Chapter 10 Notes: Sequences, Induction, and Probability
Day 31: Section 10-1: Sequences and Summation Notation
After completing section 10-1 you should be able to do the following:
1. Find particular terms of a sequence from the general term.
2. Use recursion formulas.
3. Use factorial notation.
4. Use summation notation.
Definition of a Sequence
A sequence is a function whose domain is a set of consecutive integers. If the domain is not specified,
it is understood that the domain starts with 1. The values in the range are called terms of the sequence.
Domain:
1
2
3
4……. n
The relative position of each term
Range:
a1
a2
a3
a4……. an
Terms of the sequence
A finite sequence has a limited number of terms. An infinite sequence continues without stopping.
Finite Sequence: 2, 4, 6, 8
Infinite Sequence: 2, 4, 6, 8, ……
A sequence can be specified by an equation, or rule. For example both sequences above can be described by the rule:
an = 2n
Finding the term of a sequence if given a rule for the term is as simple as evaluating the rule for the term number.
Practice: 1-2, Write the first four terms of each sequence whose rule is given.
1. an   1
n
n
2
2. an 
2n
n4
Recursion Formulas
A recursion formula defines the nth term of a sequence as a function of the previous term. When you use a recursion formula
you can only get the next term of the sequence and you must be given the prior term.
Example of using a recursion formula: Write the first four terms of the sequence that are defined by using the recursion
formula of: a1 = 7 and an = 2an−1 − 5 for n ≥ 2.
Math Analysis Notes prepared by Mrs. Atkinson
1
Practice: 1-2, are defined using recursion formulas. Write the first four terms of each sequence.
1. a1 = 4 and an = an−1 + 3 for n ≥ 2.
2. a1 = 0 and an = −an−1 + 3
Factorial Notation
Products of consecutive positive integers occur quite often in sequences. These products can be expressed in a special notation,
called factorial notation.
Factorial Notation
If n is a positive integer, the notation n! (read “n factorial”) is the product of all positive integers from n
down through 1.
n! = n(n – 1)(n – 2)……(3)(2)(1)
0! (zero factorial), by definition is 1.
0! = 1
Examples of values of n! for the first 5 positive integers are:
1! = 1
2! = 2∙1 = 2
3! = 3∙2∙1 = 6
4! = 4∙3∙2∙1 = 24
5! = 5∙4∙3∙2∙1 = 120
Practice: In 1-2, evaluate the expression.
1. 3∙4!
2. (3∙4)!
Practice: In 1-2, write the first four terms of the sequence whose nth term given:
1. an 
( n  1)!
2
2.
3n !
Math Analysis Notes prepared by Mrs. Atkinson
2
Summation Notation
It is sometimes useful to find the sum of the first n terms of a sequence. A compact notation of expressing the sum of the first n
terms is called summation notation.
Summation Notation
The sum of the first n terms of a sequence ins represented by the summation notation
n
a
i 1
i
 a1  a2  a3  ......  an 1  an ,
where i is the index of summation, n is the upper limit of summation, and 1 is the lower limit of
summation.
Practice: In 1-2, Expand and evaluate the sum.
4
1.
 2i
i 1
6
2.
 2  h  1 !
h 3
Practice: In 1-2, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of
summation.
1.
1 2 3
14
   ...... 
2 3 4
14  1
4. 1 + 3 + 5 +….+(2n – 1)
Asn 31: p933#3-54 by 3’s, 62-68(even)
Chapter P Final Exam Review Worksheet #2-23 by 3’s
Math Analysis Notes prepared by Mrs. Atkinson
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Day 33: Section 10-2: Arithmetic Sequences
After completing section 10-2 you should be able to do the following:
1. Find the common difference for an arithmetic sequence.
2. Write terms of an arithmetic sequence.
3. Use the formula for the general term of an arithmetic sequence.
4. Use the formula for the sum of the first n terms of an arithmetic sequence.
Definition of an Arithmetic Sequence
An arithmetic sequence is a sequence in which each term after the first differs from the preceding term by a constant
amount. The difference between consecutive terms is called the common difference of the sequence.
Formula for the General Term (an) of an Arithmetic Sequence
The nth term (the general term) of an arithmetic sequence with first term a 1 and common difference d is:
an  a1   n 1 d
Practice: In 1-4, write the formula for the general term of the arithmetic sequence then find the indicated term using the given
information.
1. Find an and a16 when a1 = 9 and d = 2.
2. Find an and a7 when d = −4.1 and a16 = 48.2
3. Find an and a5 when a7 = 34 and a18 = 122
4. Find an and a6 when a2 = −28 and a20 = 52
The Sum of the First n Terms of an Arithmetic Sequence
The sum, Sn, of the first n terms of an arithmetic sequence is given by
Sn 
n
 a1  an 
2
In which a1 is the first term and an is the nth term.
Practice: Find the sum of the first 20 terms of the arithmetic sequence: 4, 10, 16, 22,…..
Math Analysis Notes prepared by Mrs. Atkinson
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Practice: Find the sum of the first 60 positive event integers. Practice: Find the sum of the even integers between 21 and 45.
20
Practice: Evaluate:
 2i  4
i 1
Practice: Find n given Sn = 175 for the arithmetic sequence: 4, 7, 10, 13, 16, 19,…….
Practice: Find n given Sn = 366 for the arithmetic sequence: 3, 8, 13, 18, 23,……
Asn 33: p943#3-48 by 3’s
Arithmetic Worksheet (all)
Chapter P Final Exam Worksheet #3-24 by 3’s
Math Analysis Notes prepared by Mrs. Atkinson
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Day 34: Section 10-3: Geometric Sequences and Series
After completing section 10-3 you should be able to do the following:
1. Find the common ratio of a geometric sequence.
2. Write terms of a geometric sequence.
3. Use the formula for the general term of a geometric sequence.
4. Use the formula for the sum of the first n terms of a geometric sequence.
5. Use the formula for the sum of an infinite geometric series.
Definition of a Geometric Sequence
A geometric sequence is a sequence in which each term after the first is obtained by multiplying the preceding term by a
fixed nonzero constant. The amount by which we multiply each time is called the common ratio of the sequence.
Formula for the General Term (an) of a Geometric Sequence
The nth term (the general term) of a geometric sequence with first term a 1 and common ratio r is:
an  a1 r n 1
Practice: Write the first five terms of each geometric sequence given: a 1 = 5 and r = 3.
In 1-3, Write the rule for the nth term of the geometric sequence using the given information.
1. a1 = 4 and r = 2
2. a2 = 45 and a5 = −1215
3. a3 = 18 and a6 =
2
3
Math Analysis Notes prepared by Mrs. Atkinson
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The Sum of the First n Terms of a Geometric Sequence
The sum, Sn, of the first n terms of a geometric sequence is given by
Sn 

