Question 1

Find the sum of the geometric series, if possible.
-$$\sum _ { n = 1 } ^ { 7 } 6 \left( \frac { 2 } { 3 } \right) ^ { n - 1 }$$

Accepted Answer

A)  35 
B)  18 
C)  $\frac { 4118 } { 243 }$ 
D)  28 
A)  35 
B)  18 
C)  $\frac { 4118 } { 243 }$ 
D)  28

Question 2

Solve the problem.
-A lecture hall has 11 rows of seats. The first row has 15 seats, and each row after that has 4 more seats than the previous row. How many seats are in the last row? How many seats are in the lecture
Hall?

Accepted Answer

A)  last row: 59; total seats: 407 
B)  last row: 55; total seats: 385 
C)  last row: 59; total seats: 209 
D)  last row: 55; total seats: 209 
A)  last row: 59; total seats: 407 
B)  last row: 55; total seats: 385 
C)  last row: 59; total seats: 209 
D)  last row: 55; total seats: 209

Question 3

Use mathematical induction to prove the given statement for all positive integers n.
-$$\sum _ { i = 1 } ^ { n } 2 = 2 n$$

Accepted Answer

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

Question 4

Choose the one alternative that best completes the statement or answers the question.
Determine whether the sequence is geometric. If so, find the value of r.
-$- 2 , \frac { 2 } { 3 } , - \frac { 2 } { 9 } , \frac { 2 } { 27 }$

Accepted Answer

A)  not geometric 
B)  geometric, $r = 3$ 
C)  geometric; $r = - \frac { 1 } { 3 }$ 
D)  geometric, $r = \frac { 1 } { 3 }$ 
A)  not geometric 
B)  geometric, $r = 3$ 
C)  geometric; $r = - \frac { 1 } { 3 }$ 
D)  geometric, $r = \frac { 1 } { 3 }$

Question 5

Solve the problem.
-Twenty batteries are in a drawer. There are 5 dead batteries among the 20. If five batteries are selected at random, determine the number of ways in which 4 good batteries and 1 dead battery can
Be selected.

Accepted Answer

A)  6,825 
B)  32,765 
C)  1,370 
D)  163,800 
A)  6,825 
B)  32,765 
C)  1,370 
D)  163,800

Question 6

Find the sum.
-$$\sum _ { j = 1 } ^ { 6 } \frac { j + 4 } { j }$$

Accepted Answer

A)  $\frac { 15 } { 2 }$ 
B)  45 
C)  $\frac { 54 } { 5 }$ 
D)  $\frac { 79 } { 5 }$ 
A)  $\frac { 15 } { 2 }$ 
B)  45 
C)  $\frac { 54 } { 5 }$ 
D)  $\frac { 79 } { 5 }$

Question 7

Find the indicated term of the arithmetic sequence based on the given information.
-$a _ { 15 } = 67$ and $a _ { 41 } = 223 ;$ Find $a _ { 108 }$

Accepted Answer

A)  631 
B)  671 
C)  625 
D)  637 
A)  631 
B)  671 
C)  625 
D)  637

Question 8

Solve the problem.
-Consider the sample space for a single card drawn from a standard deck. Find the probability that the card drawn is a 6 or a queen.

Accepted Answer

A)  $\frac { 7 } { 52 }$ 
B)  $\frac { 2 } { 13 }$ 
C)  $\frac { 1 } { 26 }$ 
D)  $\frac { 1 } { 52 }$ 
A)  $\frac { 7 } { 52 }$ 
B)  $\frac { 2 } { 13 }$ 
C)  $\frac { 1 } { 26 }$ 
D)  $\frac { 1 } { 52 }$

Question 9

Find the sum of the geometric series, if possible.
-$$\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { n - 1 }$$

Accepted Answer

A)  Sum does not exist 
B)  $\frac { 1 } { 4 }$ 
C)  4 
D)  3 
A)  Sum does not exist 
B)  $\frac { 1 } { 4 }$ 
C)  4 
D)  3

Question 10

Find the sum of the geometric series, if possible.
-$$10 + 2 + \frac { 2 } { 5 } + \frac { 2 } { 25 } + \frac { 2 } { 125 }$$

Accepted Answer

A)  $\frac { 18 } { 125 }$ 
B)  $\frac { 18 } { 155 }$ 
C)  $\frac { 1562 } { 125 }$ 
D)  $\frac { 16 } { 155 }$ 
A)  $\frac { 18 } { 125 }$ 
B)  $\frac { 18 } { 155 }$ 
C)  $\frac { 1562 } { 125 }$ 
D)  $\frac { 16 } { 155 }$

