Each Chapter Opener begins with a short introduction to a real data application, which is then highlighted again in one Excursions activity and in the corresponding Excursion Exercises at the end of a section.
This specific Excursion will be denoted by an icon. A section-by-section table of contents is accompanied by a brief summary of the topics that will be covered in the chapter. A section called Problem-Solving Strategies in Chapter 1 introduces students to the inductive and deductive reasoning strategies they will use throughout the text.
The answer is located in a footnote on the same page. Carefully developed Exercise Sets, emphasizing skill building, skill maintenance, concepts, and applications, range from drill and practice exercises to engaging challenge problems. Extension exercises placed near the end of each exercise set present a combination of Critical Thinking, Cooperative Learning, and Exploration exercises to provide further challenge and concept extension. Take Note boxes in the margins alert students to a point requiring special attention or amplify a concept being developed.
Math Matters essay boxes throughout the text help motivate students by demonstrating how and why math is applicable to contemporary, real-life situations. Accompanying graphs and figures help students visually interpret the material. Point of Interest notes provide relevant, contemporary information that helps motivate learning by giving context to concepts being presented. Historical Notes offer additional context by highlighting important mathematical developments or famous individuals who have made major advancements in their fields.
Calculator Notes offer point-of-use tips on solving select problems with various kinds of calculators. Go beyond the answers--see what it takes to get there and improve your grade! This manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. Written responses will vary. For For any natural number n, the two natural current guests move to the room with the next numbers preceding 3n are not multiples higher natural number.
This will allow the new of 3. Pair these two numbers, 3n — 2 and guest to occupy room 1. Now the new guests can be correspondence between the multiples of 3 assigned to the empty rooms in the following set M and the set K of all natural numbers manner.
The first new arrival will get room 1, that are not multiples of 3. First find n. Since the set of integers includes Chapter 2: Sets 33 5. Solving: because 0 is not in the set of natural numbers. Counting numbers begin at 1. Proper subset. All natural numbers are whole numbers, but 0 is not a natural number so 7. Counting numbers and natural numbers The set of counting numbers is not a proper subset of the The sets are equivalent, since each set has set of natural numbers.
The set of real numbers is not a proper subset of The sets are both equal and equivalent. The set contains numbers, not sets. The set of integers includes positive and negative integers. The word small is not precise. The number of subsets of a set with n elements The number of letters is Chapter 2: Sets 35 Use a Venn diagram to represent the survey results. Total the numbers from each region to One possible one-to-one correspondence In the following figure, the line from E that 1.
A one-to-one b. Neither, the sets do not have the same number of elements and are not equal. The R R correspondence with a proper subset of itself, B sets are not equal since integers such as —3 is an infinite set. Chapter 2: Sets 37 b. Both sets have cardinality The number of subsets of a set of n elements is i: 2n. A possible correspondence: b.
Terms of a Sequence An ordered list of numbers such as 5, 14, 27, 44, 65, The numbers in a sequence that are separated by commas are the terms of the sequence. In the above sequence, 5 is the first term, 14 is the second term, 27 is the third term, 44 is the fourth term, and 65 is the fifth term. It is customary to use the subscript notation an to designate the nth term of a sequence. Terms of a Sequence We often construct a difference table, which shows the differences between successive terms of the sequence.
The following table is a difference table for the sequence 2, 5, 8, 11, 14, Each of the numbers in row 1 of the table is the difference between the two closest numbers just above it upper right number minus upper left number.
The differences in row 1 are called the first differences of the sequence. Terms of a Sequence In this case, the first differences are all the same. This prediction might be wrong; however, the pattern shown by the first differences seems to indicate that each successive term is 3 larger than the preceding term.
Terms of a Sequence The following table is a difference table for the sequence 5, 14, 27, 44, 65, In this table, the first differences are not all the same. In such a situation it is often helpful to compute the successive differences of the first differences. These are shown in row 2.
Terms of a Sequence These differences of the first differences are called the second differences. The differences of the second differences are called the third differences. To predict the next term of a sequence, we often look for a pattern in a row of differences. Example 1 — Predict the Next Term of a Sequence Use a difference table to predict the next term in the sequence. Solution: Construct a difference table as shown below. Extending row 3 so that it includes an additional 6 enables us to predict that the next second difference will be Adding 36 to the first difference 89 gives us the next first difference, Adding to the sixth term yields Using the method of extending the difference table, we predict that is the next term in the sequence.
In some cases we can use patterns to predict a formula, called an nth-term formula, that generates the terms of a sequence. In this Book. Beautifully illustrated, the book explores the milestones in elementary mathematics that enliven the understanding of these buildings and combines this Cited by: 2. Mathematical Excursions, 4th edition, by Aufmann, Lockwood, Nation, and Clegg, explores various topics that exemplify the power and diversity of text teaches students that mathematics is a system of understanding our surroundings by connecting concepts to contemporary applications.
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