a1 1  r n

1 r
In which a1 is the first term and r is he common ratio (r ≠ 1).
Practice: In 1-2, Find the indicated sum for the geometric sequence.
1. Find the sum of the first 10 terms of: 1, 5, 25, 125, 625,…….
10
2.
 6  2
i 1
i 1
Practice: In 1-2, Find the number of terms n need to get the given sum S n.
2. S n 
1. Sn = 3906 given 1, 5, 25, 125, 625,……
1023
given 4, 2, 1, ……
128
The Sum of an Infinite Geometric Series
The sum of an infinite geometric series with first term a1 and common ratio r is given by
S
a1
1 r
provided r  1 . If r  1 , the series has no infinite sum.
Practice: In 1-2, Find the sum of each infinite geometric series.
1 1 1
 .....
1. 1   
3 9 27

2.
1
4 

i 1  2 
i 1
Math Analysis Notes prepared by Mrs. Atkinson
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Practice: Find the common ratio of an infinite geometric series with first term a 1 = 
4
and has a sum of −2.
3
Asn 34: p955#3-42 by 3’s
Geometric Sequence Worksheet
Chapter 1 Final Exam Worksheet #1-16 by 3’s
Day 35: Section 10-4: Mathematical Induction
After completing section 10-4 you should be able to do the following:
1. Write the statement for Sk and Sk + 1.
2. Use mathematical induction to prove statements about all positive integers.
Mathematical Induction works as follows. If you know from the basis step that a statement is true for n = 1, then the
inductive step implies that it is true for n = 2, and therefore, n = 3, and so on for all positive integers n.
Mathematical Induction
To show that a statement is true for all positive integers n, perform these steps:
1. Show that the statement is true for n = 1.
2. Assume that the statement is true for n = k.
3. Prove that the statement is true for n = k + 1.
Example Use mathematical induction to prove the statement.
4 + 7 + 10 + 13 + …..+ (3n + 1)=
n  3n  5
2
Math Analysis Notes prepared by Mrs. Atkinson
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Practice: A statement for Sn about the positive integers is given. Write statements S1, S2, and S3 and show that each of these
statements is true.
Sn: 3 + 4 + 5 +……+ (n + 2) =
n  n  5
2
Practice: A statement for Sn about the positive integers is given. Write statements Sk, and Sk+1, make sure to simplify
statement Sk+1 completely.
Sn: 4 + 8 + 12 +…..+ 4n = 2n(n + 1)
Practice: Use mathematical induction to prove that each statement is true for every positive integer n.
1 + 2 + 22 +……+ 2n − 1 = 2n – 1
Asn 35: p965#2-18(even)
Chapter 1 Final Exam Worksheet #2-17 by 3’s
Study for Chapter 10 Quiz Sections 1-3
Math Analysis Notes prepared by Mrs. Atkinson
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Day 36: Quiz Chapter 5 Sections 1-3
Asn 36: p966#20-24(even)
Chapter 1 Final Exam Worksheet #3-15 by 3’s
Day 37: Section 10-5: The Binomial Theorem
After completing section 10-5 you should be able to do the following:
1. Expand a binomial raised to a power using the Binomial Theorem
The Binomial Theorem
The binomial expansion of (a + b)n for any positive integer n is:
(a + b)n = nC0anb0 + nC1an – 1b1 + nC2an – 2b2 +…….+ nCna0bn
The red parts of the binomial expansion are called the binomial coefficients.
You can find the binomial coefficients by using Pascal’s Triangle.
Pascal’s Triangle:
Pascal's Triangle is an arithmetical triangle you can use for some neat things in mathematics. Here's how
you construct it:
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
1
7 21 35
35 21 7
1
.
.
.
You start out with the top two rows: 1, and 1 1. Then to construct each entry in the next row, you look at
the two entries above it (i.e. the one above it and to the right, and the one above it and to the left). At the
beginning and the end of each row, when there's only one number above, put a 1. You might even think of
this rule (for placing the 1's) as included in the first rule: for instance, to get the first 1 in any line, you add
up the number above and to the left (since there is no number there, pretend it's zero) and the number
above and to the right (1), and get a sum of 1.
