Math 110 Chapter 8 (page 1 of 29)
8.1 Sequences and Series
Sequences
A sequence is commonly thought of as being an ordered collection so that
there is an identifiable first element, second element, third element, etc.
Mathematically, a sequence is a function whose domain is the set of positive
integers.
Definition of a Sequence
An infinite sequence is a function whose domain is the set of positive
integers. Sequences are written in subscript notation instead of function
notation. The function values are the terms of the sequence; denoted
a1 , a2 , a3 ,…, an ,…. If the domain of the function is only the first n positive
integers, the sequence is a finite sequence. It is sometimes convenient to let
the subscripting begin with zero instead of one so that the terms are denoted
a0 , a1 , a2 , a3 ,…, an ,….
TI-83 functions needed.
LIST / OPS / 5:seq(expression, variable, begin, end, [increment])
LIST / MATH / 5:sum(list)
Example A
Find the first four terms of each sequence. Express rational numbers in
fractional form rather than decimal form.
n +1
(a) an =
n
(b) bn = (−1) n (3n − 2)
(−1) n
(c) an =
2n +1
Math 110 Chapter 8 (page 2 of 29)
Example 3
Write an expression for the nth term of each sequence.
1 1 1 1 1
(a) , , , , ,
2 4 8 16 32
(b) 1, 3, 5, 7, 9, . . .
(c) 2, 5, 10, 17, . . .
Recursive Sequences
A recursive sequence is a sequence where the first term, or the first few
terms, are given, and the following terms are defined in terms of the preceding
terms.
Example 4
The Fibonacci Sequence
The Fibonacci sequence is defined recursively as follows
a0 = 1,
a1 = 1,
ak = ak−2 + ak−1 .
Write the first six terms of this sequence.
Math 110 Chapter 8 (page 3 of 29)
Factorial Notation
For each positive integer n, n factorial is
defined as
n!= 1⋅ 2 ⋅ 3 (n −1) ⋅ n
Furthermore, zero factorial is defined as
0!= 1.
0!= 1
1!= 1
2!= 1⋅ 2 = 2
3!= 1⋅ 2 ⋅ 3 = 6
4!= 1⋅ 2 ⋅ 3⋅ 4 = 3!⋅4 = 24
5!= 1⋅ 2 ⋅ 3⋅ 4 ⋅ 5 = 4!⋅5 = 120
etc 
Example 5
Factorial in a Sequence
2n
List the first five terms (in fractional form) of the sequence a n = . Begin
n!
with n = 0.
Example 6
Evaluating Factorial Expressions
Evaluate each expression without the use of your calculators.
8!
(a)
2!5!
(b)
2!7!
4!5!
(c)
n!
(n −1)!
Math 110 Chapter 8 (page 4 of 29)
Summation Notation
The sum of the first n terms of a sequence is denoted by summation
notation as
n
∑ ai = a1 + a2 + a3 + … + an
i=1
where i is the index of summation, n is the upper limit of summation, and
1 is the lower index of summation (the sum may begin at any nonnegative
integer less than n).
Example 7
Summation Notation for Sums
Evaluate the following sums.
5
(a) ∑ 2i
i=1
5
(b) 2∑ i
i=1
5
(c) ∑ (k 2 − 3)
k=2
8
1
n=0 n!
(d) ∑
10
1
n=0 n!
(e) ∑
Math 110 Chapter 8 (page 5 of 29)
Properties of Sums
1.
n
n
i=1
i=1
∑ cai = c∑ ai , where c is a constant
5
For example: ∑ 2i = 2 ⋅1+ 2 ⋅ 2 + 2 ⋅ 3+ 2 ⋅ 4 + 2 ⋅ 5
i=1
= 2(1+ 2 + 3+ 4 + 5)
5
= 2∑ i
i=1
2.
n
n
n
i=1
i=1
i=1
∑ (ai + bi ) = ∑ ai + ∑ bi
4
For example: ∑ (i + i 2 ) = (1+12 ) + (2 + 2 2 ) + (3+ 3 2 ) + (4 + 4 2 )
i=1
= (1+ 2 + 3+ 4) + (12 + 2 2 + 32 + 4 2 )
4
4
i=1
i=1
= ∑ i + ∑ i2
3.
