chapter 8 student notes

Chapter 8 

Sequences, Series, and 

Probability 

Section 8.1 Sequences and Series 

Objective: In this lesson you learned how to use sequence, factorial, and 

summation notation to write the terms and sums of 

sequences. 

Course Number 

Instructor 

Date 

I. Sequences (Pages 580−582) 

An infinite sequence is . . . 

set of positive integers. 

a function whose domain is the 

What you should learn 

How to use sequence 

notation to write the 

terms sequences 

The function values a 1 , a 2 , a 3 , a 4 , . . . , a n , . . . are the 

of an infinite sequence. 

terms 

A finite sequence is . . . a sequence whose domain 

consists of the first n positive integers only. 

To find the first three terms of a sequence, given an expression 

for its nth term, . . . evaluate the expression for the nth term 

at n = 1 to find the first term, at n = 2 for the second term, and so 

on. 

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 . 

Example 1: Find the first five terms of the sequence given by 

a = 5 + 2n( 

−1) 

. 

n 

3, 9, − 1, 13, − 5 

n 

II. Factorial Notation (Pages 582−583) 

If n is a positive integer, n factorial is defined by 

n! = 1 ⋅ 2 ⋅ 3 ⋅ 4 ⋅ ⋅ ⋅ (n − 1) ⋅ n 


How to use factorial 

notation 

Zero factorial is defined as 0! = 1 . 

Larson/Hostetler/Edwards Precalculus with Limits: A Graphing Approach, Fifth Edition Student Notetaking Guide IAE 

Copyright © Houghton Mifflin Company. All rights reserved. 135

136 Chapter 8 Sequences, Series, and Probability 

Example 2: Evaluate the factorial expression 

n! 

. 

( n + 1)! 

1/(n + 1) 

III. Summation Notation (Pages 584−585) 

The sum of the first n terms of a sequence is represented by the 

summation or sigma notation, 

n 

∑ a i 

i= 

1 

= a 1 + a 2 + a 3 + a 4 + . . . + a n 

where i is called the index of summation , n is the 

upper limit of summation , and 1 is the lower 

limit of summation . 


How to use summation 

notation to write sums 

Example 3: Find the following sum: ∑ (2 + 3i ) . 

93 

7 

i= 

2 

IV. Series (Page 585) 

The sum of the terms of a finite or infinite sequence is called a 

series . 


How to find sums of 

infinite series 

Consider the infinite sequence a 1 , a 2 , a 3 , . . . , a i , . . . . The sum of the 

first n terms of the sequence is called a(n) finite series or 

the partial sum of the sequence and is denoted by 

n 

∑ 

a1 + a2 

+ a3 

+ + a n = a i . The sum of all terms of the infinite 

i= 

1 

sequence is called a(n) infinite series and is denoted by 

a 

1 

+ a 

2 

+ a 

3 

= ∑ ∞ + + a i + a i . 

= 

Homework Assignment 

Page(s) 

Exercises 

i 1 


Copyright © Houghton Mifflin Company. All rights reserved.

Section 8.2 Arithmetic Sequences and Partial Sums 137 

Section 8.2 Arithmetic Sequences and Partial Sums 

Objective: In this lesson you learned how to recognize, write, and use 

arithmetic sequences. 

Course Number 

Instructor 

Date 

Important Vocabulary 

Define each term or concept. 

Arithmetic sequence A sequence in which the differences between consecutive terms 

are the same. 

I. Arithmetic Sequences (Pages 592−594) 

The common difference of an arithmetic sequence is . . . 

difference d between consecutive terms of an arithmetic 

sequence. That is, d = a 2 – a 1 = a 3 – a 2 = a 4 – a 3 = … 

the 


How to recognize, write, 

and find the nth terms of 

arithmetic sequences 

The nth term of an arithmetic sequence has the form 

a n = dn + c 

, where d is the common difference 

between consecutive terms of the sequence, and c = a 1 − d. An 

arithmetic sequence an 

= dn+ c can be thought of as counting 

by d’s after a shift of c units from d . 

Example 1: Determine whether or not the following sequence 

is arithmetic. If it is, find the common difference. 

7, 3, − 1, − 5, − 9, . . . 

Yes; − 4 

Example 2: Find a formula for the nth term of the arithmetic 

sequence whose common difference is 2 and 

whose first term is 7. 

a n = 2n + 5 

The nth term of an arithmetic sequence has the alternative 

recursion formula a n = a 1 + (n − 1)d . 

Example 3: Find the sixth term of the arithmetic sequence that 

begins with 15 and 12. 

0 




II. The Sum of a Finite Arithmetic Sequence 

(Pages 595−596) 

The sum of a finite arithmetic sequence with n terms is given by 

S n = n/2(a 1 + a n ) . 


