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Sequences
Sequences and Series
Precalculus 9.1
Example 1
Write the first five terms of each sequence:
a) an  2n  1
b) an  2  (1) n
Example 3
Write an expression for the most apparent nth
term of each sequence
a) 1,5,9,13,17…
b) 2,-9,28,-65,126
Example 2
Write the first five terms of an 
2  ( 1) n
n
Sequences
Some sequences are defined recursively. To
define a sequence recursively, you need to be
given one or more of the first few terms.
All other terms of the sequence are then
defined using previous terms.
A well-known recursive sequence is the
Fibonacci sequence.
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Example 4
Factorial Notation
Write the first five terms of the recursivelydefined sequence
a1  6
ak 1  ak  1
Example 5
Write the first five terms of the sequence given
below. Begin with n=0. Then graph the terms
on a set of coordinate axis.
3n  1
an 
n!
Example 6
Evaluate each factorial expression:
a)
9!
3!7!
Summation Notation
There is a convenient notation for the sum of the terms of a
finite sequence. It is called summation notation or sigma
notation because it involves the use of the uppercase Greek
letter sigma, written as .
b)
3!8!
4!4!
c)
( n  1)!
n!
Example 7
Find each sum.
4
a)  ( 4i  1)
i 1
5
b) ( 2  k 3 )
k 2
6
c) 
i 1
2
i!
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Summation Notation
Series
Example 8
Example 9
For the series, find a) the 3rd partial sum and
b) the sum

5

i
10
i 1
A deposit of $1500 is made in an account that
earns 2.5% interest compounded monthly.
The balance in the account after n months is
given by An  15001  0.12025 n
a) Write the first three terms of the sequence.
b) Find the balance in the account after four
years by computing the 48th term of the
sequence.
Example 10
The number of employees in physicians’ offices
in the US from 2002 through 2007 is
approximated by an  2.52n 2  24.8n  1908
where an is the number of employees (in
thousands) and n represents the year, with
n=2 corresponding to 2002. Find the last
three terms of this sequence, which represent
the number of physicians’ office employees
for the years 2005 through 2007.
Arithmetic Sequences & Partial
Sums
Precalculus 9.2
Source: US Bureau of Labor Statistics
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Arithmetic Sequences
A sequence whose consecutive terms have a common
difference is called an arithmetic sequence.
Example 1
For each of the arithmetic sequences, find the
common difference.
a) 5,8,11,14,17…
b) 9,5,1,-3,-7
c) 1, 76 , 43 , 32 , 53 ...
Arithmetic Sequences
Example 2
Find a formula for the nth term of the arithmetic
sequence whose common difference is 5 and
whose first term is -1.
Example 3
The eighth term of an arithmetic sequence is 25,
and the 12th term is 41. Write the first five
terms of this sequence.
Example 4
Find the tenth term of the arithmetic sequence
that begins at 8 and 20.
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The Sum of a Finite Arithmetic Sequence
Example 5
Find the sum: 40+37+34+31+28+25+22
Example 6
Find the sum of the given integers:
a) 1 to 35
b) 1 to 57
Example 8
Determine the seating capacity of an auditorium
with 30 rows of seats if there are 20 seats in
the first row, 22 seats in the second row, 24
seats in the third row, and so on.
Example 7
Find the 50th partial sum of the arithmetic
sequence -6, -2, 2, 6…
Example 9
Consider a job offer with a starting salary of
$32,500 and an annual raise of $2500.
Determine the total compensation from the
company through six full years of
employment.
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Geometric Sequences
Geometric Sequences and Series
Precalculus 9.3
• A geometric sequence may be thought of as an
exponential function whose domain is the set of
natural numbers.
Example 1
Geometric Sequences
For each of the geometric sequences, find the
common ratio:
a) 3,9,27,81,243… b) 10,20,40,80,160…
c)
1 1
4 16
1
1
1
, , 64
, 256
, 1024
...
If you know the nth term of a geometric sequence, you
can find the (n + 1)th term by multiplying by r. That is,
an+1 = anr.