n
n
n
i=1
i=1
i=1
∑ (ai − bi ) = ∑ ai − ∑ bi
4
For example: ∑ (i − i 2 ) = (1−12 ) + (2 − 2 2 ) + (3 − 3 2 ) + (4 − 4 2 )
i=1
= (1+ 2 + 3+ 4) − (12 + 2 2 + 3 2 + 4 2 )
4
4
i=1
i=1
= ∑ i − ∑i2
Math 110 Chapter 8 (page 6 of 29)
Definition of Series
1. The sum of the infinite sequence a0 , a1 , a2 , a3 ,…, ai ,… is called an
infinite series, or simply a series, and is denoted by
∞
S = ∑ ai = a1 + a2 + a3 + … + ai + …
i=1
2. The sum of the finite sequence a0 , a1 , a2 , a3 ,…, an is called a finite
series, or the nth partial sum, and is denoted by
n
Sn = ∑ ai = a1 + a2 + a3 + … + an
i=1
Example 8
Evaluating Series
∞
3
Consider the series ∑ k .
k=1 10
(a) Find the third partial sum S3.
(b) Find the sum S.
Math 110 Chapter 8 (page 7 of 29)
Example 9
Population Model
From 1960 to 2000, the resident population of the United States can be
approximated by the model
an = 33,282 + 801.3n + 6.12n 2 ,
n = 0, 1, 2, . . ., 40,
where an is the population in millions and n = 0 corresponds to the year
1960. Find the last five terms of the sequence and interpret their meanings.
Math 110 Chapter 8 (page 8 of 29)
8.2 Arithmetic Sequences and Partial Sums
Definition of an Arithmetic Sequence
A sequence is an arithmetic sequence if the difference between any two
consecutive terms is a constant. So, the sequence
a1 , a2 , a3 ,…, an ,…
is arithmetic if there is a number d such that
a2 − a1 = a3 − a2 = … = an − an−1 = … = d .
The number d is called the common difference of the arithmetic sequence.
Example 1
(i) Find the first six terms of each sequence, (ii) determine if the sequence
is arithmetic, and (iii) find the common difference of each arithmetic
sequence.
a. an = 4n + 3
b.
an = 12 − 7n
c.
an = 15 (n + 4)
Example A
Consider the sequence 1, 4, 9, 16, 25, . . .
a. What is the expression for the nth term?
b. Is the sequence arithmetic? Why / why not?
Math 110 Chapter 8 (page 9 of 29)
The nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence with common difference d is given by
an = a1 + (n −1)d .
Example B
Find the formula for the nth term of the arithmetic sequence whose first term is
5 and whose common difference is 3.
Example C
If a4 = 20 and a13 = 65 . Write a formula for the nth term of the sequence.
Example 3
Write a formula for the nth term of the sequence if the first two terms are 2
and 9.
Math 110 Chapter 8 (page 10 of 29)
The Sum of an Arithmetic Sequence
The sum of a finite arithmetic sequence with n terms is
n
Sn = (a1 + an ) .
2
When Sn is thought of as the sum of the first n terms of an infinite sequence,
then Sn is called the nth partial sum.
Example 5 and 6
a. Find the sum of the first 11 odd integers.
b. Find the sum of the first 100 positive integers.
c. Find the sum of the first 25 positive multiples of 4.
Math 110 Chapter 8 (page 11 of 29)
Example 7A
Find the 125th partial sum of the sequence 5, 16, 27, 38, 49, . . .
Example 8A
A theater has 25 rows of seats with 20 seats in the first row, 21 seats in the
second row, 22 seats in the third row, etc. How many seats are in the
theater?
Example 9A
A small businessman estimates he will sell $20,000 worth of product his first
year and will increase annual sales by $12,000 per year for the next 10 years.
Find the total sales for the first 11 years in business.