How to find nth partial 

sums of arithmetic 

sequences 

The sum of the first n terms of an infinite sequence is called the 

nth partial sum . 

Example 4: Find the sum of the first 20 terms of the sequence 

with nth term a n = 28 − 5n 

. 

− 490 

III. Applications of Arithmetic Sequences (Pages 596−597) 

Describe a real-life problem that could be solved by finding the 

sum of a finite arithmetic sequence. 

Answers will vary. 


How to use arithmetic 

sequences to model and 

solve real-life problems 

Additional notes 


Page(s) 

Exercises 



Section 8.3 Geometric Sequences and Series 139 

Section 8.3 Geometric Sequences and Series 

Objective: In this lesson you learned how to recognize, write, and use 

geometric sequences. 

Course Number 

Instructor 

Date 



Geometric sequence A sequence in which the ratios of consecutive terms are the 

same. 

Infinite geometric series or geometric series The sum of the terms of an infinite 

geometric sequence. 

I. Geometric Sequences (Pages 601−603) 

The common ratio of a geometric sequence is . . . the ratio r 

between consecutive terms of a geometric sequence. That is, 

r = a 2 /a 1 = a 3 /a 2 = a 4 /a 3 = . . . , r ≠ 0. 


How to recognize, write, 

and find the nth terms of 

geometric sequences 

The nth term of a geometric sequence has the form 

a n = a 1 r n−1 , where r is the common ratio of 

consecutive terms of the sequence. So, every geometric sequence 

can be written in the following form: 

a 1 , a 1 r, a 1 r 2 , a 1 r 3 , a 1 r 4 , . . . , a 1 r n−1 , . . . . 

If you know the nth term of a geometric sequence, you can find 

the (n + 1)th term by multiplying by r . That is, 

a + 

= a n r . 

n 1 

Example 1: Determine whether or not the following sequence 

is geometric. If it is, find the common ratio. 

60, 30, 0, − 30, − 60, . . . 

No 

Example 2: Write the first five terms of the geometric 

sequence whose first term is a 1 = 5 and whose 

common ratio is − 3. 

5, − 15, 45, − 135, 405 

Example 3: Find the eighth term of the geometric sequence 

that begins with 15 and 12. 

3.145728 




II. The Sum of a Finite Geometric Sequence (Page 604) 

The sum of the geometric sequence a 1 , a 1 r, a 1 r 2 , a 1 r 3 , a 1 r 4 , . . ., 

a 1 r n−1 with common ratio r ≠ 1 is given by 

S n = a 1 (1 − r n )/(1 − r) . 


How to find nth partial 

sums of geometric 

sequences 

When using the formula for the sum of a geometric sequence, be 

careful to check that the index begins with i = 1. If the index 

begins at i = 0, . . . you must adjust the formula for the nth 

partial sum. 

Example 4: Find the sum ∑ 2(0.5) 

1.998046875 

10 

i= 

1 

i . 

III. Geometric Series (Pages 605−606) 

If | r | < 1, then the infinite geometric series a 1 + a 1 r + a 1 r 2 + a 1 r 3 

+ a 1 r 4 + . . . + a 1 r n−1 + . . . has the sum S = a 1 /(1 − r) . 


How to find sums of 

infinite geometric series 

If r ≥ 1 , the series does not have a sum. 

Example 5: If possible, find the sum: ∑ ∞ 

= 

12 

i 1 

9(0.25) 

i−1 

. 

IV. Applications of Geometric Sequences (Page 606) 

Describe a real-life problem that could be solved by finding the 

sum of a finite geometric sequence. 

Answers will vary. 


How to use geometric 

sequences to model and 

solve real-life problems 


Page(s) 

Exercises 



Section 8.4 Mathematical Induction 141 

Section 8.4 Mathematical Induction 

Objective: In this lesson you learned how to use mathematical induction 

to prove statements involving a positive integer n. 

Course Number 

Instructor 

Date 



Mathematical induction A form of mathematical proof in which it must be shown 

for a statement, P n , involving the positive integer n, that if P 1 is true and that the truth 

of P k implies the truth of P k+1 for every positive k, then P n must be true for all positive 

integers n. 

I. Introduction (Pages 611−614) 

To apply the Principle of Mathematical Induction, you need to 

be able to determine the statement P k+1 for a 

given statement P k . 


How to use mathematical 

induction to prove 

statements involving a 

positive integer n 

When using mathematical induction to prove a summation 

formula, it is helpful to think of S k+1 as . . . 

S k+1 = S k + a k+1 where a k+1 is the (k + 1)th term of the original 

sum. 

Describe the process needed to prove a formula using 

mathematical induction. 

