Question 1

Evaluate the sum.
-$$\sum _ { n = 1 } ^ { 14 } ( 24 - 4 n )$$

Accepted Answer

A)  $- 88$ 
B)  $- 84$ 
C)  $- 92$ 
D)  $- 80$ 
A)  $- 88$ 
B)  $- 84$ 
C)  $- 92$ 
D)  $- 80$

Question 2

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The notation $\bar { E }$ represents the________ of event E. Furthermore, $$P ( E ) + P ( \bar { E } ) =$$________ .

Accepted Answer

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

Question 3

Write the given rational number as the quotient of two integers in simplest form.
-$2.2 \overline { 31 }$

Accepted Answer

A)  $\frac { 2231 } { 1000 }$ 
B)  $\frac { 2209 } { 999 }$ 
C)  $\frac { 2209 } { 1000 }$ 
D)  $\frac { 2209 } { 990 }$ 
A)  $\frac { 2231 } { 1000 }$ 
B)  $\frac { 2209 } { 999 }$ 
C)  $\frac { 2209 } { 1000 }$ 
D)  $\frac { 2209 } { 990 }$

Question 4

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The number of ways that n distinguishable items can be arranged in various order is ________!

Accepted Answer

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

Question 5

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-The sum $S _ { n }$ of the first n terms of a geometric sequence with first term a1 and common ratio r is
given by the formula $S _ { n } =$________

Accepted Answer

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

Question 6

Use trial-and-error to determine the smallest integer n for which the given statement is true.
-$$( n + 1 ) ! > 2 ^ { n }$$

Accepted Answer

A)  $n = 4$ 
B)  $n = 1$ 
C)  $n = 3$ 
D)  $n = 2$ 
A)  $n = 4$ 
B)  $n = 1$ 
C)  $n = 3$ 
D)  $n = 2$

Question 7

Find the sum.
-$\sum _ { j = 1 } ^ { 4 } j ( j + 8 )$

Accepted Answer

A)  110 
B)  36 
C)  48 
D)  120 
A)  110 
B)  36 
C)  48 
D)  120

Question 8

Write the word or phrase that best completes each statement or answers the question.
Provide the missing information.
-Consider a set of n elements of which one element is repeated $$r _ { 1 }$$ times. Then the number of
permutations of the elements of the set is given by $\frac { \square } { \square } \text {. }$

Accepted Answer

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

Question 9

Write the sum using summation notation.
-$$\frac { x ^ { 2 } } { 1 } + \frac { x ^ { 4 } } { 2 } + \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 8 } } { 24 } + \frac { x ^ { 10 } } { 120 }$$

Accepted Answer

A)  $\sum _ { i = 1 } ^ { 5 } \frac { x ^ { 2 i } } { i ! }$ 
B)  $\sum _ { i = 0 } ^ { 4 } \frac { x ^ { 2 i } } { i ! }$ 
C)  $\sum _ { i = 1 } ^ { 5 } \frac { x ^ { 2 i } } { 2 ^ { i - 1 } }$ 
D)  $\sum _ { i = 0 } ^ { 4 } \frac { x ^ { 2 i } } { ( i + 1 ) ! }$ 
A)  $\sum _ { i = 1 } ^ { 5 } \frac { x ^ { 2 i } } { i ! }$ 
B)  $\sum _ { i = 0 } ^ { 4 } \frac { x ^ { 2 i } } { i ! }$ 
C)  $\sum _ { i = 1 } ^ { 5 } \frac { x ^ { 2 i } } { 2 ^ { i - 1 } }$ 
D)  $\sum _ { i = 0 } ^ { 4 } \frac { x ^ { 2 i } } { ( i + 1 ) ! }$

Question 10

Use mathematical induction to prove the given statement for all positive integers n and real numbers x
and y.
-$$( x y ) ^ { n } = x ^ { n } y ^ { n }$$

Accepted Answer

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

Question 11

Use mathematical induction to prove the given statement for all positive integers n.
-$$\left( 1 - \frac { 1 } { 3 } \right) \left( 1 - \frac { 1 } { 4 } \right) \left( 1 - \frac { 1 } { 5 } \right) \ldots \left( 1 - \frac { 1 } { n + 2 } \right) = \frac { 2 } { n + 2 }$$

Accepted Answer

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

Question 12

Find the sum of the geometric series, if possible.
-$$\sum _ { n = 1 } ^ { 9 } 5 ( 2 ) ^ { n }$$

Accepted Answer

A)  5,110 
B)  2,550 
C)  1,022 
D)  2,555 
A)  5,110 
B)  2,550 
C)  1,022 
D)  2,555

Question 13

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 elements in a group from which r elements will be selected
in no particular order. Then each group selected is called a________ of n elements taken ________at a
time.

