Question 1

Use mathematical induction to prove the property for all positive integers $n$.
$$\left[ a ^ { n } \right] ^ { 4 } = a ^ { 4 n }$$

Accepted Answer

1) $n = 1$ :
$$\begin{array} { r } 
{ \left[ a ^ { 1 } \right] ^ { 4 } \stackrel { ? } { = } a ^ { 4 \cdot 1 } } \
{ [ a ] ^ { 4 } \stackrel { ? } { = } a ^ { 4 } } \
a ^ { 4 } = a ^ { 4 }
\end{array}$$
The statement is true for $n = 1$.
2) Assume $\left[ a ^ { k } \right] ^ { 4 } = a ^ { 4 k }$. Then,
$$\begin{aligned}
( a ) ^ { 4 } \left[ a ^ { k } \right] ^ { 4 } & = ( a ) ^ { 4 } \left( a ^ { 4 k } \right) \
{ \left[ a \cdot a ^ { k } \right] ^ { 4 } } & = a ^ { 4 + 4 k } \
{ \left[ a ^ { k + 1 } \right] ^ { 4 } } & = a ^ { 4 ( k + 1 ) }
\end{aligned}$$
By mathematical induction, the property is true for all positive values of $n$.

Question 2

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing four green, six yellow, and five red marbles such that both marbles are yellow.

Accepted Answer

A)  $\frac { 1 } { 7 }$ 
B)  $\frac { 4 } { 25 }$ 
C)  $\frac { 6 } { 35 }$ 
D)  $\frac { 1 } { 3 }$ 
E)  $\frac { 2 } { 3 }$ 
A)  $\frac { 1 } { 7 }$ 
B)  $\frac { 4 } { 25 }$ 
C)  $\frac { 6 } { 35 }$ 
D)  $\frac { 1 } { 3 }$ 
E)  $\frac { 2 } { 3 }$ A

Question 3

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)  $\frac { 3 } { 64 }$ D

Question 4

Determine the sample space for the experiment.
Four coins are flipped and the number of heads observed is recorded.

Accepted Answer

A)  $S = \{ 1,2,3,4 \}$ 
B)  $S = \{ 4 \}$ 
C)  $S = \{ 0,1,2,3,4 \}$ 
D)  $S = \{ \mathrm { HT } , \mathrm { TH } \}$ 
E)  $S = \{ \mathrm { HHHH } \}$ 
A)  $S = \{ 1,2,3,4 \}$ 
B)  $S = \{ 4 \}$ 
C)  $S = \{ 0,1,2,3,4 \}$ 
D)  $S = \{ \mathrm { HT } , \mathrm { TH } \}$ 
E)  $S = \{ \mathrm { HHHH } \}$

Question 5

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 6

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

Accepted Answer

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

Question 7

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 8

Evaluate: ${ } _ { 8 } P _ { 5 }$

Accepted Answer

A)  56 
B)  336 
C)  6720 
D)  40 
E)  undefined 
A)  56 
B)  336 
C)  6720 
D)  40 
E)  undefined

Question 9

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 10

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 11

Use mathematical induction to prove the following for every positive integer $n$.
$$\sum _ { i = 1 } ^ { n } \frac { 15 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 15 n } { 2 n + 1 }$$

Accepted Answer

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

Question 12

Use mathematical induction to prove the following inequality for all $n \geq 2$.
$$\frac { 1 } { \sqrt { 23 } } + \frac { 1 } { \sqrt { 46 } } + \frac { 1 } { \sqrt { 69 } } + \ldots + \frac { 1 } { \sqrt { 23 n } } > \frac { \sqrt { n } } { \sqrt { 23 } }$$

Accepted Answer

i. Show true for @#LAT-DLM&
@#LAT-DLM& @#LAT-DLM& @#LAT-DLM& @#LAT-DLM& ii. Assume true fo

Question 13

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 14

Use the Binomial Theorem to expand the following complex number. Write your answer in standard form.
$$\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$$

Accepted Answer

A)  $\frac { 36 } { 343 } - \frac { 20 \sqrt { 7 } } { 343 } i$ 
B)  $\frac { 36 } { 343 } + \frac { 20 \sqrt { 7 } } { 49 } i$ 
C)  $- \frac { 36 } { 49 } - \frac { 20 \sqrt { 7 } } { 343 } i$ 
D)  $- \frac { 27 } { 343 } + \frac { \sqrt { 7 } } { 49 } i$ 
E)  $- \frac { 27 } { 343 } - \frac { \sqrt { 7 } } { 49 } i$ 
A)  $\frac { 36 } { 343 } - \frac { 20 \sqrt { 7 } } { 343 } i$ 
B)  $\frac { 36 } { 343 } + \frac { 20 \sqrt { 7 } } { 49 } i$ 
C)  $- \frac { 36 } { 49 } - \frac { 20 \sqrt { 7 } } { 343 } i$ 
D)  $- \frac { 27 } { 343 } + \frac { \sqrt { 7 } } { 49 } i$ 
E)  $- \frac { 27 } { 343 } - \frac { \sqrt { 7 } } { 49 } i$

