Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1 Homework, Page 749 Write each sum using summation notation, assuming the suggested pattern continues. 1.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 2 Homework, Page 749 Write each sum using summation notation, assuming the suggested pattern continues. 5.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 3 Homework, Page 749 Find the sum of the arithmetic sequence. 9.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 4 Homework, Page 749 Find the sum of the geometric sequence. 13.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 5 Homework, Page 749 Find the sum of the first n terms of the sequence. 17.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 6 Homework, Page 749 Find the sum of the first n terms of the sequence. 21.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 7 Homework, Page 749 Determine whether the infinite geometric series converges. If it does, find its sum. 25.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 8 Homework, Page 749 Determine whether the infinite geometric series converges. If it does, find its sum. 29.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 9 Homework, Page 749 Expression the rational number as a fraction of integers. 33.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Mr. O’Hara deposits $120 at the end of each month into an account paying 7% annual interest, compounded quarterly. after ten years, the balance is a. This is a geometric series. What is the first term? What is r? The first term is 120 and r = b.Use the sum of the finite series to find the balance.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page If all terms in a series are positive, the series sums to a positive number. Justify your answer. False. The series might diverge.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page The sum of an infinite series whose first term is 3 and second term is 0.75 is a.3.75 b.2.4 c.4 d.5 e.12

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.6 Mathematical Induction

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about The Tower of Hanoi Problem Principle of Mathematical Induction Induction and Deduction … and why The principle of mathematical induction is a valuable technique for proving combinatorial formulas.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide The Tower of Hanoi Solution The minimum number of moves required to move a stack of n washers in a Tower of Hanoi game is 2 n – 1.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Principle of Mathematical Induction Let P n be a statement about the integer n. Then P n is true for all positive integers n provided the following conditions are satisfied: 1.(the anchor) P 1 is true; 2.(inductive step) if P k is true, then P k+1 is true.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Proving a Statement Using Mathematical Induction Use mathematical induction to prove the statement holds for all positive integers.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Induction and Deduction Induction - the process of using evidence from a particular example to draw conclusions about general principles Deduction - the process of using general principles to draw conclusions about specific examples

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework Homework Assignment #35 Read Section 9.7 Page 756, Exercises: 1 – 41 (EOO) Quiz next time

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.7 Statistics and Data (Graphical)

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Quick Review Solutions

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Statistics Displaying Categorical Data Stemplots Frequency Tables Histograms Time Plots … and why Graphical displays of data are increasingly prevalent in professional and popular media. We all need to understand them.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Terminology Sets of data tabulate one or more characteristics, called variables, of each member of a population, called objects or individuals. Categorical variable - identifies an individual as a member of a distinct class, such as male or female, or freshman, sophomore, junior, or senior Quantitative variables have numerical values, such as height or weight

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Leading Causes of Death in the United States in 2001 Cause of DeathNumber of DeathsPercentage Heart Disease700, Cancer553, Stroke163, Other1,018, The causes of death noted above are categorical variables. Source: National Center for Health Statistics, as reported in The World Almanac and Book of Facts 2005.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Bar Chart, Pie Chart, Circle Graph bar graph 3-D pie chart circle chart In the bar graph, the x-axis is labeled by category. Since the categories are unconnected, the bars are not adjacent to one another. The pie and circle charts demonstrate how the data in the individual categories add up to the total

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Stemplots Stemplots - a presentation of numerical data where the data is split into a stem consisting of the initial digit or digits, and a leaf, which is the final digit

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Making a Stemplot Make a stemplot for the given data

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Frequency Table A table showing the frequency with which data in a specified range appear in the total data set. An example would be a list of heights of students with the number of students who are at each height along side HeightGirlsBoys 5’ 8” ’9” ’10”17198

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Histogram A histogram is a bar chart that shows the data of a frequency table in graphic form

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Time Plots Time plots show us how some particular variable changes over time. The plot is developed by plotting the variable on the y-axis and time on the x-axis. Individual data points are connected by straight lines to complete the time plot

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Time Plot

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example, Page Make a back-to-back stemplot comparing the annual home run production of Roger Maris to that of Hank Aaron.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example, Page Draw a histogram of the frequency table in exercise 8.

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example, Page Make a time plot for Mickey Mantle’s annual home run totals.