Accepted Answer

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

Question 14

Find the indicated term.
-$a _ { 1 } = \frac { 1 } { 2 } , d = \frac { 1 } { 5 } ;$ Find $a _ { 11 }$

Accepted Answer

A)  $\frac { 27 } { 10 }$ 
B)  $\frac { 29 } { 10 }$ 
C)  $\frac { 19 } { 10 }$ 
D)  $\frac { 5 } { 2 }$ 
A)  $\frac { 27 } { 10 }$ 
B)  $\frac { 29 } { 10 }$ 
C)  $\frac { 19 } { 10 }$ 
D)  $\frac { 5 } { 2 }$

Question 15

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.
-$$a _ { n } = 9 n - 7$$

Accepted Answer

A)  $2,11,20,29$ 
B)  $2,4,6,8$ 
C)  $- 2 , - 11 , - 20 , - 29$ 
D)  $- 2 , - 4 , - 6 , - 8$ 
A)  $2,11,20,29$ 
B)  $2,4,6,8$ 
C)  $- 2 , - 11 , - 20 , - 29$ 
D)  $- 2 , - 4 , - 6 , - 8$

Question 16

Expand the binomial by using the binomial theorem.
-$$( 4 p - 5 q ) ^ { 5 }$$

Accepted Answer

A)  $1,024 p ^ { 5 } - 6,400 p ^ { 4 } q + 16,000 p ^ { 3 } q ^ { 2 } - 20,000 p ^ { 2 } q ^ { 3 } + 12,500 p q ^ { 4 } - 3,125 q ^ { 5 }$ 
B)  $1,024 p ^ { 5 } - 3,125 q ^ { 5 }$ 
C)  $1,024 p ^ { 5 } - 6,400 p ^ { 4 } q + 8,000 p ^ { 3 } q ^ { 2 } - 10,000 p ^ { 2 } q ^ { 3 } + 12,500 p q ^ { 4 } - 3,125 q ^ { 5 }$ 
D)  $1,024 p ^ { 5 } - 12,500 p ^ { 4 } q + 20,000 p ^ { 3 } q ^ { 2 } - 16,000 p ^ { 2 } q ^ { 3 } + 6,400 p q ^ { 4 } - 3,125 q ^ { 5 }$ 
A)  $1,024 p ^ { 5 } - 6,400 p ^ { 4 } q + 16,000 p ^ { 3 } q ^ { 2 } - 20,000 p ^ { 2 } q ^ { 3 } + 12,500 p q ^ { 4 } - 3,125 q ^ { 5 }$ 
B)  $1,024 p ^ { 5 } - 3,125 q ^ { 5 }$ 
C)  $1,024 p ^ { 5 } - 6,400 p ^ { 4 } q + 8,000 p ^ { 3 } q ^ { 2 } - 10,000 p ^ { 2 } q ^ { 3 } + 12,500 p q ^ { 4 } - 3,125 q ^ { 5 }$ 
D)  $1,024 p ^ { 5 } - 12,500 p ^ { 4 } q + 20,000 p ^ { 3 } q ^ { 2 } - 16,000 p ^ { 2 } q ^ { 3 } + 6,400 p q ^ { 4 } - 3,125 q ^ { 5 }$

Question 17

Find the sum of the geometric series, if possible.
-$$9 + 45 + 225 + 1,125 + 5,625 + \ldots$$

Accepted Answer

A)  $- \frac { 5 } { 8 }$ 
B)  Sum does not exist 
C)  $\frac { 9 } { 4 }$ 
D)  $- \frac { 9 } { 4 }$ 
A)  $- \frac { 5 } { 8 }$ 
B)  Sum does not exist 
C)  $\frac { 9 } { 4 }$ 
D)  $- \frac { 9 } { 4 }$

Question 18

Find the indicated term of the binomial expansion.
-$\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }$; eighth term

Accepted Answer

A)  $- 329,472 x ^ { 15 } y ^ { 16 }$ 
B)  $- 3,432 x ^ { 18 } y ^ { 14 }$ 
C)  $- 2,574 x ^ { 15 } y ^ { 16 }$ 
D)  $- 219,648 x ^ { 18 } y ^ { 14 }$ 
A)  $- 329,472 x ^ { 15 } y ^ { 16 }$ 
B)  $- 3,432 x ^ { 18 } y ^ { 14 }$ 
C)  $- 2,574 x ^ { 15 } y ^ { 16 }$ 
D)  $- 219,648 x ^ { 18 } y ^ { 14 }$

