Question 1

Use mathematical induction to prove the given statement for all positive integers n.
-$$1 \cdot 4 + 2 \cdot 7 + 3 \cdot 10 + \ldots + n ( 3 n + 1 ) = n ( n + 1 ) ^ { 2 }$$

Accepted Answer

The answer of Use mathematical induction to prove the given...

Question 2

Write the sum using summation notation.
-$$\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }$$

Accepted Answer

A)  $\sum _ { i = 1 } ^ { 5 } \frac { i ! } { x + 3 i }$ 
B)  $\sum _ { i = 1 } ^ { 5 } \frac { i ! } { x + i }$ 
C)  $\sum _ { i = 1 } ^ { 5 } \frac { 2 ^ { i - 1 } } { x + 3 i }$ 
D)  $\sum _ { i = 1 } ^ { 5 } \frac { i ! } { x + 3 }$

Question 3

Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
-A quiz consists of 5 multiple-choice questions, each with five possible responses, and 5 true/false questions. In how many ways can a student answer the questions on the test?

Accepted Answer

A)  3,150 
B)  100,000 
C)  78,125 
D)  3,157

Question 4

One property of summation indicates that $$\sum _ { i = 1 } ^ { n } c =\_\_\_\_. $$

Accepted Answer

The answer of One property of summation indicates that \[\sum...

Question 5

Evaluate the expression.
-$\frac { 9 ! } { 5 ! \cdot 4 ! }$

Accepted Answer

A)  24 
B)  1 
C)  126 
D)  362,736

Question 6

Choose the one alternative that best completes the statement or answers the question.
The nth term of a sequence is given. Write the first four terms of the sequence.
-$$a _ { n } = n \left( n ^ { 3 } - 8 \right)$$

Accepted Answer

A)  $- 7,0,57,224$ 
B)  $- 6,2,22,60$ 
C)  $- 7 , - 4,3,16$ 
D)  $- 6,0,4,8$

Question 7

Evaluate the expression.
-$\left( \begin{array} { l } 8 \ 3 \end{array} \right)$

Accepted Answer

A)  56 
B)  336 
C)  11 
D)  24

Question 8

Use mathematical induction to prove the given statement for all positive integers n.
-$$1 + 4 + 4 ^ { 2 } + 4 ^ { 3 } + \ldots + 4 ^ { n - 1 } = \frac { 4 ^ { n } - 1 } { 3 }$$

Accepted Answer

The answer of Use mathematical induction to prove the given...

Question 9

The expression $a _ { n }$ is called the________ term or general term of a sequence.

Accepted Answer

The answer of The expression $a _ { n }$...

Question 10

Choose the one alternative that best completes the statement or answers the question.
Use Pascal's triangle to write the expansion of the the given expression.
-$$( x - y ) ^ { 4 } \text {. }$$

Accepted Answer

A)  $x ^ { 4 } + 4 x ^ { 3 } y - 6 x ^ { 2 } y ^ { 2 } + 4 x y ^ { 3 } - y ^ { 4 }$ 
B)  $x ^ { 4 } - 4 x ^ { 3 } y + 6 x ^ { 2 } y ^ { 2 } - 4 x y ^ { 3 } + y ^ { 4 }$ 
C)  $x ^ { 4 } - y ^ { 4 }$ 
D)  $x ^ { 4 } - x y + y ^ { 4 }$

Question 11

Solve the problem.
-In how many ways can a manager assign 7 employees at a coffee shop to 7 different tasks? Assume that each employee is assigned to exactly one task.

Accepted Answer

A)  5,040 
B)  823,543 
C)  128 
D)  49

Question 12

Find the sum.
-$$\sum _ { k = 2 } ^ { 5 } \left( \frac { 1 } { 3 } \right) ^ { k }$$

Accepted Answer

A)  $\frac { 121 } { 1024 }$ 
B)  $\frac { 40 } { 243 }$ 
C)  $\frac { 121 } { 243 }$ 
D)  $\frac { 5 } { 128 }$

Question 13

Use mathematical induction to prove the given statement for all positive integers n and real numbers x
and y.
-$$\text { If } 0 < x < 1 \text {, then } x ^ { n } < x ^ { n - 1 } \text {. }$$

Accepted Answer

The answer of Use mathematical induction to prove the given...