Example 2
Example 3
Write the first five terms of the geometric
sequence whose first term is 2 and whose
common ratio is 4. Then graph the terms on a
set of coordinate axes.
Find the ninth term of the geometric sequence
whose first term is 4 and whose common ratio
is ½.
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Example 4
Find the 10th term of the geometric sequence
2
3
6,2, ...
Example 5
The second term of a geometric sequence is -18,
and the fifth term is 2/3. Find the sixth term.
Example 6
The Sum of a Finite Geometric Sequence
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Find the sum  2
n 1
n 1
Geometric Series
What do we do differently for these sums:
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a)  2 n 1
n 0
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b)
2
n
n 1
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Example 7
Find each sum

a)  5 12 
n
b) 5+0.5+0.05+0.005+…
n 0
Example 8
A deposit of $100 is made on the first day of
each month in a savings account that pays 6%
compounded monthly. Find the balance of
this annuity after 4 years.
Introduction
In this section, we will study a form of mathematical
proof called mathematical induction.
Mathematical Induction
S1 = 1 = 12
S2 = 1 + 3 = 22
Precalculus 9.4
S3 = 1 + 3 + 5 = 32
S4 = 1 + 3 + 5 + 7 = 42
S5 = 1 + 3 + 5 + 7 + 9 = 52
Introduction
Judging from the pattern formed by these first five sums, it
appears that the sum of the first n odd integers is
Sn = 1 + 3 + 5 + 7 + 9 + …. + (2n – 1) = n2.
Although this particular formula is valid, it is important for you
to see that recognizing a pattern and then simply jumping
to the conclusion that the pattern must be true for all
values of n is not a logically valid method of proof.
There are many examples in which a pattern appears to be
developing for small values of n and then at some point the
pattern fails.
Introduction
One of the most famous cases of this was the conjecture by
the French mathematician Pierre de Fermat, who
speculated that all numbers of the form
Fn = 22n + 1,
n = 0, 1, 2, . . .
are prime.
For n = 0, 1, 2, 3, and 4, the conjecture is true.
F0 = 3
F1 = 5
F2 = 17
F3 = 257
F4 = 65,537
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Introduction
The size of the next Fermat number (F5 = 4,294,967,297)
is so great that it was difficult for Fermat to determine
whether it was prime or not.
However, another well-known mathematician, Leonhard
Euler, later found the factorization
Introduction
Just because a rule, pattern, or formula seems to work for
several values of n, you cannot simply decide that it is valid for
all values of n without going through a legitimate proof.
Mathematical induction is one method of proof.
F5 = 4,294,967,297
= 641(6,700,417)
which proved that F5 is not prime and therefore Fermat’s
conjecture was false.
Example 1
Find the statement Pk+1 for each given statement
Pk.
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a) Pk 
k ( k  3)
Example 1 con’t
c) Pk : k  4  6k 2
d) Pk : 4 k  3k  1
b) Pk : S k  2  5  8  ...  (3k  2)
Example 2
Use mathematical induction to prove the
following formula.
5  7  9  11  ...  (3  2n)  n( n  4)
Example 3
Use mathematical induction to prove the
following formula.
1(1  1)  2(2  1)  3(3  1)  ...  n(n  1) 
n( n  1)(n  1)
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Example 4
Prove that ( n  1)! 2 n for all integers n such
that n  2.
Example 5
Prove that 4 is a factor of 5n  1 for all positive
integers n.
Pattern Recognition
Example 6
Find the formula for the finite sum and prove its
validity.
3  7  11  15  ...  4n  1
Sums of Powers of Integers
Formulas dealing with the sums of various powers of the first n
positive integers are as follows.
Example 7
Find each sum.
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a)  i 4
i 1
5
b)
 3i
2
 5i

i 1
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Finite Differences
Finite Differences
The first differences of a sequence are found by
subtracting consecutive terms. The second
differences are found by subtracting
consecutive first differences.