Question 11

Find the sum.
-$$\sum _ { k = 1 } ^ { 5 } ( k + 3 ) ( k + 5 )$$

Accepted Answer

A)  130 
B)  250 
C)  70 
D)  1,200 
A)  130 
B)  250 
C)  70 
D)  1,200

Question 12

Write a recursive formula to define the sequence.
-$$a _ { 1 } = 5 , d = 4$$

Accepted Answer

A)  $a _ { 1 } = 5$ and $a _ { n } = a _ { n - 1 } - 4$ for $n \geq 2$ 
B)  $a _ { 1 } = 5$ and $a _ { n } = a _ { n + 1 } + 4$ for $n \geq 2$ 
C)  $a _ { 1 } = 5$ and $a _ { n } = a _ { n - 1 } + 4$ for $n \geq 2$ 
D)  $a _ { 1 } = 5$ and $a _ { n } = 4 a _ { n - 1 }$ for $n \geq 2$ 
A)  $a _ { 1 } = 5$ and $a _ { n } = a _ { n - 1 } - 4$ for $n \geq 2$ 
B)  $a _ { 1 } = 5$ and $a _ { n } = a _ { n + 1 } + 4$ for $n \geq 2$ 
C)  $a _ { 1 } = 5$ and $a _ { n } = a _ { n - 1 } + 4$ for $n \geq 2$ 
D)  $a _ { 1 } = 5$ and $a _ { n } = 4 a _ { n - 1 }$ for $n \geq 2$

Question 13

Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
-Consider the set of integers from 1 to 23, inclusive. If one number is selected, in how many ways can we obtain a number that is a multiple of 10?

Accepted Answer

A)  0 
B)  3 
C)  13 
D)  2 
A)  0 
B)  3 
C)  13 
D)  2

Question 14

Find the nth term $a _ { n }$ of a sequence whose first four terms are given.
-$$- \frac { 6 } { 12 } , - \frac { 7 } { 24 } , - \frac { 8 } { 36 } , - \frac { 9 } { 48 } , \ldots$$

Accepted Answer

A)  $a _ { n } = \frac { n - 7 } { 12 n }$ 
B)  $a _ { n } = \frac { n + 5 } { 12 n }$ 
C)  $a _ { n } = - \frac { n + 5 } { 12 n }$ 
D)  $a _ { n } = \frac { 7 - n } { 12 n }$ 
A)  $a _ { n } = \frac { n - 7 } { 12 n }$ 
B)  $a _ { n } = \frac { n + 5 } { 12 n }$ 
C)  $a _ { n } = - \frac { n + 5 } { 12 n }$ 
D)  $a _ { n } = \frac { 7 - n } { 12 n }$

Question 15

Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
-Suppose you wish to prove the statement that follows using mathematical induction. $$4 + 9 + 14 + \ldots + ( 5 n - 1 ) = \frac { n } { 2 } ( 5 n + 3 ) , \text { for all positive integers } n$$ Let $S _ { n }$ be the statement $4 + 9 + 14 + \ldots + ( 5 n - 1 ) = \frac { n } { 2 } ( 5 n + 3 )$. Show that $S _ { 1 }$ is true.

Accepted Answer

A)  Since $4 + 9 = 13 = \frac { 2 } { 2 } ( 5 ( 2 ) + 3 ) , S _ { 1 }$ is true. 
B)  Since $( 5 ( 1 ) - 1 ) = 4 , S _ { 1 }$ is true. 
C)  Since $4 + ( 5 ( 1 ) - 1 ) = \frac { 2 } { 2 } ( 4 + 5 ( 1 ) - 1 ) , S _ { 1 }$ is true. 
D)  Since $\frac { 1 } { 2 } ( 5 ( 1 ) + 3 ) = \frac { 1 } { 2 } ( 8 ) = 4 , S _ { 1 }$ is true. 
A)  Since $4 + 9 = 13 = \frac { 2 } { 2 } ( 5 ( 2 ) + 3 ) , S _ { 1 }$ is true. 
B)  Since $( 5 ( 1 ) - 1 ) = 4 , S _ { 1 }$ is true. 
C)  Since $4 + ( 5 ( 1 ) - 1 ) = \frac { 2 } { 2 } ( 4 + 5 ( 1 ) - 1 ) , S _ { 1 }$ is true. 
D)  Since $\frac { 1 } { 2 } ( 5 ( 1 ) + 3 ) = \frac { 1 } { 2 } ( 8 ) = 4 , S _ { 1 }$ is true.

Question 16

Solve the problem.
-Three biology books, 4 math books, and 2 physics books are to be placed on a book shelf where the books in each discipline are grouped together. In how many ways can the books be arranged on the
Book shelf?

Accepted Answer

A)  1,728 
B)  24 
C)  72 
D)  288 
A)  1,728 
B)  24 
C)  72 
D)  288

Question 17

Given $\sum _ { i = 1 } ^ { n } a _ { i }$ , the variable i is called the ________of________ . The value 1 is called the________limit of summation. The value n is called the upper ________of summation.