Practice: What is the 8th row of Pascal’s Triangle.
Practice: In 1-2, Use the Binomial Theorem to expand each binomial.
1. (2x – 3y)4
2. (x + 2y)5
Asn 37: p972#10-30(even)
Chapter 2 Final Exam Worksheet #1, 5, 10
Math Analysis Notes prepared by Mrs. Atkinson
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Day 38: Section 10-5: The Binomial Theorem(continued) Section 10-6: Counting Principles,
Permutations, and Combinations
After completing section 10-5 and 10-6 you should be able to do the following:
1. Find a particular term in a binomial expansion.
2. Use the Fundamental Counting Principle.
3. Use the permutations formula.
4. Use the combinations formula.
5. Distinguish between permutation problems and combination problems.
Finding a Particular Term in a Binomial Expansion
By observing the terms in the formula for expanding binomials, we can find a formula for finding a particular term without
writing the entire expansion: Second
(a + b)n = nC0anb0
+
nC1a
n–1 1
b
nth Term
Third
Term
Term
First
Term
+
nC2a
n–2 2
b
+…….+
nCna
0 n
b
The exponent of b is 1 less than the term number
Use the following formula to find a particular term in a Binomial Expansion
The (r + 1)st term of the expansion of (a + b) n is:
n–r r
b
nCra
Practice: In 1-2, find the indicated term of the binomial expansion.
1. Find the 3rd term of (x – 2y)5
2. Find the sixth term of (2x + 3y)9
The Fundamental Counting Principle
The number of ways in which a series of successive things can occur is found by multiplying the number of ways each thing
can occur.
Example of Fundamental Counting Principle:
A pizza can be ordered with three choices of size (small, medium, or large), four choices of crust (thin, think, crispy, or
regular), and six choices of toppings(ground beef, sausage, pepperoni, bacon, mushrooms, or onions). How many different
one-topping pizzas can be ordered?
Event #1: Size
3
Event #2: Crust
4
Event #3: Toppings
6
Total number of one-topping pizzas
that can be ordered: 3∙4∙6 = 72
Math Analysis Notes prepared by Mrs. Atkinson
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Practice: You are taking a multiple-choice test that has six questions. Each of the questions has three answer choices, with
one correct answer per question. If you select one of these three choices for each questions and leave nothing blank, in homw
many ways can you answer the questions?
Practice: License plates in a state display two letters followed by three digits. How many different license plates can be
manufactured (a) if you are allowed to repeat letters and digits and (b) if you are not allowed to repeat letters and digits.
Permutations
A permutation is an ordered arrangement of items.
The number of possible permutations if r items are taken from n items is
n!
n Pr 
 n  r !
Practice: A corporation has seven members on its board of directors. In how many different ways can it elect a president,
vice-president, secretary, and treasurer?
Practice: In how many ways can 6 different books be lined up?
Combinations
Order does not matter.
The number of possible combinations if r items are taken form n items is
n!
n Cr 
n  r !r !
Practice: From a group of 10 doctors, in how many was can four of them be selected to attend a conference?
Math Analysis Notes prepared by Mrs. Atkinson
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Practice: How many different 5-card poker hands are possible from a standard 52-card deck?
Asn 38: p972#40-46(even)
p981#3-66 by 3’s
Chapter 2 Final Exam Worksheet #2, 6, 11
Day 39: Section 10-7: Probability
After completing section 10-7 you should be able to do the following:
1. Compute theoretical probability.
2. Find the probability of one event or a second event occurring.
3. Find the probability of one event and a second event occurring.
Theoretical Probability
If an event E has n(E) equally likely outcomes and its sample space S has n(S) equally likely outcomes, the theoretical
probability of event (E), denoted by P(E), is:
number of outcomes in event E
n( E )
P( E ) 

