homework, page 728

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

Homework, Page 728A red die and a green die have been rolled. What is the probability of the event?

1. The sum is nine.

1

2

3

4

2 3 4 5 6 7

3 4 5 6 7 8

4 5 6 7 8 9

5 6 7 8 9 10

6 7 8 9 10 11

7 8

1 2 3

9 10

5

6

4 5 6

11 12

49

36P

1

9


Homework, Page 728A red die and a green die have been rolled. What is the probability of the event?

5. Both numbers are even.

9Both even

36P 1

2

3

4

2 3 4 5 6 7

3 4 5 6 7 8

4 5 6 7 8 9

5 6 7 8 9 10

6 7 8 9 10 11

7 8

1 2 3

9 10

5

6

4 5 6

11 12

1

4

or

1 1Both even

2 2P

1

4


Homework, Page 7289. Alrik’s gerbil cage has four compartments, A, B, C, and D. After careful observation, he estimates the proportion of time spent in each compartment and constructs the following table.

(a) Is this a valid probability function? Explain.No, this is not a valid probability function because the proportions total more than 1.(b) Is there a problem with Alrik’s reasoning? Explain.The table has the gerbil spending 110% of its time in the compartments. This cannot be, as it only has 100% of the time to apportion to the various compartments.

Compartment A B C D

Proportion 0.25 0.20 0.35 0.30


Homework, Page 728Candy is produced in the following proportions:

13. A single candy is randomly selected from a newly-opened bag. What is the probability that the candy is red?

Red 0.2P

Color Brown Red Yellow Green Orange Tan

Proportion 0.3 0.2 0.2 0.1 0.1 0.1

1

5 or 20%


Homework, Page 728A peanut candy is produced in the following proportions:

17. A single peanut candy is randomly selected from each of two newly-opened bags. What is the probability that both are brown?

Brown 0.3 0.3P

Color Brown Red Yellow Green Orange

Proportion 0.3 0.2 0.2 0.2 0.1

0.09 9%


Homework, Page 728A peanut candy is produced in the following proportions:

21. A single peanut candy is randomly selected from each of two newly-opened bags. What is the probability that neither is yellow?

Not yellow 0.8 0.8P

Color Brown Red Yellow Green Orange

Proportion 0.3 0.2 0.2 0.2 0.1

0.64


Homework, 28A card game uses a 24-card deck, containing 9 through ace of the usual four suits. Each hand has six cards. Find the probability.

25. A hand contains all four aces.

4 4 20 2

24 6

Four AcesC C

PC

0.0014115

3542


Homework, Page 72829. If it rains tomorrow, the probability is 0.8 that John will practice his piano lesson. If it does not rain, there is only a 0.4 chance John will practice. Suppose the chance of rain tomorrow is 60%. What is the probability that John will practice tomorrow?

Practice Rain Practice|rainP P P No rain Practice|no rainP P

0.6 0.8 0.4 0.4 0.48 0.16 0.64


Homework, Page 72833. Floppy Jalopy Rent-a-Car has 25 cars available for rental, 20 big bombs and five midsize cars. If two cars are selected at random, what is the probability that both are big bombs?

Big bombsP 20 2

25 2

C

C 190

300

19

30


Homework, Page 72837. Explain why the following statement cannot be true. The probabilities that a computer salesperson will sell zero, one, two, or three computers in any one day are 0.12, 0.45, 0.38, and 0.15, respectively.

This statement cannot be true because the probabilities add to more than one.


Homework, Page 72841. To complete the kinesiology requirement at Palpitation Tech, you must pass two classes chosen from aerobics, aquatics, defense arts, racket sports, recreational activities, rhythmic activities, soccer, gymnastics, and volleyball. If you decide to choose your two classes at random by drawing two class names from a box, what is the probability you will take racket sports and rhythmic activities?

racket & rhythmicP 2 2

9 2

C

C

1

36


Homework, Page 728Ten dimes dated 1990 through 1999 are tossed. Find the probability.

45. Heads on all ten dimes

ten headsP10

1

2

1

1024


Homework, Page 728Ten dimes dated 1990 through 1999 are tossed. Find the probability.

49. At least one head

at least one headP10

10

2 1

2

1023

1024

11

1024

1023

1024


Homework, Page 72853. The probability of rolling a five on a pair of fair dice is:

A.

B.

C.

D.

E.

1

4

1

5

1

6

1

91

11

1 2 3 4 5 6

1

2

3

4

2 3 4 5 6 7

3 4 5 6 7 8

4 5 6 7 8 9

5 6 7 8 9 10

6 7 8 9 10 11

7 8 9 1

5

6 0 11 12

45

36P

1

9

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

9.4

Sequences


What you’ll learn about

Infinite Sequences Limits of Infinite Sequences Arithmetic and Geometric Sequences Sequences and Graphing Calculators

… and whyInfinite sequences, especially those with finite limits, are involved in some key concepts of calculus.


Sequence

Sequence - an ordered progression of numbers Finite sequence - a sequence with a finite number of entries Infinite sequence - a sequence that continues without bound Explicitly defined sequence - a sequence for which any entry

may be written directly using the definition Recursively defined sequence - a sequence defined in such a

manner that one must know the prior entry before being able to write the next entry using the definition


Example of an Explicitly Defined Sequence

th

2

Find the first 4 and the 50 term of the sequence

in which 3.

k

k

a

a k


Example of a Recursively Defined Sequence

th

1 1

Find the first 4 and the 50 term of the sequence

in which 2, 2.k

n n

a

a a a


Limit of a Sequence

Let be a sequence of real numbers, and consider lim .