Question 15

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 16

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 17

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 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

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

Question 19

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 20

Find the sum of the following infinite geometric series.
$$- 14 + 13 - \frac { 169 } { 14 } + \frac { 2197 } { 196 } - \ldots$$

Accepted Answer

A)  $- \frac { 27 } { 196 }$ 
B)  $- \frac { 1 } { 196 }$ 
C)  $\frac { 27 } { 196 }$ 
D)  $- 14$ 
E)  $- \frac { 196 } { 27 }$ 
A)  $- \frac { 27 } { 196 }$ 
B)  $- \frac { 1 } { 196 }$ 
C)  $\frac { 27 } { 196 }$ 
D)  $- 14$ 
E)  $- \frac { 196 } { 27 }$

Use mathematical induction to prove the property for all positive integers $n$ . $\left[ a ^ { n } \right] ^ { 4 } = a ^ { 4 n }$

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing four green, six yellow, and five red marbles such that both marbles are yellow.

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

Determine the sample space for the experiment. Four coins are flipped and the number of heads observed is recorded.

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

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

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

Evaluate: ${ } _ { 8 } P _ { 5 }$

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

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

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 15 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 15 n } { 2 n + 1 }$

Use mathematical induction to prove the following inequality for all $n \geq 2$ . $\frac { 1 } { \sqrt { 23 } } + \frac { 1 } { \sqrt { 46 } } + \frac { 1 } { \sqrt { 69 } } + \ldots + \frac { 1 } { \sqrt { 23 n } } > \frac { \sqrt { n } } { \sqrt { 23 } }$

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 the Binomial Theorem to expand the following complex number. Write your answer in standard form. $\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$

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

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 of the integers from $- 1$ to 27 .

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

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

Find the sum of the following infinite geometric series. $- 14 + 13 - \frac { 169 } { 14 } + \frac { 2197 } { 196 } - \ldots$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Additional Topics in Trigonometry

Linear Systems and Matrices

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Analytic Geometry in Three Dimensions

Calculus Practice Problems

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Exam 8: Sequences, Series, and Probability

Use mathematical induction to prove the property for all positive integers nnn . [an]4=a4n\left[ a ^ { n } \right] ^ { 4 } = a ^ { 4 n }[an]4=a4n

Find the probability for the experiment of drawing two marbles (without replacement) from a bag containing four green, six yellow, and five red marbles such that both marbles are yellow.

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​

Determine the sample space for the experiment. Four coins are flipped and the number of heads observed is recorded.

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

Use mathematical induction to prove the following for every positive integer nnn . ∑i=1n8(2i−1)(2i+1)=8n2n+1\sum _ { i = 1 } ^ { n } \frac { 8 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 8 n } { 2 n + 1 }∑i=1n​(2i−1)(2i+1)8​=2n+18n​

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,…

Evaluate: 8P5{ } _ { 8 } P _ { 5 }8​P5​

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

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

Use mathematical induction to prove the following for every positive integer nnn . ∑i=1n15(2i−1)(2i+1)=15n2n+1\sum _ { i = 1 } ^ { n } \frac { 15 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 15 n } { 2 n + 1 }∑i=1n​(2i−1)(2i+1)15​=2n+115n​

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 the Binomial Theorem to expand the following complex number. Write your answer in standard form. (−37−77i)3\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }(−73​−77​​i)3

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

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 of the integers from −1- 1−1 to 27 .

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)​

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

Find the sum of the following infinite geometric series. −14+13−16914+2197196−…- 14 + 13 - \frac { 169 } { 14 } + \frac { 2197 } { 196 } - \ldots−14+13−14169​+1962197​−…

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 mathematical induction to prove the property for all positive integers $n$ . $\left[ a ^ { n } \right] ^ { 4 } = a ^ { 4 n }$

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

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

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

Evaluate: ${ } _ { 8 } P _ { 5 }$

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

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

Use mathematical induction to prove the following for every positive integer $n$ . $\sum _ { i = 1 } ^ { n } \frac { 15 } { ( 2 i - 1 ) ( 2 i + 1 ) } = \frac { 15 n } { 2 n + 1 }$

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

Use the Binomial Theorem to expand the following complex number. Write your answer in standard form. $\left( - \frac { 3 } { 7 } - \frac { \sqrt { 7 } } { 7 } i \right) ^ { 3 }$

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

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 of the integers from $- 1$ to 27 .

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

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

Find the sum of the following infinite geometric series. $- 14 + 13 - \frac { 169 } { 14 } + \frac { 2197 } { 196 } - \ldots$