Math 110 Chapter 8 (page 12 of 29)
8.3 Geometric Sequences and Series
Definition of a Geometric Sequences
A sequence is a geometric sequence if the ratio of any two consecutive
terms is a constant r, called the common ratio. That is,
a2 a 3 a4
a
= =
= = n = = r
a1 a2 a3
an−1
Example 1
(i) Find the first five terms of each sequence, (ii) determine if the sequence
is geometric, and (iii) find the common ratio r of each geometric sequence.
a. an = 3n
b.
⎛ 1⎞
an = 4 ⎜ − ⎟
⎝ 2⎠
c.
an =
n
n +1
n
Math 110 Chapter 8 (page 13 of 29)
Example A
Consider the sequence 1, 4, 9, 16, 25, . . .
a. What is the expression for the nth term?
b. Is the sequence geometric? Why / why not?
The nth Term of a Geometric Sequence
The nth term of a geometric sequence with common ratio r can be expressed
in two useful ways.
1. an = an−1r
2. an = a1r n−1
Example B
Finding Terms of a Geometric Sequence
Write the first five terms of the geometric sequence whose first term is 2 and
whose common ratio is 5.
Example C
Finding a Term of a Geometric Sequence
Find the 15th term of the geometric sequence whose first term is 18 and
whose common ratio is 1.075.
Math 110 Chapter 8 (page 14 of 29)
Example D
Finding a Formula for a Geometric Sequence
Find the formula for the nth term of the geometric sequence 6, 18, 54, . . ..
Then find the 8th term.
Example 5
Finding a Formula for a Geometric Sequence
The 4th term of a geometric sequence is 125, and the 10th term is 125/64.
a. Find the formula for the nth term of the sequence.
b. Find the 15th term of the sequence.
Example G
Finding a Formula for a Geometric Sequence
Suppose m and n are integers and m > n. Show that am = an r m−n .
Math 110 Chapter 8 (page 15 of 29)
The Sum of a Finite Geometric Sequence
The sum of the finite geometric sequence a1 , a1r, a1r 2 , a1r 3 ,…, a1r n−1, where
r ≠ 1 is the common ratio, is
Sn = a1 + a1r + a1r + a1r +…+ a1r
2
3
n−1
⎛1− r n ⎞
= a1 ⎜
⎟
⎝ 1− r ⎠
Example 6
Finding the Sum of a Finite Geometric Sequence
Find the following sums.
1.
12
∑ 4(0.3) n
n=1
2.
12
∑ 4(0.3) n
n=0
Math 110 Chapter 8 (page 16 of 29)
Geometric Series
The sum of the terms of an infinite geometric sequence is a geometric series.
The formula for the sum of an infinite geometric series requires that r < 1
because if r < 1, then r n → 0 as x → ∞.
Example G
Finding the Sum of a Finite Geometric Sequence
a. Let r = 0.25 to illustrate the fact that if r < 1, then r n → 0 as x → ∞.
b. Let r = 0.25 and a1 = 5 to illustrate the fact that if r < 1, then
⎛1− r n ⎞
⎛1− 0 ⎞
⎛ 1 ⎞ a1
Sn = a1 ⎜
as x → ∞.
⎟ = a1 ⎜
⎟=
⎟ → a1 ⎜
⎝ 1− r ⎠
⎝1− r ⎠ 1− r
⎝ 1− r ⎠
The Sum of an Infinite Geometric Series
If the common ratio r < 1, then the sum of the terms of the infinite geometric
sequence a1 , a1r, a1r 2 , a1r 3 ,…, a1r n−1 ,… is
S=
a1
.
1− r
Math 110 Chapter 8 (page 17 of 29)
The Sum of an Infinite Geometric Series
If the common ratio r < 1, then the sum of the terms of the infinite geometric
sequence a1 , a1r, a1r 2 , a1r 3 ,…, a1r n−1 ,… is
S=
a1
.
1− r
Example 7 Finding the Sum of an Infinite Geometric Sequence
Find the following sums.
a.
∞
∑ 4(0.3) n−1
n=1
b.
∞
∑ 4(0.3) n−1
n=0
c.
