Chapter 9 & 10 FITZGERALD MAT 150/51 1 SECTION 9.1 Sequence Definition An infinite sequence is a function whose domain is the set of positive integers. The function values ,...,...,,,, 4321 naaaaa are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence. ( ) 52 += xxf is the same as 52 += nan

Find the first five terms of 52 += nan . Find the first five terms of ( )12

1+

!=n

an

n .

Finding an sequence function. 1, 3, 5, 7, …. 2, -5, 10, -17, …

Chapter 9 & 10 FITZGERALD MAT 150/51 2 Recursive sequence is a sequence function where the next term in the sequence is determined by the values of the previous terms and not the value of n.

,10 =a ,11 =a 12 !! += nnn aaa where .2!n Factorial Notation If n is a positive integer, n factorial is defined as ( ) ( ) 1221! !!!!"!"!= nnnn As special cases, zero factorial is defined as 0! = 1and 1! = 1.

=======

!6!5!4!3!21!11!0

Write the first five terms given by

!2n

an

n =

Begin with n = 0.

Chapter 9 & 10 FITZGERALD MAT 150/51 3 Simplify the factorials by hand.

Summation Notation. The sum of the first n terms of a sequence is represented as

nn

n

kk aaaaaaa ++++++= !

=" 143211

...

Where k is called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation. Find each sum. Write in summation notation.

2222 9321 +!!!+++ 121

81

41

211 !+"""++++ n

!6!2!8! !5!3

!6!2!!

( )!1!!nn

!=

5

13

kk ( )!

=

+6

3

21k

k

Chapter 9 & 10 FITZGERALD MAT 150/51 4 Properties of Sums c is a constant.

1. cncn

k=!

=1 2. !!

==

=n

kk

n

kk acca

11

3. ( ) !!!===

+=+n

kk

n

kk

n

kkk baba

111 4. ( ) !!!

===

"="n

kk

n

kk

n

kkk baba

111

5. !!!==+=

"=j

kk

n

kk

n

jkk aaa

111 6.

( )21321

1

+=+!!!+++="=

nnnkn

k

7. ( )( )

6121321 2222

1

2 ++=+!!!+++="=

nnnnkn

k

8. ( ) 2

3333

1

3

21321 !"#

$%& +=+'''+++=(

=

nnnkn

k

Find each sum.

!=

10

13

kk ( )!

=

+8

1

3 1kk

( )!=

+"24

1

2 27k

kk ( )!=

20

6

24k

k

Chapter 9 & 10 FITZGERALD MAT 150/51 5 SECTION 9.2 A sequence is arithmetic if the differences between consecutive terms are the same. The sequence ,...,...,,,, 4321 naaaaa is arithmetic if there is a number d such that

1342312 ... !!==!=!=!= nn aaaaaaaad . If so, the arithmetic sequence equation can be defined as ( )11 !+= ndaan . Find the 40th term of the sequence 2, 6, 10, 14, 18… Find the arithmetic sequence function if the 4th term is 20 and the 13th term is 65. Arithmetic Series Find the sum of the first 100 positive integers. 1 + 2 + 3 + 4 + …+ 96 + 97 + 98 + 99 + 100 =? Find the 150th partial sum of 5, 16, 27, 38, 49 … Find the sum of 1000 + 950 + 900 + … + 250 =

Chapter 9 & 10 FITZGERALD MAT 150/51 6 SECTION 9.3 A sequence is geometric if the common ratios between consecutive terms are the same. The sequence ,...,...,,,, 4321 naaaaa is geometric if there is a number r such that

13

4

2

3

1

2 ...!

=====n

n

aa

aa

aa

aar . If so, the geometric sequence equation can be defined as

( )11

!"= nn raa .

Find the 10th term of the sequence 2, 6, 18, 54… Find the geometric sequence function if the 3rd term is 36 and the 5th term is16. Geometric Series ( )1

12

111!+"""+++= n

n rararaaS Find the formula. Find the 10th partial sum of 5, 10, 20, 40 …

Chapter 9 & 10 FITZGERALD MAT 150/51 7 Find the infinite sum of 1000 + 500 + 250 + 125 + … = Prove that the repeating decimal 0.999… equals 1. Determine when a series diverge or converge.

Chapter 9 & 10 FITZGERALD MAT 150/51 8 SECTION 10.1 Subsets and Counting Principles. If every element in set A is also an element of set B, we say that A is a subset of B and write BA ! . If BA ! and BA ! , we say that A is a proper subset of B and write BA! . We should also agree that the empty set, ! or { }, is a subset of every set. Write down all the possible subsets of the set {a, b, c}. Finding the number of subsets and the cardinal number of a set. Define Union and Intersection of sets. In a survey of 100 college students, 35 were registered in College Algebra, 52 were registered in English, and 18 were registered in both courses. How many were registered in College Algebra or English? How many were registered in neither class?