For this sequence, the second differences are all
• The first and second differences of the
sequence 3, 5, 8, 12, 17, 23, . . . are as follows.
the same. When this happens, the sequence
has a perfect quadratic model.
If the first differences are all the same, the
sequence has a linear model. That is, it is
arithmetic.
Example 8
Find a quadratic model for the sequence
0, 4, 10, 18, 28, …
Quiz 9.1 to 9.4
Find first 8 terms of sequence
Find sum of sequence
Find term/formula/sum arithmetic/geometric
sequence
Find Pk+1.
Proof Math Induction
Binomial Coefficients
Recall that a binomial is a polynomial that has two terms.
The Binomial Theorem
Precalculus 9.5
In this section, you will study a formula that provides a quick
method of raising a binomial to a power. To begin, look at
the expansion of (x + y)n for several values of n.
(x + y)0 = 1
(x + y)1 = x + y
(x + y)2 = x2 + 2xy + y2
(x + y)3 = x3 + 3x2y + 3xy2 + y 3
(x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y 4
(x + y)5 = x5 + 5x 4y + 10x3y2 + 10x2y 3 + 5xy 4 + y 5
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Binomial Coefficients
There are several observations you can make about these
expansions.
1. In each expansion, there are n + 1 terms.
2. In each expansion, x and y have symmetrical roles. The
powers of x decrease by 1 in successive terms, whereas
the powers of y increase by 1.
Binomial Coefficients
4. The coefficients increase and then decrease in a
symmetric pattern.
The coefficients of a binomial expansion are called binomial
coefficients. To find them, you can use the Binomial
Theorem.
3. The sum of the powers of each term is n. For instance, in
the expansion of (x + y)5, the sum of the powers of each
term is 5.
4+1=5
3+2=5
(x + y)5 = x5 + 5x4y1 + 10x3y2 + 10x2y3 + 5x1y4 + y5
Example 1
Example 2
Find each binomial coefficient.
a) 9 C 2
b) 11
4
 
c) 8 C 0
Find each binomial coefficient.
 5
d)  
 5
Binomial Coefficients
a) 8 C 5
b)  8 
 3
 
c) 14 C 1
14 
d)  
 13 
Pascal’s Triangle
In general, it is true that
nCr
= nCn–r.
This shows the symmetric property of binomial
coefficients.
The first and last numbers in each row of Pascal’s Triangle
are 1. Every other number in each row is formed by
adding the two numbers immediately above the
number.
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Pascal’s Triangle
Pascal noticed that numbers in this triangle are precisely the
same numbers that are the coefficients of binomial
expansions, as follows.
Binomial Expansions
When you write out the coefficients for a binomial that is raised
to a power, you are expanding a binomial.
Example 3
Use the 6th row of Pascal’s Triangle to find the
binomial coefficients.
6 C 0 , 6 C 1, 6 C 2 , 6 C 3 , 6 C 4 , 6 C 5 , 6 C 6
Example 4
Write the expansion for the expression  x  2 
4
The formulas for binomial coefficients give you an easy way to
expand binomials, as demonstrated in the next example.
Example 5
Write the expansion for each expression.
Example 6
Write the expansion for 3  x 2 3
a)  y  2 4
4
b) 2 x  y 
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Example 7
8
a) Find the fifth term of a  2b 
b) Find the coefficient of the term a 4b 7 in the
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expansion of 2a  3b 
Counting Principals
Precalculus 9.6
Example 1
Determine in how many ways a computer can
randomly generate two integers from 1 to 12
whose sum is 6.
The Fundamental Counting Principle
Example 2
Determine in how many ways a computer can
randomly generate two distinct integers from
1 to 12 whose sum is 10.
Example 3
A combination lock will open when the right
choice of three numbers (from 1 to 30,
inclusive) is selected. How many different lock
combinations are possible?
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Example 4
Permutations
In a certain state, each automobile license plate
number consists of two letters followed by a
four-digit number. How many distinct license
plate numbers can be formed?
Example 5
Permutations
In how many ways can eight children line up in a
row?
