Question 1

Calculate the binomial coefficient:
$\left( \begin{array} { c } 11 \ 6 \end{array} \right)$

Accepted Answer

A)  332,640 
B)  66 
C)  462 
D)  1 
E)  0 
A)  332,640 
B)  66 
C)  462 
D)  1 
E)  0

Question 2

Find a formula for the $n t h$ term of the following geometric sequence, then find the 4 th term of the sequence.
$7,28,112 , \ldots$

Accepted Answer

A)  $a _ { n } = 7 ( 4 ) ^ { n } ; a _ { 4 } = 448$ 
B)  $a _ { n } = 7 ( 4 ) ^ { n } ; a _ { 4 } = 1792$ 
C)  $a _ { n } = 7 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 1792$ 
D)  $a _ { n } = 7 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 448$ 
E)  $a _ { n } = 7 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 112$ 
A)  $a _ { n } = 7 ( 4 ) ^ { n } ; a _ { 4 } = 448$ 
B)  $a _ { n } = 7 ( 4 ) ^ { n } ; a _ { 4 } = 1792$ 
C)  $a _ { n } = 7 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 1792$ 
D)  $a _ { n } = 7 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 448$ 
E)  $a _ { n } = 7 ( 4 ) ^ { n - 1 } ; a _ { 4 } = 112$

Question 3

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

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

Question 4

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 5

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 6

Find the sum.
$$\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 5 }$$

Accepted Answer

A)  $\frac { 22 } { 63 }$ 
B)  $\frac { 1 } { 14 }$ 
C)  1 
D)  $\frac { 73 } { 168 }$ 
E)  $\frac { 12 } { 11 }$ 
A)  $\frac { 22 } { 63 }$ 
B)  $\frac { 1 } { 14 }$ 
C)  1 
D)  $\frac { 73 } { 168 }$ 
E)  $\frac { 12 } { 11 }$

Question 7

Find the specified $n$th term in the expansion of the binomial. (Write the expansion in descending powers of $x$.)
$$( 2 x + 3 y ) ^ { 6 } , n = 3$$

Accepted Answer

A)  $160 x ^ { 3 } y ^ { 3 }$ 
B)  $2160 x ^ { 4 } y ^ { 2 }$ 
C)  $20 x ^ { 3 } y ^ { 3 }$ 
D)  $729 y ^ { 6 }$ 
E)  $120 x ^ { 3 } y ^ { 3 }$ 
A)  $160 x ^ { 3 } y ^ { 3 }$ 
B)  $2160 x ^ { 4 } y ^ { 2 }$ 
C)  $20 x ^ { 3 } y ^ { 3 }$ 
D)  $729 y ^ { 6 }$ 
E)  $120 x ^ { 3 } y ^ { 3 }$

Question 8

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 9

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 10

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 11

You are given the probability that an event will happen. Find the probability that the event will not happen.
$$P ( E ) = 0.33$$

Accepted Answer

A)  $0.33$ 
B)  $0.67$ 
C)  $0.335$ 
D)  0 
E)  1 
A)  $0.33$ 
B)  $0.67$ 
C)  $0.335$ 
D)  0 
E)  1

Question 12

Find the sum of the integers from - 1 to 27 .

Accepted Answer

A)  378 
B)  28 
C)  754 
D)  377 
E)  756 
A)  378 
B)  28 
C)  754 
D)  377 
E)  756

Question 13

Use mathematical induction to prove the following for every positive integer $n$.
$$\sum _ { i = 1 } ^ { n } 36 i ^ { 5 } = 3 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$$

Accepted Answer

i. Show true for @#LAT-DLM&
\[\begin{aligned}
36 \

Question 14

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 15

Determine whether the sequence is geometric. If so, find the common ratio. A) $3,6,12,24 , \ldots$

Accepted Answer

A)  2 
B)  3 
C)  $\frac { 1 } { 2 }$ 
D)  - 2 
E)  not geometric 
A)  2 
B)  3 
C)  $\frac { 1 } { 2 }$ 
D)  - 2 
E)  not geometric

Question 16

Find the coefficient $a$ of the term in the expansion of the binomial.
$ \begin{array}{ll}\text { Binomial } & \text { Term } \ (4 x+3 y)^{6} & a x^{2} y^{4}\end{array} $

Accepted Answer

A)  $a = 10$ 
B)  $a = 19,440$ 
C)  $\quad a = 12$ 
D)  $\quad a = 729$ 
E)  $\quad a = 360$ 
A)  $a = 10$ 
B)  $a = 19,440$ 
C)  $\quad a = 12$ 
D)  $\quad a = 729$ 
E)  $\quad a = 360$

Question 17

Find the indicated $n$th partial sum of the arithmetic sequence. $3.6,5.7,7.8,9.9 , \ldots , n = 60$

Accepted Answer

A)  6498 
B)  4059 
C)  3933 
D)  $3931.5$ 
E)  $3934.5$ 
A)  6498 
B)  4059 
C)  3933 
D)  $3931.5$ 
E)  $3934.5$

Question 18

Find the indicated $n$th term of the geometric sequence.
5 th term: $a _ { 3 } = - \frac { 3 } { 16 } , a _ { 9 } = - \frac { 3 } { 65,536 }$

Accepted Answer

A)  $\frac { 3 } { 1024 }$ 
B)  $- \frac { 4 } { 81 }$ 
C)  $- \frac { 3 } { 4096 }$ 
D)  $- \frac { 3 } { 256 }$ 
E)  $\frac { 3 } { 64 }$ 
A)  $\frac { 3 } { 1024 }$ 
B)  $- \frac { 4 } { 81 }$ 
C)  $- \frac { 3 } { 4096 }$ 
D)  $- \frac { 3 } { 256 }$ 
E)  $\frac { 3 } { 64 }$

Question 19

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 20

Use mathematical induction to prove the property for all positive integers $n$.
$$\left[ a ^ { n } \right] ^ { 3 } = a ^ { 3 n }$$

Accepted Answer

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

Calculate the binomial coefficient: $\left( \begin{array} { c } 11 \\ 6 \end{array} \right)$

Find a formula for the $n t h$ term of the following geometric sequence, then find the 4 th term of the sequence. $7,28,112 , \ldots$

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

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.