Chapter 9 & 10 FITZGERALD MAT 150/51 9 Make a tree diagram for two appetizers, 4 entrees, and 2 desserts. Counting Principle. The Student Union is having a lunch special value meal. You get to choose one of 4 sandwiches, one of 5 bags of chips, one of 7 drinks and one of 2 desserts. How many different lunch specials can you make? SECTION 10.2 Permutations and Combinations. A permutation is an ordered arrangement of r objects chosen from n objects. Type 1. Distinct Objects with Repetition. The number of ordered arrangements of r objects chosen from n objects, in which the n objects are distinct and repetition is allowed, is rn . How many 3 letter Airport abbreviations exist when the letters are allowed to repeat? Type 2. Distinct Objects without Repetition. The number of ordered arrangements of r objects chosen from n objects, in which the n objects are distinct and repetition is

not allowed, is ( ) nrrnnPrnP rn !"

== ;)!(!, .

How many 3 letter Airport abbreviations exist when the letters are not allowed to repeat?

Chapter 9 & 10 FITZGERALD MAT 150/51 10 A combination is a grouping, arrangement where order doesn’t matter, of r objects selected from n distinct objects without repetition, where r < n. The notation is

( ) nrrrn

nCrn

rnC rn !"#

==$$%

&''(

)= ;

!)!(!,

Given the first 5 letters of the alphabet, how many 3 letter groups are there? How many 3 person committees of 3 people can be formed from 8 people? How many groups of 4 people, 2 men and 2 women, if there are 5 men and 6 women to choose from? Distinct Permutations. The number of permutations on n objects of which r1 are of one kind, r2 are of a second kind, r3 are of a third kind, …, rk are of a kth kind is given by ( ) ( ) ( ) ( )kk rrrrnCrrrnCrrnCrnC ,......,,, 121221211 !!!!!""!!"!" or this

simplifies to !...!!!!

321 krrrrn

!!!! where nrrrr k =++++ ...321

How many distinct letter arrangements are there for the word MISSISSIPPI?

Chapter 9 & 10 FITZGERALD MAT 150/51 11 SECTION 9.5 Pascal’s Triangle and Binomial Expansion.

How to use the graphing calculator to find a row of Pascal’s triangle! Binomial Expansion.

Expand ( )432 !x

Chapter 9 & 10 FITZGERALD MAT 150/51 12

Find the coefficient of 7x of ( )1032 !x .

Find the 7th term of ( )1032 !x . 10.3 Probability. Any happening for which the result is uncertain is called an experiment. The possible results of the experiment are outcomes, the set of all possible outcomes of the experiment is the sample space of the experiment, and any sub-collection of a sample space is an event. Finding Sample space. Tree Diagram, Matrix, Counting Principles. Sample space for… … Flipping a coin three times. …Rolling two die. … A deck of cards.

Chapter 9 & 10 FITZGERALD MAT 150/51 13 Probability will always be the number of outcomes of the desired event out of the total number of outcomes in the sample space. P(Event) will have the following range…

0 < P(E) < 1

Find the following probabilities… P(2 tails in 3 flips) P(at least 2 tails in 3 flips) P(rolling a 7) P(rolling a value > 8) P(face card) Probability with “and” and “or”. P(A and B) are the outcomes that satisfy both A and B. ( ) ( ) ( )BPAPBAP !=" P(A or B) are the outcomes that satisfy A or B or both. ( ) ( ) ( ) ( )BAPBPAPBAP !"+=# P(Hearts and Face Cards) P(Hearts or Face Cards)

Chapter 9 & 10 FITZGERALD MAT 150/51 14 Complement Probability. Let A be an event and let A be its complement. P(A) + P( A ) = 1 or P( A ) = 1 – P(A). P(at least one tails in 5 flips) In a group of 10 people, what is the probability that at least 2 people have the same birthday? Binomial Probability. This probability has only two outcomes, success or failure. Binomial Probability Formula The spirit club is 75% girls. What is the probability of randomly picking 5 people, where there are 3 boys and 2 girls? A manufacture has determined that a bulb machine will produce 1 bad bulb for every 2000 bulbs it produces. What is the probability that an order of 200 bulbs is all perfect? There is at least one bad bulb? There is one bad bulb? There are two bad bulbs?

Chapter 9 & 10 FITZGERALD MAT 150/51 15 Probabilities of actual experiments In a bag of M & Ms, the candies are colored red, green, blue, brown, yellow, and orange. Say we open a bag and counted all the colors to get the following table. Probability with & without replacement. Arrangements vs Groups. A bag contains 5 red chips, 3 white chips, and 2 blue chips. Find the following probabilities. P( red, white, and blue in order with replacement) P( red, white, and blue in order without replacement) P( one of each color with replacement) P( one of each color without replacement)

Color Number Probability

Red 6

Green 3

Blue 1

Brown 3

Yellow 5

Orange 2