Accepted Answer

index; sum

Question 18

Write the given rational number as the quotient of two integers in simplest form.
-$0 . \overline { 77 }$

Accepted Answer

A)  $\frac { 77 } { 100 }$ 
B)  $\frac { 77 } { 999 }$ 
C)  $\frac { 7 } { 9 }$ 
D)  $\frac { 77 } { 1000 }$ 
A)  $\frac { 77 } { 100 }$ 
B)  $\frac { 77 } { 999 }$ 
C)  $\frac { 7 } { 9 }$ 
D)  $\frac { 77 } { 1000 }$

Question 19

Use mathematical induction to prove the given statement for all positive integers n.
-$$8 + 11 + 14 + \ldots + ( 3 n + 5 ) = \frac { n } { 2 } ( 3 n + 13 )$$

Accepted Answer

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

Question 20

Solve the problem.
-Consider the sample space when two fair dice are rolled. Determine the probability for the following event. The sum of the numbers on the dice is not 2.

Accepted Answer

A)  $\frac { 17 } { 18 }$ 
B)  $\frac { 35 } { 36 }$ 
C)  $\frac { 1 } { 36 }$ 
D)  $\frac { 1 } { 18 }$ 
A)  $\frac { 17 } { 18 }$ 
B)  $\frac { 35 } { 36 }$ 
C)  $\frac { 1 } { 36 }$ 
D)  $\frac { 1 } { 18 }$

Find the sum of the geometric series, if possible. - $\sum _ { n = 1 } ^ { 7 } 6 \left( \frac { 2 } { 3 } \right) ^ { n - 1 }$

Solve the problem. -A lecture hall has 11 rows of seats. The first row has 15 seats, and each row after that has 4 more seats than the previous row. How many seats are in the last row? How many seats are in the lecture Hall?

Use mathematical induction to prove the given statement for all positive integers n. - $\sum _ { i = 1 } ^ { n } 2 = 2 n$

Choose the one alternative that best completes the statement or answers the question. Determine whether the sequence is geometric. If so, find the value of r. - $- 2 , \frac { 2 } { 3 } , - \frac { 2 } { 9 } , \frac { 2 } { 27 }$

Solve the problem. -Twenty batteries are in a drawer. There are 5 dead batteries among the 20. If five batteries are selected at random, determine the number of ways in which 4 good batteries and 1 dead battery can Be selected.

Find the sum. - $\sum _ { j = 1 } ^ { 6 } \frac { j + 4 } { j }$

Find the indicated term of the arithmetic sequence based on the given information. - $a _ { 15 } = 67$ and $a _ { 41 } = 223 ;$ Find $a _ { 108 }$

Solve the problem. -Consider the sample space for a single card drawn from a standard deck. Find the probability that the card drawn is a 6 or a queen.

Find the sum of the geometric series, if possible. - $\sum _ { n = 1 } ^ { \infty } \left( \frac { 3 } { 4 } \right) ^ { n - 1 }$

Find the sum of the geometric series, if possible. - $10 + 2 + \frac { 2 } { 5 } + \frac { 2 } { 25 } + \frac { 2 } { 125 }$

Find the sum. - $\sum _ { k = 1 } ^ { 5 } ( k + 3 ) ( k + 5 )$

Write a recursive formula to define the sequence. - $a _ { 1 } = 5 , d = 4$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Consider the set of integers from 1 to 23, inclusive. If one number is selected, in how many ways can we obtain a number that is a multiple of 10?

Find the nth term $a _ { n }$ of a sequence whose first four terms are given. - $- \frac { 6 } { 12 } , - \frac { 7 } { 24 } , - \frac { 8 } { 36 } , - \frac { 9 } { 48 } , \ldots$

Solve the problem. -Three biology books, 4 math books, and 2 physics books are to be placed on a book shelf where the books in each discipline are grouped together. In how many ways can the books be arranged on the Book shelf?

Given $\sum _ { i = 1 } ^ { n } a _ { i }$ , the variable i is called the of . The value 1 is called thelimit of summation. The value n is called the upper of summation.

Write the given rational number as the quotient of two integers in simplest form. - $0 . \overline { 77 }$

Use mathematical induction to prove the given statement for all positive integers n. - $8 + 11 + 14 + \ldots + ( 3 n + 5 ) = \frac { n } { 2 } ( 3 n + 13 )$

Solve the problem. -Consider the sample space when two fair dice are rolled. Determine the probability for the following event. The sum of the numbers on the dice is not 2.

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

Find the sum of the geometric series, if possible. - ∑n=176(23)n−1\sum _ { n = 1 } ^ { 7 } 6 \left( \frac { 2 } { 3 } \right) ^ { n - 1 }∑n=17​6(32​)n−1