Question 14

Write the first five terms of the sequence defined recursively.
-$$b _ { 1 } = 5 ; b _ { n } = 5 b _ { n - 1 } - 3$$

Accepted Answer

A)  $b _ { 1 } = 5 , b _ { 2 } = 25 , b _ { 3 } = 110 , b _ { 4 } = 535 , b _ { 5 } = 2,660$ 
B)  $b _ { 1 } = 5 , b _ { 2 } = 10 , b _ { 3 } = 15 , b _ { 4 } = 20 , b _ { 5 } = 25$ 
C)  $b _ { 1 } = 5 , b _ { 2 } = 22 , b _ { 3 } = 107 , b _ { 4 } = 532 , b _ { 5 } = 2,657$ 
D)  $b _ { 1 } = 5 , b _ { 2 } = 2 , b _ { 3 } = - 1 , b _ { 4 } = - 4 , b _ { 5 } = - 7$

Question 15

Find the sum.
-$$\sum _ { n = 2 } ^ { 4 } \left( n ^ { 2 } + 5 \right)$$

Accepted Answer

A)  35 
B)  50 
C)  59 
D)  74

Question 16

Write the sum using summation notation.
-$$- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }$$

Accepted Answer

A)  $\sum _ { i = 1 } ^ { 5 } \left( ( - 1 ) ^ { i + 1 } \frac { 1 } { 3 i } \right)$ 
B)  $\sum _ { i = 1 } ^ { 5 } \left( ( - 1 ) ^ { i } \frac { 1 } { 3 i } \right)$ 
C)  $\sum _ { i = 1 } ^ { 5 } \left( ( - 1 ) ^ { i } \frac { 1 } { 3 ^ { i } } \right)$ 
D)  $\sum _ { i = 1 } ^ { 5 } \left( ( - 1 ) ^ { i + 1 } \frac { 1 } { 3 ^ { i } } \right)$

Question 17

Find the nth term $a _ { n }$ of a sequence whose first four terms are given.
-$$\frac { 5 } { 16 } , \frac { 10 } { 25 } , \frac { 15 } { 36 } , \frac { 20 } { 49 } , \ldots$$

Accepted Answer

A)  $a _ { n } = \frac { 5 ^ { n } } { ( n + 3 ) ^ { 2 } }$ 
B)  $a _ { n } = \frac { 5 n } { ( n + 2 ) ^ { 2 } }$ 
C)  $a _ { n } = \frac { 5 ^ { n } } { ( n + 2 ) ^ { 2 } }$ 
D)  $a _ { n } = \frac { 5 n } { ( n + 3 ) ^ { 2 } }$

Question 18

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the
formula $S_{n}=$_______

Accepted Answer

The answer of Write the word or phrase that best...

Question 19

Find the number of terms of the finite arithmetic sequence.
-$12,11.7,11.4,11.1 , \ldots , - 2.4$

Accepted Answer

A)  48 
B)  50 
C)  51 
D)  49

Question 20

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Suppose that n represents the number of distinct elements in a group from which r elements will be
chosen in a particular order. Then each arrangement is called a________ of n items taken ________ at atime.

Accepted Answer

The answer of Write the word or phrase that best...

Exam 8: Sequences, Series, Induction, and Probability

Use mathematical induction to prove the given statement for all positive integers n. - $1 \cdot 4 + 2 \cdot 7 + 3 \cdot 10 + \ldots + n ( 3 n + 1 ) = n ( n + 1 ) ^ { 2 }$

Write the sum using summation notation. - $\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -A quiz consists of 5 multiple-choice questions, each with five possible responses, and 5 true/false questions. In how many ways can a student answer the questions on the test?

