Homework, Page 756

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1

Homework, Page 756Use mathematical induction to prove that the statement holds for all positive integers.

1. 22 4 6 2n n n

22 4 6 2 2 1 2 1n n n n n 2 3 2n n 2 2 1 1n n n

2 2 1 1n n n

21 1n n


Homework, Page 756State an explicit rule for the nth term of the recursively defined sequence. Use mathematical induction to prove the rule.

5. 1 15; 3n na a a 1 3a 2 1 5a a 3 5 8

3 2 5a a 8 5 13 3 5 1na n 5 2 for 1n n

1 5 1 2 3a 5 2ka k 1 5 2 5ka k 5 5 2k 5 1 2k


Homework, Page 756Write the statement P1, Pk, and Pk+1.

9. 1

1 22n

n nP n

1

1 1 1:1

2P

1:1 2

2k

k kP k

1

1:1 2 1 1

2k

k kP k k k


Homework, Page 756Use mathematical induction to prove the statement for all positive integers.

13. 1 5 9 4 3 2 1n n n

1 :1 1 2 1 1 1P :1 5 9 4 3 2 1kP k k k

1 :1 5 9 4 3 4 1 3 2 1 4 1 3kP k k k k k 22 4 4 3k k k 22 3 1k k

1 2 1k k

1 2 1 1k k



17. 2 2n n1

1 : 2 2 1P 2 2

: 2 2kkP k

11 : 2 2 1k

kP k

: 2 2 2 2 2 from kkk k k P

: 2 2 1k k k

12 2 1k k



21. The sum of the first n terms of a geometric sequence with first term a1

and common ration r ≠ 1 is 1

1

1

nra

r

1

1 1 1

1:

1

rP a a

r

1

1 1 1 1

1:

1

kk

k

rP a a r a r a

r

11 1 1 1 1 1 1

1:

1

kk k k

k

rP a a r a r a r a a r

r

1 1

1 1

1 1

kkr r

a a rr r

1

1

1

1

k k kr r ra

r

1

1

1

1

kra

r


Homework, 56Use the results of 21 – 24 and Example 2 to find the sums.

25. 1 2 3 500

1

From Exercise 22:

2 12n

nS a n d

5002 1 499 1

2

250 501 125,250


Homework, Page 756Use the results of 21 - 24 and Example 2 to find the sums.

29. 341 2 4 8 2

1

From Exercise 21:

1

1

n

n

rS a

r

351 2

11 2

103.436 10

1

103.436 10


Homework, Page 756Use the results of 21 - 24 and Example 2 to find the sum in terms of n.

33. 3

11

n

kk

22 3

1

From Exercise #24:

1

4

n

k

n nk

3

11

n

kk

22 1

4

n nn


Homework, Page 75637. The goal of mathematical induction is to prove that a statement Pn is true for all real numbers n. Justify your answer.

nFalse, mathematical induction is used to prove that P is true for

all positive integers.


Homework, Page 756

I. Mathematical induction

II. The formula for the sum of a finite arithmetic sequence.

III. The formula for the sum of a finite geometric sequence.

A. I only.

B. I and II only.

C. I and III only.

D. II and III only.

E. I, II, and III only.

2

41. Which of the following can be used to prove that

1 3 5 2 1 for all positive integers ?n n n

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

9.7

Statistics and Data (Graphical)


What you’ll learn about

Statistics Displaying Categorical Data Stemplots Frequency Tables Histograms Time Plots

… and whyGraphical displays of data are increasingly prevalent in professional and popular media. We all need to understand them.


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


Leading Causes of Death in the United States in 2001

Cause of Death Number of Deaths Percentage

Heart Disease 700,142 29.0

Cancer 553,768 22.9

Stroke 163,538 6.8

Other 1,018,977 41.3

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.


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


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


Example Making a Stemplot

Make a stemplot for the given data.12.3

23.4

12.0

24.5

23.7

18.7

22.4

19.5

24.5

24.6


Frequency TableA table showing the frequency with which data in a

specified range appear in the total data set is called Frequency Table. An example would be a list of heights of students with the number of students who are at each height listed in adjacent columns.

Height Girls Boys

5’ 8” 125 256

5’9” 51 287

5’10” 17 198


HistogramA histogram is a bar chart that shows the data

of a frequency table in graphic form

0%

5%

10%

15%

20%

Percentage

A A- B B- C C- D F

Grades

Math Department Overall Grade Distribution

Overall


Time PlotsTime 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


Time Plot


Example, Page 7682. Make a back-to-back stemplot comparing the annual home run production of Roger Maris to that of Hank Aaron.


Example, Page 76810. Draw a histogram of the frequency table in exercise 8.


Example, Page 76814. Make a time plot for Mickey Mantle’s annual home run totals.


Homework

Homework Assignment #32Review Section 9.7Page 768, Exercises: 1 - 29 (EOO), 31

Homework, Page 756

Documents

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