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

Find the sum. $\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 5 }$

Find the specified $n$ th term in the expansion of the binomial. (Write the expansion in descending powers of $x$ .) $( 2 x + 3 y ) ^ { 6 } , n = 3$

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

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.) $- 4 , - 2,0,2,4$

You are given the probability that an event will happen. Find the probability that the event will not happen. $P ( E ) = 0.33$

Find the sum of the integers from - 1 to 27 .

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } 36 i ^ { 5 } = 3 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$

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

Determine whether the sequence is geometric. If so, find the common ratio. A) $3,6,12,24 , \ldots$

Find the coefficient $a$ of the term in the expansion of the binomial. Binomial Term (4x+3y a

Find the indicated $n$ th partial sum of the arithmetic sequence. $3.6,5.7,7.8,9.9 , \ldots , n = 60$

Find the indicated $n$ th term of the geometric sequence. 5 th term: $a _ { 3 } = - \frac { 3 } { 16 } , a _ { 9 } = - \frac { 3 } { 65,536 }$

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

Use mathematical induction to prove the property for all positive integers $n$ . $\left[ a ^ { n } \right] ^ { 3 } = a ^ { 3 n }$

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

Review of Graphs, Equations, and Inequalities

Filters

Exam 8: Sequences, Series, and Probability

Calculate the binomial coefficient: (116)\left( \begin{array} { c } 11 \\ 6 \end{array} \right)(116​)

Find a formula for the nthn t hnth term of the following geometric sequence, then find the 4 th term of the sequence. 7,28,112,…7,28,112 , \ldots7,28,112,…

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)​

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.

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

Find the sum. ∑k=131k2+5\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 5 }∑k=13​k2+51​

Find the specified nnn th term in the expansion of the binomial. (Write the expansion in descending powers of xxx .) (2x+3y)6,n=3( 2 x + 3 y ) ^ { 6 } , n = 3(2x+3y)6,n=3

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

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

You are given the probability that an event will happen. Find the probability that the event will not happen. P(E)=0.33P ( E ) = 0.33P(E)=0.33

Find the sum of the integers from - 1 to 27 .

Use mathematical induction to prove the following for every positive integer nnn . ∑i=1n36i5=3n2(n+1)2(2n2+2n−1)\sum _ { i = 1 } ^ { n } 36 i ^ { 5 } = 3 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)∑i=1n​36i5=3n2(n+1)2(2n2+2n−1)

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

Determine whether the sequence is geometric. If so, find the common ratio. A) 3,6,12,24,…3,6,12,24 , \ldots3,6,12,24,…

Find the coefficient aaa of the term in the expansion of the binomial. Binomial Term (4x+3y a

Find the indicated nnn th partial sum of the arithmetic sequence. 3.6,5.7,7.8,9.9,…,n=603.6,5.7,7.8,9.9 , \ldots , n = 603.6,5.7,7.8,9.9,…,n=60

Find the indicated nnn th term of the geometric sequence. 5 th term: a3=−316,a9=−365,536a _ { 3 } = - \frac { 3 } { 16 } , a _ { 9 } = - \frac { 3 } { 65,536 }a3​=−163​,a9​=−65,5363​

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

Use mathematical induction to prove the property for all positive integers nnn . [an]3=a3n\left[ a ^ { n } \right] ^ { 3 } = a ^ { 3 n }[an]3=a3n

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

Calculate the binomial coefficient: $\left( \begin{array} { c } 11 \\ 6 \end{array} \right)$

Find a formula for the $n t h$ term of the following geometric sequence, then find the 4 th term of the sequence. $7,28,112 , \ldots$

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

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

Find the sum. $\sum _ { k = 1 } ^ { 3 } \frac { 1 } { k ^ { 2 } + 5 }$

Find the specified $n$ th term in the expansion of the binomial. (Write the expansion in descending powers of $x$ .) $( 2 x + 3 y ) ^ { 6 } , n = 3$

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

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.) $- 4 , - 2,0,2,4$

You are given the probability that an event will happen. Find the probability that the event will not happen. $P ( E ) = 0.33$

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } 36 i ^ { 5 } = 3 n ^ { 2 } ( n + 1 ) ^ { 2 } \left( 2 n ^ { 2 } + 2 n - 1 \right)$

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

Determine whether the sequence is geometric. If so, find the common ratio. A) $3,6,12,24 , \ldots$

Find the coefficient $a$ of the term in the expansion of the binomial. Binomial Term (4x+3y a

Find the indicated $n$ th partial sum of the arithmetic sequence. $3.6,5.7,7.8,9.9 , \ldots , n = 60$

Find the indicated $n$ th term of the geometric sequence. 5 th term: $a _ { 3 } = - \frac { 3 } { 16 } , a _ { 9 } = - \frac { 3 } { 65,536 }$

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

Use mathematical induction to prove the property for all positive integers $n$ . $\left[ a ^ { n } \right] ^ { 3 } = a ^ { 3 n }$