One property of summation indicates that $\sum _ { i = 1 } ^ { n } c =\_\_\_\_.$

Evaluate the expression. - $\frac { 9 ! } { 5 ! \cdot 4 ! }$

Choose the one alternative that best completes the statement or answers the question. The nth term of a sequence is given. Write the first four terms of the sequence. - $a _ { n } = n \left( n ^ { 3 } - 8 \right)$

Evaluate the expression. - $\left( \begin{array} { l } 8 \\ 3 \end{array} \right)$

Use mathematical induction to prove the given statement for all positive integers n. - $1 + 4 + 4 ^ { 2 } + 4 ^ { 3 } + \ldots + 4 ^ { n - 1 } = \frac { 4 ^ { n } - 1 } { 3 }$

The expression $a _ { n }$ is called the________ term or general term of a sequence.

Choose the one alternative that best completes the statement or answers the question. Use Pascal's triangle to write the expansion of the the given expression. - $( x - y ) ^ { 4 } \text {. }$

Solve the problem. -In how many ways can a manager assign 7 employees at a coffee shop to 7 different tasks? Assume that each employee is assigned to exactly one task.

Find the sum. - $\sum _ { k = 2 } ^ { 5 } \left( \frac { 1 } { 3 } \right) ^ { k }$

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. - $\text { If } 0 < x < 1 \text {, then } x ^ { n } < x ^ { n - 1 } \text {. }$

Write the first five terms of the sequence defined recursively. - $b _ { 1 } = 5 ; b _ { n } = 5 b _ { n - 1 } - 3$

Find the sum. - $\sum _ { n = 2 } ^ { 4 } \left( n ^ { 2 } + 5 \right)$

Write the sum using summation notation. - $- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }$

Find the nth term $a _ { n }$ of a sequence whose first four terms are given. - $\frac { 5 } { 16 } , \frac { 10 } { 25 } , \frac { 15 } { 36 } , \frac { 20 } { 49 } , \ldots$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the formula $S_{n}=$ _______

Find the number of terms of the finite arithmetic sequence. - $12,11.7,11.4,11.1 , \ldots , - 2.4$

Exam 8: Sequences, Series, Induction, and Probability

Use mathematical induction to prove the given statement for all positive integers n. - 1⋅4+2⋅7+3⋅10+…+n(3n+1)=n(n+1)21 \cdot 4 + 2 \cdot 7 + 3 \cdot 10 + \ldots + n ( 3 n + 1 ) = n ( n + 1 ) ^ { 2 }1⋅4+2⋅7+3⋅10+…+n(3n+1)=n(n+1)2

Write the sum using summation notation. - 1x+3+2x+6+6x+9+24x+12+120x+15\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }x+31​+x+62​+x+96​+x+1224​+x+15120​

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -A quiz consists of 5 multiple-choice questions, each with five possible responses, and 5 true/false questions. In how many ways can a student answer the questions on the test?

One property of summation indicates that ∑i=1nc=____.\sum _ { i = 1 } ^ { n } c =\_\_\_\_. ∑i=1n​c=____.

Evaluate the expression. - 9!5!⋅4!\frac { 9 ! } { 5 ! \cdot 4 ! }5!⋅4!9!​

Choose the one alternative that best completes the statement or answers the question. The nth term of a sequence is given. Write the first four terms of the sequence. - an=n(n3−8) a _ { n } = n \left( n ^ { 3 } - 8 \right) an​=n(n3−8)

Evaluate the expression. - (83) \left( \begin{array} { l } 8 \\ 3 \end{array} \right) (83​)

Use mathematical induction to prove the given statement for all positive integers n. - 1+4+42+43+…+4n−1=4n−131 + 4 + 4 ^ { 2 } + 4 ^ { 3 } + \ldots + 4 ^ { n - 1 } = \frac { 4 ^ { n } - 1 } { 3 }1+4+42+43+…+4n−1=34n−1​

The expression ana _ { n }an​ is called the________ term or general term of a sequence.

Choose the one alternative that best completes the statement or answers the question. Use Pascal's triangle to write the expansion of the the given expression. - (x−y) 4. ( x - y ) ^ { 4 } \text {. }(x−y) 4.

Solve the problem. -In how many ways can a manager assign 7 employees at a coffee shop to 7 different tasks? Assume that each employee is assigned to exactly one task.