Question 19

Find the nth term $a _ { n }$ of a sequence whose first four terms are given.
-$$1 , - 16,81 , - 256$$

Accepted Answer

A)  $a _ { n } = ( - 1 ) ^ { n + 1 } \left( n ^ { 4 } \right)$ 
B)  $a _ { n } = ( - 1 ) ^ { n } \left( n ^ { 4 } \right)$ 
C)  $a _ { n } = - n ^ { 4 }$ 
D)  $a _ { n } = n ^ { 4 }$ 
A)  $a _ { n } = ( - 1 ) ^ { n + 1 } \left( n ^ { 4 } \right)$ 
B)  $a _ { n } = ( - 1 ) ^ { n } \left( n ^ { 4 } \right)$ 
C)  $a _ { n } = - n ^ { 4 }$ 
D)  $a _ { n } = n ^ { 4 }$

Question 20

Solve the problem.
-The initial swing (one way) of a pendulum makes an arc of 20 in. Each swing (one way) thereafter makes an arc of 98% of the length of the previous swing. What is the total arc length that the
Pendulum travels?

Accepted Answer

A)  1,000 in. 
B)  500 in. 
C)  1,100 in. 
D)  1,400 in. 
A)  1,000 in. 
B)  500 in. 
C)  1,100 in. 
D)  1,400 in.

Evaluate the sum. - $\sum _ { n = 1 } ^ { 14 } ( 24 - 4 n )$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The notation $\bar { E }$ represents the of event E. Furthermore, $P ( E ) + P ( \bar { E } ) =$ .

Write the given rational number as the quotient of two integers in simplest form. - $2.2 \overline { 31 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The number of ways that n distinguishable items can be arranged in various order is ________!

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The sum $S _ { n }$ of the first n terms of a geometric sequence with first term a1 and common ratio r is given by the formula $S _ { n } =$ ________

Use trial-and-error to determine the smallest integer n for which the given statement is true. - $( n + 1 ) ! > 2 ^ { n }$

Find the sum. - $\sum _ { j = 1 } ^ { 4 } j ( j + 8 )$

Write the sum using summation notation. - $\frac { x ^ { 2 } } { 1 } + \frac { x ^ { 4 } } { 2 } + \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 8 } } { 24 } + \frac { x ^ { 10 } } { 120 }$

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. - $( x y ) ^ { n } = x ^ { n } y ^ { n }$

Use mathematical induction to prove the given statement for all positive integers n. - $\left( 1 - \frac { 1 } { 3 } \right) \left( 1 - \frac { 1 } { 4 } \right) \left( 1 - \frac { 1 } { 5 } \right) \ldots \left( 1 - \frac { 1 } { n + 2 } \right) = \frac { 2 } { n + 2 }$

Find the sum of the geometric series, if possible. - $\sum _ { n = 1 } ^ { 9 } 5 ( 2 ) ^ { n }$

Find the indicated term. - $a _ { 1 } = \frac { 1 } { 2 } , d = \frac { 1 } { 5 } ;$ Find $a _ { 11 }$

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. - $a _ { n } = 9 n - 7$

Expand the binomial by using the binomial theorem. - $( 4 p - 5 q ) ^ { 5 }$

Find the sum of the geometric series, if possible. - $9 + 45 + 225 + 1,125 + 5,625 + \ldots$

Find the indicated term of the binomial expansion. - $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }$ ; eighth term

Find the nth term $a _ { n }$ of a sequence whose first four terms are given. - $1 , - 16,81 , - 256$

Solve the problem. -The initial swing (one way) of a pendulum makes an arc of 20 in. Each swing (one way) thereafter makes an arc of 98% of the length of the previous swing. What is the total arc length that the Pendulum travels?

Equations and Inequalities

Functions and Relations

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants and Applications

Analytic Geometry

Review of Prerequisites

Filters

Exam 8: Sequences, Series, Induction, and Probability

Evaluate the sum. - ∑n=114(24−4n)\sum _ { n = 1 } ^ { 14 } ( 24 - 4 n )∑n=114​(24−4n)

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The notation Eˉ\bar { E }Eˉ represents the________ of event E. Furthermore, P(E)+P(Eˉ)=P ( E ) + P ( \bar { E } ) =P(E)+P(Eˉ)= ________ .

