine the coefficients. (5x−4y)6( 5 x - 4 y ) ^ { 6 }(5x−4y)6

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 . f ( x ) = \frac { 5 } { 2 x }hf(x+h)−f(x)​,h=0.f(x)=2x5​

Determine whether the sequence is geometric. If so, find the common ratio. −1,3,7,11,…- 1,3,7,11 , \ldots−1,3,7,11,…

Use mathematical induction to prove the following for every positive integer nnn . ∑i=1n9i(i+1)(i+2)=9n(n+3)4(n+1)(n+2)\sum _ { i = 1 } ^ { n } \frac { 9 } { i ( i + 1 ) ( i + 2 ) } = \frac { 9 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }∑i=1n​i(i+1)(i+2)9​=4(n+1)(n+2)9n(n+3)​

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

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 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

Find a quadratic model for the sequence with the indicated terms. a0=7,a2=3,a5=12a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12a0​=7,a2​=3,a5​=12

Use mathematical induction to prove the formula for every positive integer nnn . Show all your work. 7+10+13+16+…+(3n+4)=n2(3n+11)7 + 10 + 13 + 16 + \ldots + ( 3 n + 4 ) = \frac { n } { 2 } ( 3 n + 11 )7+10+13+16+…+(3n+4)=2n​(3n+11)

Find the number of distinguishable permutations of the group of letters. G, A, U, S, S

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

Find the coefficient $a$ of the term in the expansion of the binomial. Binomial Term (2x-5y a

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

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

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $( 3 ) ^ { n + 3 } > \frac { 27 } { 2 } n ( 2 ) ^ { n + 1 }$

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 coefficient $a$ of the term in the expansion of the binomial. Binomial Term (x-3y a

Find the sum of the following infinite geometric series. $- 10 + 9 - \frac { 81 } { 10 } + \frac { 729 } { 100 } - \ldots$

Expand the binomial by using Pascal's triangle to determine the coefficients. $( 5 x - 4 y ) ^ { 6 }$

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

Determine whether the sequence is geometric. If so, find the common ratio. $- 1,3,7,11 , \ldots$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 9 } { i ( i + 1 ) ( i + 2 ) } = \frac { 9 n ( n + 3 ) } { 4 ( n + 1 ) ( n + 2 ) }$

Write the first five terms of the sequence defined recursively. Use the pattern to write the $n$ th 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 binomial by using Pascal's triangle to determine the coefficients. $( 3 x + 2 y ) ^ { 6 }$

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

Find a quadratic model for the sequence with the indicated terms. $a _ { 0 } = 7 , a _ { 2 } = 3 , a _ { 5 } = 12$

Use mathematical induction to prove the formula for every positive integer $n$ . Show all your work. $7 + 10 + 13 + 16 + \ldots + ( 3 n + 4 ) = \frac { n } { 2 } ( 3 n + 11 )$