Example 6
Permutations
A coin club has five members. In how many ways
can a president and a vice-president be
selected from its members?
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Permutations
Example 7
Find the number of distinguishable
permutations of this group of letters:
A, A,E, E, E, J, P, P, R
Combinations
Example 8
In how many different ways can two letters be
chosen from the letters A, B, C, D, E, F, and G?
(The order of the two letters is not important)
Example 9
For an experiment, four students are randomly
selected from a class of 20. How many
different groups of four students are possible?
Example 10
You are forming a 10 person committee from 9
women and 12 men. The committee must
consist of 5 women and 5 men. How many
different 10-member committees are
possible?
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The Probability of an Event
Any happening for which the result is uncertain
is called an experiment.
Probability
Precalculus 9.7
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 subcollection of a sample
space is an event.
Example 1
Find the sample space for each of the following.
a) A letter is chosen at random from the
alphabet.
b) A six-sided die is tossed.
c) A coin is flipped once, and a six-sided die is
tossed once.
The Probability of an Event
If P(E) = 0, event E cannot occur, and E is called
an impossible event.
If P(E) = 1, event E must occur, and E is called a
certain event.
The Probability of an Event
To calculate the probability of an event, count
the number of outcomes in the event and in
the sample space.
Example 2
a) Two coins are tossed. What is the probability
that one will land heads up and that the
other will land tails up?
b) A card is drawn from a standard deck of 52
playing cards. What is the probability that
the card is a heart?
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Example 3
A six-sided die is tossed twice. What is the
probability that the total of the two tosses is
5?
Example 4
Show that the probability of drawing a card that
is a heart from a standard deck of playing
cards is the same as the probability of drawing
the ace of hearts from the set consisting of
the aces of hearts, diamonds, clubs, and
spades.
Example 5
Example 6
A class is given a list of 20 study problems from
which 10 will be part of an upcoming exam. If
a given student knows how to solve 17 of the
problems, find the probability that that
student will be able to answer all 10 questions
on the exam.
A college sends a survey to selected members of
the class of 2009. Of the 1354 people who
graduated that year, 672 are female, and 682
are male. If a graduate is selected at random
for the survey, what is the probability that the
person is female?
Mutually Exclusive Events
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.
To find the probability that one or the other of two mutually
exclusive events will occur, you can add their individual
probabilities.
In the terminology of sets, the intersection of A and B is
the empty set, which implies that
P(A ∩ B) = 0.
For instance, if two dice are tossed, the event A of rolling
a total of 6 and the event B of rolling a total of 9 are
mutually exclusive.
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Example 7
One card is selected at random from a standard
deck of 52 playing cards. What is the
probability that the card is either a club or an
ace?
Example 8
A wind turbine plant has a total of 158 wage and salary workers.
The circle graph gives the age profile of these workers.
Age of Wage and Salary Workers
What is the probability
that a person selected at
random from the plant is in
the 35-44 age group?
3%
9%
16%
16-24
25-34
20%
35-44
45-54
25%
35-54 age group?
55-64
65 and over
27%
Independent Events
Two events are independent if the occurrence of one has no
effect on the occurrence of the other.
For instance, rolling a total of 12 with two six-sided dice has no
effect on the outcome of future rolls of the dice.
Example 10
A sales representative makes a sale on
approximately one-third of all calls. If, on a
given day, the representative contacts five
potential clients, what is the probability that a
sale will be made with each of the five
contacts?
Example 9
Two integers from 1 to 30 are chosen by a
random number generator. What is the
probability that both numbers are less than
12?
The Complement of an Event
The complement of an event A is the collection of
all outcomes in the sample space that are not in
A.
The complement of event A is denoted by A.
Because P(A or A) = 1 and because A and A are
mutually exclusive, it follows that P(A) + P(A) = 1.
So, the probability of A is P(A) = 1 – P(A).
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The Complement of an Event
Example 11
A sales representative makes a sale on
approximately one-third of all calls. If, on a
given day, the representative contacts five
potential clients, what is the probability that a
sale will be made with at least one contact?
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