Mathematical induction consists of two distinct parts. First, you 

must show that the formula is true when n = 1. The second part 

of mathematical induction has two steps. The first step is to 

assume that the formula is valid for some integer k. The second 

step is to use this assumption to prove that the formula is valid 

for the next integer, k + 1. Combining the results of the first and 

second parts, you can conclude by mathematical induction that 

the formula is valid for all positive integer values of n. 

The extended principle of mathematical induction is . . . 

a slight variation of the Principle of Mathematical Induction in 

which a statement P n is proved true for n ≥ k by verifying P k , 

rather than P 1 . 




II. Sums of Powers of Integers (Page 615) 

List the formulas for the following sums of powers of integers. 

1. 1 + 2 + 3 + 4 + + n = n(n + 1)/2 


How to find the sums of 

powers of integers 

2 

2 

2 

2 

2. 1 + 2 + 3 + 4 + + n = n(n + 1)(2n + 1)/6 

3 

3 

3 

3 

3. 1 + 2 + 3 + 4 + + n = 

n 2 (n + 1) 2 /4 

4 

4 

4 

4 

3 

2 

4 

4. 1 + 2 + 3 + 4 + + n = n(n + 1)(2n + 1)(3n 2 + 3n − 1)/30 

5 

5 

5 

5 

5 

5. 1 + 2 + 3 + 4 + + n = n 2 (n + 1) 2 (2n 2 + 2n − 1)/12 

III. Finite Differences (Page 616) 

First differences are . . . found by subtracting consecutive 

terms of a sequence. 


How to find finite 

differences of sequences 

If the first differences of a sequence are all the same nonzero 

number, the sequence has a linear model, that is 

arithmetic . 

Second differences are . . . 

consecutive first differences. 

are found by subtracting 

When the second differences of a sequence are all the same and 

nonzero, the sequence has a perfect quadratic model. 


Page(s) 

Exercises 



Section 8.5 The Binomial Theorem 143 

Section 8.5 The Binomial Theorem 

Objective: In this lesson you learned how to use the Binomial Theorem 

and Pascal’s Triangle to calculate binomial coefficients and 

write binomial expansions. 

Course Number 

Instructor 

Date 



Binomial coefficients The coefficients of a binomial expansion. 

Pascal’s Triangle A triangular pattern, named for the French mathematician Blaise 

Pascal, in which the first and last numbers in each row are 1and every other number in 

each row is formed by adding the two numbers immediately above the number. These 

numbers are precisely the same as the coefficients of binomial expansions. 

I. Binomial Coefficients (Pages 619−620) 

List four general observations about the expansion of 

for various values of n. 

1) In each expansion, there are n + 1 terms. 

( x + y) 

n 


How to use the Binomial 

Theorem to calculate 

binomial coefficients 

2) In each expansion, x and y have symmetric roles. The powers of x 

decrease by 1 in successive terms, whereas the powers of y increase by 1. 

3) The sum of the powers in each term is n. 

4) The coefficients increase and then decrease in a symmetric pattern. 

The Binomial Theorem states that in the expansion of (x + y) n = 

x n + nx n – 1 y + . . . + n C r x n – r y r + . . . + nxy n – 1 + y n , the coefficient 

of x n – r y r is nC r = n!/((n − r)!r!) . 

Example 1: Find the binomial coefficient 12 C 5 . 

792 

II. Binomial Expansions (Pages 621−622) 

Writing out the coefficients for a binomial that is raised to a 

power is called expanding a binomial . 


How to use binomial 

coefficients to write 

binomial expansions 

Example 2: Write the expansion of the expression 

x 5 + 10x 4 + 40x 3 + 80x 2 + 80x + 32 

5 

( x + 2) . 




III. Pascal’s Triangle (Page 623) 

Construct rows 0 through 6 of Pascal’s Triangle. 

1 

1 1 

1 2 1 

1 3 3 1 

1 4 6 4 1 

1 5 10 10 5 1 

1 6 15 20 15 6 1 


How to use Pascal’s 

Triangle to calculate 

binomial coefficients 

Additional notes 


Page(s) 

Exercises 



Section 8.6 Counting Principles 145 

Section 8.6 Counting Principles 

Objective: In this lesson you learned how to solve counting problems 

using the Fundamental Counting Principle, permutations, 

and combinations. 

Course Number 

Instructor 

Date 



Fundamental Counting Principle Let E 1 and E 2 be two events. The first event E 1 

can occur in m 1 different ways. After E 1 has occurred, E 2 can occur in m 2 different 

ways. The number of ways that the two events can occur is m 1 ⋅ m 2 . 

Permutation An ordering of n different elements such that one element is first, one is 

second, one is third, and so on. 

Distinguishable permutations Suppose a set of n objects has n 1 of one kind of 

object, n 2 of a second kind, n 3 of a third kind, and so on, with n = n 1 + n 2 + n 3 + . . . + 

n k . Then the number of distinguishable permutations of the n objects is given by 

n!/(n 1 !⋅n 2 !⋅n 3 !⋅ ⋅ ⋅ ⋅ n k !). 

