CHAPTER 10

Sequences, Induction, & Probability

10.1 Sequences & Summation Notation

• Objectives– Find particular terms of sequence from the

general term– Use recursion formulas– Use factorial notation– Use summation notation

What is a sequence?

• An infinite sequence is a function whose domain is the set of positive integers. The function values, terms, of the sequences are represented by

• Sequences whose domains are the first n integers, not ALL positive integers, are finite sequences.

...,...,, 321 naaaa

Recursive Sequences

• A specific term is given.• Other terms are determined based on the value of

the previous term(s)• Example:•

42113 ,,:,13,10 aaafindaaa nn

311)10(3133

2

3

13

1333

110

1310

34

1

12

23

aa

a

aa

aa

Find the 1st 3 terms of the sequence:

• 1)4, 5/2, 6

• 2)4, 5/2, 1

• 3)1, 2, 3

• 4)4, 5, 6

!

1

n

nan

Summation Notation

• The sum of the first n terms, as i goes from 1 to n is given as:

• Example:

10233282318

]2)85[(]2)75[(]2)65[(]2)55([]2)45[(258

4

i

i

nn

n

ii aaaaaa

13211

...

10.2 Arithmetic Sequences

• Objectives– Find the common difference for an arithmetic

sequence– Write terms of an arithmetic sequence– Use the formula for the general term of an

arithmetic sequence– Use the formula for the sum of the first n

terms of an arithmetic sequence

What is an arithmetic sequence?

• A sequence where there is a common difference between every 2 terms.

• Example: 5,8,11,14,17,…..

• The common difference (d) is 3

• If a specific term is known and the difference is known, you can determine the value of any term in the sequence

• For the previous example, find the 20th term

• (continue on next slide)

Example continued

• The first term is 5 and d=3

• Notice between the 1st & 2nd terms there is 1 (3). Between the 1st & 4th terms there are 3 (3’s). Between the 1st & nth terms there would be (n-1) 3’s

• 20th term would be the 1st term + 19(3’s)

62)3(19520 a

The sum of the 1st n terms of an arithmetic sequence

• Since every term is increasing by a constant (d), the sequence, if plotted on a graph (x=the indicated term, y=the value of that term), would be a line with slope= d

• The average of the 1st & last terms would be greater than the 1st term by k and less than the last term by k. The same is true for the 2nd term & the 2nd to last term, etc

• Therefore, you can find the sum by replacing each term by the average of the 1st & last terms (continue next slide)

Sum of an arithmetic sequence

• If there are n terms in the arithmetic sequence and you replace all of them with the average of the 1st & last, the result is:

21 n

n

aanS

Find the sum of the 1st 30 terms of the arithmetic sequence if

• 1) 81• 2) 3430• 3) 2430• 4) 168

6,61 da

10.3 Geometric Sequences & Series

• Objectives– Find the common ratio of a geometric sequence– Write terms of a geometric sequence– Use the formula for the general term of a geometric

sequence– Use the formula for the sum of the 1st n terms of a

geometric sequence– Find the value of an annuity– Use the formula for the sum of an infinite geometric

series

What is a geometric sequence?

• A sequence of terms that have a common multiplier (r) between all terms

• The multiplier is the ratio between the (n+1)th term & the nth term

• Example: -2,4,-8,16,-32,…

• The ratio between any 2 terms is (-2) which is the value you multiply any term by to find the next term

Given a term in a geometric sequence, find a specified other

term• Example: If 1st term=3 and r=4, find the 14th term• Notice to find the 2nd term, you multiply 3(4)• To find the 3rd term, you would multiply 3(4)(4)• To find the 4th term, multiply 3(4)(4)(4)• To find the nth term, multiply: 3(4)(4)(4)…

(n-1 times)• 14th term =

• (in a geometric sequence, terms get large quickly!)

592,326,201)4(3 13

Sum of the 1st n terms of a geometric sequence

r

raS

n

n

1

)1(1

What if 0<r<1 or -1<r<0?

• Examine an example:• If 1st term=6 and r=-1/3

• Even though the terms are alternating between pos. & neg., their magnitude is getting smaller & smaller

• Imagine infinitely many of these terms: the terms become infinitely small

,...243

2,

81

2,

27

2,

9

2,3

2,2,6

Find the Sum of an Infinite Geometric Series

• If -1<r<1 and r not equal zero, then we CAN find the sum, even with infinitely many terms (remember, after a while the terms become infinitely small, thus we can find the sum!)

• If and n is getting very large, then r raised to the n, recall, is getting very, very small…so small it approaches zero, which allows us to replace r raised to the nth power with a zero!

• This leads to:

r

raS

n

n 1

)1(1

r

aSn

1

1

Repeating decimals can be considered as infinite sums

• Example: Write .34444444…as an infinite sum• Separate the .3 from the rest of the number:• .3444….. = .3 + .044444…..• .044444….. = .04 + .004 + .0004 + .0004+…• This is an infinite sum with 1st term=.04,r=.1

• .3444….=

45

2

90

4

9.

04.

1.1

04.

S

90

31

90

4

90

27

45

2

10

3

10.4 Mathematical Induction

• Objectives– Understand the principle of mathematical

induction– Prove statements using mathematical

induction

What is mathematical induction?

• A method of proof. To prove something holds true for all values of n, you cannot simply plug in numbers. Why not? Because you could never verify something is true for ALL numbers.

