Exam 8: Sequences, Series, Induction, and Probability

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -A school cafeteria has 5 choices for the main dish, 3 choices of vegetable, 4 choices of beverage, and 3 choices of dessert. How many different meals can be formed if a student chooses one item From each category?

(Multiple Choice)

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Question 181

Simplify the difference quotient: $\frac { f ( x + h ) - f ( x ) } { h }$ - $6 x ^ { 3 } + 6$

(Multiple Choice)

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Question 182

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 face card or a black card.

(Multiple Choice)

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Question 183

Use trial-and-error to determine the smallest integer n for which the given statement is true. - $2 n < 2 ^ { n }$

(Multiple Choice)

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Question 184

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If P(E) = 0, then E is called an event. If P(E) = 1, then E is called a event.

(Short Answer)

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Question 185

Evaluate the expression. - $\frac { ( 6 n ) ! } { ( 6 n + 1 ) ! }$

(Multiple Choice)

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Question 186

Use mathematical induction to prove the given statement for all positive integers n. - $\sum _ { i = 1 } ^ { n } 2 i ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 3 }$

(Short Answer)

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Question 187

Solve the problem. -Suppose a die is rolled followed by the flip of a coin. Find the probability that the outcome is a 3 on the die followed by the coin turning up heads.

(Multiple Choice)

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Question 188

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If two events A and B are mutually exclusive, then $P ( A \cap B ) =$ . As a result, $P ( A \cup B )$ ) can be computed from the formula $P ( A \cup B ) =$ .

(Short Answer)

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Question 189

Solve the problem. -A test has 11 questions. Seven questions are true/false and four are multiple-choice. Each multiple-choice question has 3 possible responses of which exactly one is correct. Find the Probability that a student guesses on each question and gets a perfect score.

(Multiple Choice)

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Question 190

Solve the problem. -An American roulette wheel has 38 slots, numbered 1 through 36, 0, and 00. Eighteen slots are red, 18 are black, and 2 are green. The dealer spins the wheel in one direction and rolls a small ball in The opposite direction until both come to rest. The ball is equally likely to fall in any one of the 38 Slots. For a given spin of the wheel, find the a. The ball lands on a number that is a multiple of 5 (do not include 0 and 00) b. The ball does not land on the number 4.

(Multiple Choice)

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Question 191

Solve the problem. -Liza is a basketball coach and must select 5 players out of 17 players to start a game. In how many ways can she select the 5 players if each player is equally qualified to play each position?

(Multiple Choice)

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Question 192

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Two events A and B are________ if neither event affects the probability of the other.

(Short Answer)

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Question 193

Provide the missing information. -Consider $( a + b ) ^ { n }$ where n is a whole number. What is the degree of each term in the expansion?

(Short Answer)

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Question 194

Use mathematical induction to prove the given statement for all positive integers n. - $\frac { 2 } { 3 } + \frac { 2 } { 9 } + \frac { 2 } { 27 } + \ldots + \frac { 2 } { 3 ^ { n } } = 1 - \left( \frac { 1 } { 3 } \right) ^ { n }$

(Short Answer)

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Question 195

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The nth partial sum $S _ { n }$ of the first n terms of a geometric sequence is a (finite/infinite) geometric series.

(Short Answer)

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Question 196

Solve the problem. -A baseball player with a batting average of 0.304 has a probability of 0.304 of getting a hit for a given time at bat. What is the probability that the player will not get a hit for a given time at bat?

(Multiple Choice)

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Question 197

Write the sum using summation notation. - $\frac { 1 } { 5 } + \frac { 4 } { 3 } + \frac { 27 } { 7 } + \ldots + \frac { n ^ { 3 } } { n + 4 }$

(Multiple Choice)

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Question 198

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -In how many ways can 5 students stack their homework assignments?

(Multiple Choice)

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Question 199

Expand the binomial by using the binomial theorem. - $\left( 3 - y ^ { 3 } \right) ^ { 5 }$

(Multiple Choice)

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Question 200

Showing 181 - 200 of 270

Simplify the difference quotient: $\frac { f ( x + h ) - f ( x ) } { h }$ - $6 x ^ { 3 } + 6$

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 face card or a black card.

Use trial-and-error to determine the smallest integer n for which the given statement is true. - $2 n < 2 ^ { n }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If P(E) = 0, then E is called an event. If P(E) = 1, then E is called a event.

Evaluate the expression. - $\frac { ( 6 n ) ! } { ( 6 n + 1 ) ! }$

Use mathematical induction to prove the given statement for all positive integers n. - $\sum _ { i = 1 } ^ { n } 2 i ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 3 }$

Solve the problem. -Suppose a die is rolled followed by the flip of a coin. Find the probability that the outcome is a 3 on the die followed by the coin turning up heads.

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If two events A and B are mutually exclusive, then $P ( A \cap B ) =$ . As a result, $P ( A \cup B )$ ) can be computed from the formula $P ( A \cup B ) =$ .

Solve the problem. -A test has 11 questions. Seven questions are true/false and four are multiple-choice. Each multiple-choice question has 3 possible responses of which exactly one is correct. Find the Probability that a student guesses on each question and gets a perfect score.

Solve the problem. -Liza is a basketball coach and must select 5 players out of 17 players to start a game. In how many ways can she select the 5 players if each player is equally qualified to play each position?

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Two events A and B are________ if neither event affects the probability of the other.

Provide the missing information. -Consider $( a + b ) ^ { n }$ where n is a whole number. What is the degree of each term in the expansion?