If the limit is a finite number , the sequence and

is the . If the limit is infinite or nonexistent,

the se

n nn

a a

L L

converges

limit of the sequence

quence .diverges


Example Finding Limits of Sequences

Determine whether the sequence converges or diverges. If it converges,

give the limit.

2 1 2 22,1, , , ,..., ,...

3 2 5 n


Arithmetic Sequence

A sequence is an if it can be written in the

form , , 2 ,..., ( 1) ,... for some constant .

The number is called the .

Each term in an arithmetic seque

na

a a d a d a n d d

d

arithmetic sequence

common difference

1

nce can be obtained recursively from

its preceding term by adding : (for all 2).n nd a a d n


Example Arithmetic SequencesFind (a) the common difference, (b) the tenth term, (c) a recursive rule for thenth term, and (d) an explicit rule for the nth term.-2, 1, 4, 7, …


Geometric Sequence

2 1

A sequence is a if it can be written in the

form , , ,..., ,... for some nonzero constant .

The number is called the .

Each term in a geometric sequence

n

n

a

a a r a r a r r

r

geometric sequence

common ratio

1

can be obtained recursively from

its preceding term by multiplying by : (for all 2).n nr a a r n


Example Defining Geometric SequencesFind (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, and (d) an explicit rule for the nth term.2, 6, 18,…


Sequences and Graphing Calculators

One way to graph an explicitly defined sequence is as a scatter plot of the points of the form (k,ak).

A second way is to use the sequence mode on a graphing calculator.


Example Graphing a Sequential Scatter Plot

Use you calculator to generate the first 10 terms of the sequence explicitly defined by an = 3n - 5 in a scatter plot.


Example Calculating Sequence Values

Use you calculator to generate the first 10 terms of the sequence recursively defined by a1 = 4, an = 3an-1 + 5 in a scatter plot.

Lower case u in calculator entry is obtained by pressing 2nd and then 7.


The Fibonacci Sequence

1

2

2 1

The Fibonacci sequences can be defined recursively by

1

1

for all positive integers 3.n n n

a

a

a a a

n


Homework

Homework Assignment #29 Review Section 9.4 Page 739, Exercises: 1 - 37 (EOO), 43, 45, 47 Quiz next time


Quick Review

10

1

3

10

1

3

is an arithmetic sequence. Use the given information to find .

1. 5; 4

2. 5; 2

is a geometric sequence. Use the given information to find .

3. 5; 4

4. 5; 4

5. Find the sum of

n

n

a a

a d

a d

a a

a r

a r

2 the first 3 terms of the sequence .n


Quick Review Solutions

10

1

3

10

1

3

is an arithmetic sequence. Use the given information to find .

1. 5; 4

2. 5; 2

is a geometric sequence. Use the given information to find .

3.

41

19

1,310,7205; 4

4.

n

n

a a

a d

a d

a a

a r

a

2

5; 4

5. Find the sum of the first 3 terms of the sequence

81,920

14.

r

n


What you’ll learn about

Summation Notation Sums of Arithmetic and Geometric Sequences Infinite Series Convergences of Geometric Series

… and why

Infinite series are at the heart of integral calculus.


Summation Notation

1 21

In , the sum of the terms of the sequence

, ,..., is denoted which is read "the sum of

from 1 to ." The variable is called the .

n

n k kk

a a a a a

k n k

summation notation

index of summation


Sum of a Finite Arithmetic Sequence

1 2

1 21

1

1

Let , ,..., be a finite arithmetic sequence with common

difference . Then, the sum of the terms of the sequence is

...

2

2 ( 1)2

n

n

k nk

n

a a a

d

a a a a

a an

na n d


Example Summing the Terms of an Arithmetic Sequence

A corner section of a stadium has 6 seats along the front row. Each

successive row has 3 more seats than the row preceding it. If the top

row has 24 seats, how many seats are in the entire section?


Sum of a Finite Geometric Sequence

1 2

1 21

1

Let , ,..., be a finite geometric sequence with common

ratio . Then the sum of the terms of the sequence is

...

1

1

n

n

k nk

n

a a a

r

a a a a

a r

r


Example Summing the Terms of a Finite Geometric Sequence

11

Find the summation of the geometric sequence

1 1 12, , , , ,2 0.25

2 8 32


Partial Sums

Partial sums are the sums of a finite number of terms in an infinite sequence. In some instances, the partial sums approach a finite limit and the series is said to converge.


Example Examining Partial Sums

For the following series, find the first five terms in the

sequence of partial sums. Which of the series appear

to converge?

a 0.1 0.01 0.001 0.0001 0.00001

b 10 15 22.5 33.75 50.625

c 1 1 1 1 1


Infinite Series

1 21

An infinite series is an expression of the form

... ...k nk

a a a a


Sum of an Infinite Geometric Series

1

1The geometric series converges if and only

if | | 1. If it does converge, the sum is .1

k

ka r

ar

r


Example Summing Infinite Geometric Series

1

1

Determine whether the series converges. If it converges,

give the sum.

2 0.25k

k

homework, page 728

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