• Using this method, you prove a statement is true for one term (generally the 1st) and then if you assume the statement is true for the kth term, you must prove it holds true for the (k+1)th term.

• If you can prove it is true for the (k+1)th term, it must be true for all terms.

Carefully examine examples in the book for mathematical

induction.

10.5 The Binomial Theorem

• Objectives

– Evaluate a binomial coefficient

– Expand a binomial raised to a power

– Find a particular term in a binomial expansion

What is the binomial theorem?

• The binomial theorem provides a means to expand a binomial expression.

• It provides a “shortcut” to taking an expression, such as (a+b) and raising it to a power (n) without having to continue to multiply binomials out.

nnnnn bn

nba

nba

na

nba

...

210)( 221

What is ?

• This is the binomial coefficient in the previous expansion of binomials

• It may also be considered as the combination of n objects taken r at a time.

)!(!

!

rnr

n

r

n

r

n

Use binomial expansion theorem to find

6542332456

65142

3324156

645762160432048602916729

)2(6

6)2()3(

5

6)2()3(

4

6

)2()3(3

6)2()3(

2

6)2()3(

1

6)3(

0

6

babbabababaa

bbaba

bababaa

6)23( ba

10.6 Counting Principles, Permutations, & Combinations

• Objectives

– Use the fundamental counting principle

– Use the permutations formula

– Distinguish between permutation problems & combination problems

– Use the combinations formula

What is the Fundamental Counting Principle?

• It is the notion that if one event can happen in “a” different ways and another event can occur in “b” different ways, then there are “a times b” different ways for BOTH events to happen together.

• Example: If you have 5 shirts and 3 pair of jeans, there are 15 different outfits comprised of a shirt & a pair of jeans.

What is a permutation?

• An ordered arrangement of items.

• For example, if you have a telephone number comprised of 10 different digits, the order matters. Two people could have DIFFERENT phone numbers made up of the same digits.

• 402-555-2378 is NOT 402-555-3287

• There are 2 permutations of that set of 10 digits.

How many different permutations of n objects are there?

• There are n choices for the 1st object, (n-1) choices for the 2nd object, (n-2) choices for the 3rd object, etc, until there is only 1 choice for the last object.

• Since we want all these choices to happen at the same time, we apply Fundamental Counting Principle and multiply the # choices:

• n(n-1)(n-2)(n-3)(n-4)…(2)(1)

What if you want to look at the different ordered arrangements of any 4 non-repeated letters of the

alphabet for a password?• There are 26 different objects (letters) and

we’re taking just 4 of them at a time.

• There are 26 choices for the 1st letter, 25 for the 2nd letter, 24 for the 3rd letter and 23 choices for the 4th letter.

• Thus there are 26(25)(24)(23)=358,800 different passwords.

Would it have made a difference if the letters could be repeated?

• Yes, then there would be 26 choices for the 1st letter, 26 choices for the 2nd letter, 26 choices for the 3rd letter and 26 choices for the 4th letter.

• Total=456,976 different passwords with letters possibly repeated.

Permutations of n objects taken r at a time (none repeated)

• General formula

)!(

!

rn

n

r

n

What if we don’t care about order?

• There are 25 children in a class and I’m selecting 4 of them to clean the board. Does it matter if I select Joe, Mary, Sue & Tim or Sue, Joe, Mary & Tim? NO – it is exactly the same result.

• The number of ways to select n objects r at a time, when order DOES NOT matter, is considered the COMBINATION of n objects taken r at a time.

10.7 Probability

• Objectives– Compute empirical probability– Compute theoretical probability– Find the probability that an event will not occur– Find the probability of one event or a second

event occurring– Find the probability of one event & a second

event occurring.

If order does NOT matter, we must divide out the ways that are the

same.• How many different ways are there to arrange

any “r” objects? There are r! ways.

• Therefore, the combination of n objects taken r at a time is:

!)!(

!

rrn

nCrn

What is probability?

• Probability of an event happening = outcomestotal

successes

#

#

Empirical probability• We COUNT or OBSERVE outcomes.

• Often referred to as experimental probability.

• An experiment is done, or situation exists, and various outcomes are recorded.

• The empirical (experimental) probability is the ratio of total number of successful outcomes (however it has been defined) to the total number of outcomes.

Theoretical Probability• An actual experiment is rather conducted.

Successful outcomes are determined based on known information regarding the event.

• For example, if one has a fair die and there are 6 sides to the die, the probability of rolling a 4 is 1/6.

• We don’t roll the die to count outcomes, rather we are assured the die is fair, thus there is 1 success (a “4” in this example) and 6 possible outcomes (1,2,3,4,5, or 6), the theoretical probability of rolling a 4 is 1/6.

What is the probability of selecting 3 men & 2 women for a committee

from a room of 10 men & 15 women?

• Success = select 3 men & 2 women (order does NOT matter) (use combinations for # ways to select 3 men & for selecting 2 women, THEN apply Fundamental Counting Principle)

• Total outcomes = select any 5 of the 15 people

• Probability = .24 (approx.)

600,12105120215310 CC

130,53525 C

Independent Events

• If the outcome of one event does NOT effect the outcome of the other event they are considered to be independent.

Mutually Exclusive Events

• 2 events that canNOT happen at the same time

If you know the probability of an event occurring, what is the probability it will NOT occur?

• Total probability of all options of any event occurring must equal 1.

• Therefore, if P(A)=x, P(not A) = 1 – x

• If the probability of rain on a given day is 30%, the probability it will NOT rain is 70%