Write the given rational number as the quotient of two integers in simplest form. - 2.231‾2.2 \overline { 31 }2.231

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The number of ways that n distinguishable items can be arranged in various order is ________!

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The sum SnS _ { n }Sn​ of the first n terms of a geometric sequence with first term a1 and common ratio r is given by the formula Sn=S _ { n } =Sn​= ________

Use trial-and-error to determine the smallest integer n for which the given statement is true. - (n+1)!>2n( n + 1 ) ! > 2 ^ { n }(n+1)!>2n

Find the sum. - ∑j=14j(j+8)\sum _ { j = 1 } ^ { 4 } j ( j + 8 )∑j=14​j(j+8)

Write the sum using summation notation. - x21+x42+x66+x824+x10120\frac { x ^ { 2 } } { 1 } + \frac { x ^ { 4 } } { 2 } + \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 8 } } { 24 } + \frac { x ^ { 10 } } { 120 }1x2​+2x4​+6x6​+24x8​+120x10​

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. - (xy)n=xnyn( x y ) ^ { n } = x ^ { n } y ^ { n }(xy)n=xnyn

Find the sum of the geometric series, if possible. - ∑n=195(2)n\sum _ { n = 1 } ^ { 9 } 5 ( 2 ) ^ { n }∑n=19​5(2)n

Find the indicated term. - a1=12,d=15;a _ { 1 } = \frac { 1 } { 2 } , d = \frac { 1 } { 5 } ;a1​=21​,d=51​; Find a11a _ { 11 }a11​

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=9n−7a _ { n } = 9 n - 7an​=9n−7

Expand the binomial by using the binomial theorem. - (4p−5q)5( 4 p - 5 q ) ^ { 5 }(4p−5q)5

Find the sum of the geometric series, if possible. - 9+45+225+1,125+5,625+…9 + 45 + 225 + 1,125 + 5,625 + \ldots9+45+225+1,125+5,625+…

Find the indicated term of the binomial expansion. - (x3−2y2)13\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }(x3−2y2)13 ; eighth term

Find the nth term ana _ { n }an​ of a sequence whose first four terms are given. - 1,−16,81,−2561 , - 16,81 , - 2561,−16,81,−256

Solve the problem. -The initial swing (one way) of a pendulum makes an arc of 20 in. Each swing (one way) thereafter makes an arc of 98% of the length of the previous swing. What is the total arc length that the Pendulum travels?

Equations and Inequalities

Functions and Relations

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Systems of Equations and Inequalities

Matrices and Determinants and Applications

Analytic Geometry

Review of Prerequisites

Filters

Evaluate the sum. - $\sum _ { n = 1 } ^ { 14 } ( 24 - 4 n )$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The notation $\bar { E }$ represents the of event E. Furthermore, $P ( E ) + P ( \bar { E } ) =$ .

Write the given rational number as the quotient of two integers in simplest form. - $2.2 \overline { 31 }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The sum $S _ { n }$ of the first n terms of a geometric sequence with first term a1 and common ratio r is given by the formula $S _ { n } =$ ________

Use trial-and-error to determine the smallest integer n for which the given statement is true. - $( n + 1 ) ! > 2 ^ { n }$

Find the sum. - $\sum _ { j = 1 } ^ { 4 } j ( j + 8 )$

Write the sum using summation notation. - $\frac { x ^ { 2 } } { 1 } + \frac { x ^ { 4 } } { 2 } + \frac { x ^ { 6 } } { 6 } + \frac { x ^ { 8 } } { 24 } + \frac { x ^ { 10 } } { 120 }$

Use mathematical induction to prove the given statement for all positive integers n and real numbers x and y. - $( x y ) ^ { n } = x ^ { n } y ^ { n }$

Find the sum of the geometric series, if possible. - $\sum _ { n = 1 } ^ { 9 } 5 ( 2 ) ^ { n }$

Find the indicated term. - $a _ { 1 } = \frac { 1 } { 2 } , d = \frac { 1 } { 5 } ;$ Find $a _ { 11 }$

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. - $a _ { n } = 9 n - 7$

Expand the binomial by using the binomial theorem. - $( 4 p - 5 q ) ^ { 5 }$

Find the sum of the geometric series, if possible. - $9 + 45 + 225 + 1,125 + 5,625 + \ldots$

Find the indicated term of the binomial expansion. - $\left( x ^ { 3 } - 2 y ^ { 2 } \right) ^ { 13 }$ ; eighth term

Find the nth term $a _ { n }$ of a sequence whose first four terms are given. - $1 , - 16,81 , - 256$