I. Simple Counting Problems (Page 627) 

If two balls are randomly drawn from a bag of six balls, 

numbered from 1 to 6, such that it is possible to choose two 3’s, 

the random selection occurs with replacement . If 

two balls are drawn from the bag at the same time, the random 

selection occurs without replacement , which 

eliminates the possibility of choosing two 3’s. 


How to solve simple 

counting problems 

II. The Fundamental Counting Principle (Page 628) 

The Fundamental Counting Principle can be extended to three or 

more events. For instance, if E 1 can occur in m 1 ways, E 2 in m 2 

ways, and E 3 in m 3 ways, the number of ways that three events 

E 1 , E 2 , and E 3 can occur is m 1 ⋅ m 2 ⋅ m 3 . 


How to use the 

Fundamental Counting 

Principle to solve more 

complicated counting 

problems 

Example 1: A diner offers breakfast combination plates which 

can be made from a choice of one of 4 different 

types of breakfast meats, one of 8 different styles 

of eggs, and one of 5 different types of breakfast 

breads. How many different breakfast combination 

plates are possible? 

160 different combinations 




III. Permutations (Pages 629−631) 

The number of different ways that n elements can be ordered is 

n! . 


How to use permutations 

to solve counting 

problems 

A permutation of n elements taken r at a time is . . . 

an ordering of subset of a collection of n elements, rather than 

the entire collection, for example, choosing and ordering r 

elements out of a collection of n elements. The number of n 

elements taken r at a time is given by n P r = n!/(n − r)! . 

Example 2: In how many ways can a chairperson, a vice 

chairperson, and a recording secretary be chosen 

from a committee of 14 people? 

2184 ways 

Example 3: In how many distinguishable ways can the letters 

COMMITTEE be written? 

45,360 ways 

IV. Combinations (Pages 632−633) 

A combination of n elements taken r at a time is . . . 

a subset of a larger set in which the order is not important. The 

number of combinations of n elements taken r at a time is given 

by n C r = n!/((n − r)! r!). 


How to use combinations 

to solve counting 

problems 

Example 4: In how many ways can a research team of 3 

students be chosen from a class of 14 students? 

364 ways 


Page(s) 

Exercises 



Section 8.7 Probability 147 

Section 8.7 Probability 

Objective: In this lesson you learned how to find the probability of 

events and their complements. 

Course Number 

Instructor 

Date 



Independent events Two events are independent if the occurrence of one has no 

effect on the occurrence of the other. 

Complement of an event The collection of all outcomes in the sample space that are 

not in the event. 

I. The Probability of an Event (Pages 637−640) 

Any happening whose result is uncertain is called a(n) 

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 subcollection of a sample space is a(n) 

event . 


How to find probabilities 

of events 

To calculate the probability of an event, . . . count the number 

of outcomes in the event, n(E), and in the sample space, n(S). 

If an event E has n(E) equally likely outcomes and its sample 

space S has n(S) equally likely outcomes, the probability of 

event E is given by P(E) = n(E)/n(S) . 

The probability of an event must be between 0 and 

1 . 

If P(E) = 0, the event E cannot occur, and E is called 

a(n) impossible event. If P(E) = 1, the event E 

must occur, and E is called a(n) certain 

event. 

Example 1: A box contains 3 red marbles, 5 black marbles, 

and 2 yellow marbles. If a marble is selected at 

random from the box, what is the probability that 

it is yellow? 

1/5 




II. Mutually Exclusive Events (Pages 641−642) 

Two events A and B (from the same sample space) are mutually 

exclusive if A and B have no outcomes in common. 



of mutually exclusive 

events 

If A and B are events in the same sample space, the probability of 

A or B occurring is given by P(A ∪ B) = P(A) + P(B) − P(A ∩ B) . 

To find the probability that one or the other of two mutually 

exclusive events will occur, . . . add their individual 

probabilities. 


and 2 yellow marbles. If a marble is selected at 

random from the box, what is the probability that 

it is either red or black? 

4/5 

III. Independent Events (Page 643) 

If A and B are independent events, the probability that both A 

and B will occur is P(A and B) = P(A) ⋅ P(B) . 



of independent events 

That is, to find the probability that two independent events will 

occur, . . . multiply the probabilities of each event. 


and 2 yellow marbles. If two marbles are 

randomly selected with replacement, what is the 

probability that both marbles are yellow? 

1/25 

IV. The Complement of an Event (Page 644) 

Let A be an event and let A′ be its complement. If the probability 

of A is P(A), the probability of the complement is 

P(A′) = 1 − P(A) . 



of complements of events 


Page(s) 

Exercises

chapter 8 student notes

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