3+ 0.3+ 0.03+ 0.003+…
Example 8 Compound Interest
A deposit of $50 is make on the first day of each month in a savings account
that pays 6% annual interest compounded monthly. What is the balance in the
account at the end of two years?
Math 110 Chapter 8 (page 18 of 29)
8.6 Counting Principles
Example 1 Selecting Random Pairs of Numbers
Eight pieces of paper are numbered 1 through 8 and placed in a box. One
piece of paper is drawn from the box, its number is written down, and it is
placed back into the box. Then, a piece of paper is again drawn from the
box, and its number is written down. Finally the two numbers are added
together. How many ways can a total of 12 be obtained?
Example 2 Selecting Random Pairs of Numbers
Eight pieces of paper are numbered 1 through 8 and placed in a box. Two
pieces of paper are drawn from the box, and their numbers are totaled. How
many ways can a total of 12 be obtained?
Fundamental Counting Principle
Let E1 and E 2 be two events such that, E1 can occur in n1 different ways
and E 2 can occur in n2 different ways. Then the number of ways the two
events can occur is n1 ⋅ n2 .
Math 110 Chapter 8 (page 19 of 29)
Example 3 Fundamental Counting Principle
The English alphabet has 26 letters. How many pairs of letters are possible.
Example 4 Counting Phone Numbers
How many different telephone numbers are possible in each area code?
Assume a phone number cannot start with a 0 or 1.
Permutations
A permutation of n different elements is an ordered arrangement of the
elements so that there is a first element, second element, third element, and so
on.
Number of Permutations of n Elements
The number of permutations of n different elements is n!
Example 5 Permutations
a. How many different permutations are possible of the letters A and B.
b. How many different permutations are possible of the letters A, B, and C.
Math 110 Chapter 8 (page 20 of 29)
c. How many different permutations are possible of the letters A, B, C, and
D.
d. How many different permutations are possible of the letters A, B, C, D,
and E.
Number of Permutations of n Elements Taken r at a Time
The number of permutations of n different elements taken r at a time is
n Pr =
n!
.
(n − r)!
TI-83: MATH / PRB / 2:nPr
Example 6 Horse Race Finishes
Eight horses run in a race. In how many different ways can these horses
come in first, second, and third?
Math 110 Chapter 8 (page 21 of 29)
Distinguishable Permutations
Suppose a set of n objects has n1 objects of one kind, n2 objects of a
second kind, n3 objects of a third kind, and so on, so that
n = n1 + n2 + n3 ++ nk
The number of distinguishable permutations of the n objects is
n!
n1!⋅n2!⋅n3!nk !
Example 7 Distinguishable Permutations
In how many distinguishable ways can the letters in BANANA be written.
Math 110 Chapter 8 (page 22 of 29)
Combinations
Permutations are ordered arrangements, that is, the order is important. A
combination of a set is a subset of the given set, and in sets the order is
irrelevant. For example, the combinations
{a, b, c} and {c, b, a}
are equivalent because both sets contain the same elements - the order does
not matter.
Number of Combinations of n Elements Taken r at a Time
The number of combinations of n different elements taken r at a time is
n Cr =
n!
.
(n − r)!r!
TI-83: MATH / PRB / 3:nCr
Example 8 Combinations versus Permutations
a. In how many ways can the three letters be chosen from the letters A, B,
C, D, and E?
b. In how many ways can the three letters be chosen and arranged from the
letters A, B, C, D, and E?
Math 110 Chapter 8 (page 23 of 29)
Example 9 Counting Card Hands
A standard poker hand consists of five cards dealt from a deck of 52 cards.
How many different poker hands are possible?
Example 10 Number of Subsets in a Set
Find the total number of subsets in a set that has 10 elements.
Math 110 Chapter 8 (page 24 of 29)
8.7 Probability
The set of all possible outcomes of an experiment forms the sample space
of the experiment, and any subset of the sample space is called an event.
For example, when a die is tossed there are six possible, equally likely
outcomes in the sample space S = {1, 2, 3, 4, 5, 6}. The event E = {2, 4,
6}, a subset of S, is described as rolling an even number on a die.
