Textbook Question
Use mathematical induction to prove that each statement is true for every positive integer n. 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)
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Ch. 8 - Sequences, Induction, and Probability
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 9, Problem 15
In Exercises 11–16, a die is rolled. Find the probability of getting a number greater than 4.
Verified step by step guidance1
Identify the total number of possible outcomes when rolling a standard die. Since a die has 6 faces, the total number of outcomes is \(6\).
Determine the favorable outcomes for the event "getting a number greater than 4." The numbers greater than 4 on a die are \(5\) and \(6\).
Count the number of favorable outcomes. There are \(2\) such numbers: \(5\) and \(6\).
Use the probability formula: \(\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\).
Substitute the values into the formula: \(\text{Probability} = \frac{2}{6}\). This fraction can be simplified if needed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Sample Space
The sample space is the set of all possible outcomes of an experiment. For a single roll of a standard die, the sample space consists of the numbers {1, 2, 3, 4, 5, 6}. Understanding the sample space is essential to determine the total number of possible outcomes.
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Introduction to Probability
Event
An event is a subset of the sample space that includes outcomes of interest. In this problem, the event is rolling a number greater than 4, which includes the outcomes {5, 6}. Identifying the event helps in calculating the probability.
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Probability Calculation
Probability is calculated as the ratio of favorable outcomes to the total number of possible outcomes. Here, the probability of rolling a number greater than 4 is the number of favorable outcomes (2) divided by the total outcomes (6), resulting in 2/6 or 1/3.
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5:37Introduction to Probability
Related Practice
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The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a1=3 and an=4an-1 for n≥2
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Use the formula for nCr to evaluate each expression. 4C4
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