Find the sum. - ∑k=25(13) k\sum _ { k = 2 } ^ { 5 } \left( \frac { 1 } { 3 } \right) ^ { k }∑k=25​(31​) k

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. - If 0<x<1, then xn<xn−1. \text { If } 0 < x < 1 \text {, then } x ^ { n } < x ^ { n - 1 } \text {. } If 0<x<1, then xn<xn−1.

Write the first five terms of the sequence defined recursively. - b1=5;bn=5bn−1−3b _ { 1 } = 5 ; b _ { n } = 5 b _ { n - 1 } - 3b1​=5;bn​=5bn−1​−3

Find the sum. - ∑n=24(n2+5) \sum _ { n = 2 } ^ { 4 } \left( n ^ { 2 } + 5 \right) ∑n=24​(n2+5)

Write the sum using summation notation. - −13+19−127+181−1243- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }−31​+91​−271​+811​−2431​

Find the nth term ana _ { n }an​ of a sequence whose first four terms are given. - 516,1025,1536,2049,…\frac { 5 } { 16 } , \frac { 10 } { 25 } , \frac { 15 } { 36 } , \frac { 20 } { 49 } , \ldots165​,2510​,3615​,4920​,…

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the formula Sn=S_{n}=Sn​= _______

Find the number of terms of the finite arithmetic sequence. - 12,11.7,11.4,11.1,…,−2.412,11.7,11.4,11.1 , \ldots , - 2.412,11.7,11.4,11.1,…,−2.4

Use mathematical induction to prove the given statement for all positive integers n. - $1 \cdot 4 + 2 \cdot 7 + 3 \cdot 10 + \ldots + n ( 3 n + 1 ) = n ( n + 1 ) ^ { 2 }$

Write the sum using summation notation. - $\frac { 1 } { x + 3 } + \frac { 2 } { x + 6 } + \frac { 6 } { x + 9 } + \frac { 24 } { x + 12 } + \frac { 120 } { x + 15 }$

One property of summation indicates that $\sum _ { i = 1 } ^ { n } c =\_\_\_\_.$

Evaluate the expression. - $\frac { 9 ! } { 5 ! \cdot 4 ! }$

Choose the one alternative that best completes the statement or answers the question. The nth term of a sequence is given. Write the first four terms of the sequence. - $a _ { n } = n \left( n ^ { 3 } - 8 \right)$

Evaluate the expression. - $\left( \begin{array} { l } 8 \\ 3 \end{array} \right)$

Use mathematical induction to prove the given statement for all positive integers n. - $1 + 4 + 4 ^ { 2 } + 4 ^ { 3 } + \ldots + 4 ^ { n - 1 } = \frac { 4 ^ { n } - 1 } { 3 }$

The expression $a _ { n }$ is called the________ term or general term of a sequence.

Choose the one alternative that best completes the statement or answers the question. Use Pascal's triangle to write the expansion of the the given expression. - $( x - y ) ^ { 4 } \text {. }$

Find the sum. - $\sum _ { k = 2 } ^ { 5 } \left( \frac { 1 } { 3 } \right) ^ { k }$

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. - $\text { If } 0 < x < 1 \text {, then } x ^ { n } < x ^ { n - 1 } \text {. }$

Write the first five terms of the sequence defined recursively. - $b _ { 1 } = 5 ; b _ { n } = 5 b _ { n - 1 } - 3$

Find the sum. - $\sum _ { n = 2 } ^ { 4 } \left( n ^ { 2 } + 5 \right)$

Write the sum using summation notation. - $- \frac { 1 } { 3 } + \frac { 1 } { 9 } - \frac { 1 } { 27 } + \frac { 1 } { 81 } - \frac { 1 } { 243 }$

Find the nth term $a _ { n }$ of a sequence whose first four terms are given. - $\frac { 5 } { 16 } , \frac { 10 } { 25 } , \frac { 15 } { 36 } , \frac { 20 } { 49 } , \ldots$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Given an arithmetic sequence with first term a1 and nth term an, the nth partial sum is given by the formula $S_{n}=$ _______

Find the number of terms of the finite arithmetic sequence. - $12,11.7,11.4,11.1 , \ldots , - 2.4$