Use mathematical induction to prove the given statement for all positive integers n. - $\frac { 2 } { 3 } + \frac { 2 } { 9 } + \frac { 2 } { 27 } + \ldots + \frac { 2 } { 3 ^ { n } } = 1 - \left( \frac { 1 } { 3 } \right) ^ { n }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The nth partial sum $S _ { n }$ of the first n terms of a geometric sequence is a (finite/infinite) geometric series.

Solve the problem. -A baseball player with a batting average of 0.304 has a probability of 0.304 of getting a hit for a given time at bat. What is the probability that the player will not get a hit for a given time at bat?

Write the sum using summation notation. - $\frac { 1 } { 5 } + \frac { 4 } { 3 } + \frac { 27 } { 7 } + \ldots + \frac { n ^ { 3 } } { n + 4 }$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -In how many ways can 5 students stack their homework assignments?

Expand the binomial by using the binomial theorem. - $\left( 3 - y ^ { 3 } \right) ^ { 5 }$

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

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Exam 8: Sequences, Series, Induction, and Probability

Simplify the difference quotient: f(x+h)−f(x)h\frac { f ( x + h ) - f ( x ) } { h }hf(x+h)−f(x)​ - 6x3+66 x ^ { 3 } + 66x3+6

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 face card or a black card.

Use trial-and-error to determine the smallest integer n for which the given statement is true. - 2n<2n2 n < 2 ^ { n }2n<2n

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If P(E) = 0, then E is called an________ event. If P(E) = 1, then E is called a ________event.

Evaluate the expression. - (6n)!(6n+1)!\frac { ( 6 n ) ! } { ( 6 n + 1 ) ! }(6n+1)!(6n)!​

Use mathematical induction to prove the given statement for all positive integers n. - ∑i=1n2i2=n(n+1)(2n+1)3\sum _ { i = 1 } ^ { n } 2 i ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 3 }∑i=1n​2i2=3n(n+1)(2n+1)​

Solve the problem. -Suppose a die is rolled followed by the flip of a coin. Find the probability that the outcome is a 3 on the die followed by the coin turning up heads.

Solve the problem. -A test has 11 questions. Seven questions are true/false and four are multiple-choice. Each multiple-choice question has 3 possible responses of which exactly one is correct. Find the Probability that a student guesses on each question and gets a perfect score.

Solve the problem. -Liza is a basketball coach and must select 5 players out of 17 players to start a game. In how many ways can she select the 5 players if each player is equally qualified to play each position?

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -Two events A and B are________ if neither event affects the probability of the other.

Provide the missing information. -Consider (a+b)n( a + b ) ^ { n }(a+b)n where n is a whole number. What is the degree of each term in the expansion?

Use mathematical induction to prove the given statement for all positive integers n. - 23+29+227+…+23n=1−(13)n\frac { 2 } { 3 } + \frac { 2 } { 9 } + \frac { 2 } { 27 } + \ldots + \frac { 2 } { 3 ^ { n } } = 1 - \left( \frac { 1 } { 3 } \right) ^ { n }32​+92​+272​+…+3n2​=1−(31​)n

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The nth partial sum SnS _ { n }Sn​ of the first n terms of a geometric sequence is a (finite/infinite) geometric series.

Solve the problem. -A baseball player with a batting average of 0.304 has a probability of 0.304 of getting a hit for a given time at bat. What is the probability that the player will not get a hit for a given time at bat?

Write the sum using summation notation. - 15+43+277+…+n3n+4\frac { 1 } { 5 } + \frac { 4 } { 3 } + \frac { 27 } { 7 } + \ldots + \frac { n ^ { 3 } } { n + 4 }51​+34​+727​+…+n+4n3​

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -In how many ways can 5 students stack their homework assignments?

Expand the binomial by using the binomial theorem. - (3−y3)5\left( 3 - y ^ { 3 } \right) ^ { 5 }(3−y3)5

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

Simplify the difference quotient: $\frac { f ( x + h ) - f ( x ) } { h }$ - $6 x ^ { 3 } + 6$

Use trial-and-error to determine the smallest integer n for which the given statement is true. - $2 n < 2 ^ { n }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -If P(E) = 0, then E is called an event. If P(E) = 1, then E is called a event.

Evaluate the expression. - $\frac { ( 6 n ) ! } { ( 6 n + 1 ) ! }$

Use mathematical induction to prove the given statement for all positive integers n. - $\sum _ { i = 1 } ^ { n } 2 i ^ { 2 } = \frac { n ( n + 1 ) ( 2 n + 1 ) } { 3 }$

Provide the missing information. -Consider $( a + b ) ^ { n }$ where n is a whole number. What is the degree of each term in the expansion?

Use mathematical induction to prove the given statement for all positive integers n. - $\frac { 2 } { 3 } + \frac { 2 } { 9 } + \frac { 2 } { 27 } + \ldots + \frac { 2 } { 3 ^ { n } } = 1 - \left( \frac { 1 } { 3 } \right) ^ { n }$

Write the word or phrase that best completes each statement or answers the question. Provide the missing information. -The nth partial sum $S _ { n }$ of the first n terms of a geometric sequence is a (finite/infinite) geometric series.

Write the sum using summation notation. - $\frac { 1 } { 5 } + \frac { 4 } { 3 } + \frac { 27 } { 7 } + \ldots + \frac { n ^ { 3 } } { n + 4 }$

Expand the binomial by using the binomial theorem. - $\left( 3 - y ^ { 3 } \right) ^ { 5 }$