Example 1
Find the sample space for each of the following experiments.
a. One coin is tossed.
b. Two coins are tossed.
c. Three coins are tossed.
Math 110 Chapter 8 (page 25 of 29)
The Probability of an Event (All Outcomes are Equally Likely)
Let n(E) be the number of outcomes in event E, and n(S) be the number of
equally likely outcomes in the sample space S. The probability of event E is
P(E) =
n(E)
,
n(S)
0 ≤ P(E) ≤ 1.
If P(E) = 1, then event E is a certain event. If P(E) = 0 , then event E is an
impossible event.
Example 2
a. Two coins are tossed. What is the probability that both will land heads
up?
b. If one card is drawn from a standard deck of 52 cards, what is the
probability that it is an ace?
Example 3
Two dice are tossed. What is the probability that the total is 7?
Example 5
In a state lottery six different numbers are chosen between 1 and 40. What is
the probability of choosing all six matching numbers.
Math 110 Chapter 8 (page 26 of 29)
Mutually Exclusive Events
Two events A and B (from the same sample space) are mutually exclusive if
A and B have no outcomes in common. In set terminology, the intersection of
sets A and B is the empty set ( A ∩ B = ∅ ) and P(A and B) = P(A ∩ B) = 0 .
Example A
Determine if the two events are mutually exclusive.
a. A die is tossed once. Let A be the event of rolling a 3 and B be the
event of rolling a 5.
b. One card is drawn from a standard deck of 52 playing cards. Let A be
the event of drawing a Jack and B be the event of drawing a red card.
Probability of Event A or Event B, P(A∪ B)
If events A and B are from the same sample space, the probability of event A
or event B occurring is
P(A or B) = P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Furthermore, if events A and B are mutually exclusive, then P(A ∩ B) = 0 and
P(A or B) = P(A ∪ B) = P(A) + P(B) .
Example 7
One card is drawn from a standard deck of 52 cards.
a. What is the probability that the card is a spade?
b. What is the probability that the card is a face card?
Math 110 Chapter 8 (page 27 of 29)
c. What is the probability that the card is a spade and a face card?
d. What is the probability that the card is a spade or a face card?
Example 8
The personnel department of a company has compiled data on number of
years employees’ have worked for the company. The data is below. If an
employee is chosen at random, what is the probability that the employee has
nine or fewer years of service?
Years
0-4
5-9
10-14
15-19
20-24
25-29
30-34
35-39
40-44
# Employees
157
89
74
63
42
38
37
21
8
Math 110 Chapter 8 (page 28 of 29)
Independent Events
Two events A and B are independent if the occurrence of one has no effect
on the occurrence of the other. For instance, each roll of a die is independent
of any other roll.
Probability of Event A and Event B
If events A and B are independent, then the probability that both A and B
will occur is
P(A and B) = P(A ∩ B) = P(A) ⋅ P(B).
Example 9
A die is rolled three times.
a. What is the probability that all three rolls are a 3?
b. What is the probability that a 3 is rolled first, a 5 second, and a 1 third?
c. What is the probability that all three rolls result in a number less than 5?
Example 10
Suppose 58% of the U.S. population is 30 years old or older. If 10 people
are chosen at random, what is the probability that all 10 are 30 years old or
older.
Math 110 Chapter 8 (page 29 of 29)
Complement of an Event
The complement of event A, denoted A c , is the set of all outcomes in the
sample space that are not in A.
Example A
Suppose a card is drawn from a standard deck of 52 playing cards. Let A be
the event drawing a face card.
a. Describe the event A c . (answer: A c is the event of drawing any card other than a
face card)
b. Find n(A), n(A c ), P(A) and P(A c ).
c. Verify P(A c ) = 1− P(A).
Probability of the Complement of an Event
If A is an event with complement A c , then P(A c ) = 1− P(A).
Example 11
A machine averages one faulty unit for every 1000 it produces. What is the
probability that an order of 200 units will have one or more faulty units?