number of outcomes in sample space E n( s )
Example: A die is rolled. Find the probability of getting a number less than 5.
Solution: n(E) = 4 since the numbers on a die less than 5 are: 1, 2, 3, and 4; n(S) = 6 since there are six sides of the dice each
one with a different number so:
4 2
P(rolling a number less than 5)= 
6 3
Practice: What is the probability of getting a sum of 5 when two six-sided dice are rolled?
Practice: If you are dealt one card from a standard 52-card deck, find the probability of being dealt a king.
Math Analysis Notes prepared by Mrs. Atkinson
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Practice: What is the probability of getting a 5-card hand that contains two aces and three other cards (not aces) from a
standard deck of 52 cards.
Practice: What is the probability of getting a 5-card hand that contains 3-cards of the same suit and 2-cards of a different suit.
Or Probabilities with Mutually Exclusive Events (events that have nothing in common).
If A and B are mutually exclusive events, then
P(A or B) = P(A) + P(B)
Practice: If one card is randomly selected from a deck of 52 cards, what is the probability of selecting a king or a seven?
Practice: If you roll a single, six-sided die, what is the probability of getting either a 4 or a 5?
Or Probabilities with Events That are Not Mutually Exclusive (events that have something in common).
If A and B are not mutually exclusive events, then
P(A or B) = P(A) + P(B) – P(A and B)
where P(A and B) is the overlap when both events happen at the same time.
Practice: If one card is randomly selected from a deck of 52 cards, what is the probability of selecting a king or a heart?
Math Analysis Notes prepared by Mrs. Atkinson
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Practice: If you roll a single, six-sided die, what is the probability of getting either a 4 or an even number?
And Probabilities with Independent Events
If A and B are independent events, then
P(A and B) = P(A)∙P(B)
Practice: Find the probability of a family having four boys in a row.
Practice: Find the probability a single die that is rolled twice will have an even number the first time it is rolled and a number
greater than 2 the second time rolled.
Asn 39: p994#1-26(all), 29, 30, 37-42(all), 47-52(all)
Chapter 2 Final Exam Worksheet #3, 4, 7, 8, 9
Chapter 10 Review Sheet
Complete each problem on a separate sheet of paper. Show all of your work! NO WORK = NO
CREDIT!
For question 1-3, write a rule for the nth term of the arithmetic sequence. Then find a20 .
9 11 13
1. 1, 7, 13, 19, 25, …
2. ,5, , 6, ,…
3. -4, -2, 0, 2, 4,…
2
2
2
For questions 4-6, write a rule for the nth term of the arithmetic sequence.
4. d = -4, a1 =7
5. a4  18, a10  48
6. a7  22, a11  34
For questions 7-8, find the sum of the first n terms of the arithmetic series and
5
7
7. 2   3   4  ... , n = 50
8. 32, 24, 16, 8, 0, …,
n = 30
2
2
For questions 9-10, find n for the given sum, S n .
Math Analysis Notes prepared by Mrs. Atkinson
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9. 2 
5
7
 3   4  ... , S n =1005
2
2
10. 32, 24, 16, 8, 0, …, Sn = -880
For questions 11-12, write a rule for the nth term of the geometric sequence.
1
11. a3  18, a6  486
12. a8  , a15  243
9
For questions 13-14, find the sum of the first n terms of the geometric series and
9
2 4
 1    ... , n = 20
13. 1  (4)  16  (64)  ... , n = 8
14.
2
9 81
For questions 15-16, find n for the given sum, S n .
15. 1  (4)  16  (64)  ... , Sn =-819
16.
9
2 4
 1    ... , S n =5.5
2
9 81
For questions 17-19, find the sum of the series.
11
17.
i
1
 7  2 
i 0

36
18.
 (5  3i)
19.
i 1
2
 4  3 
n 0
n
For questions 20-22, find the sum of the infinite geometric series if it has one.

20.
1
 2  4 
n 1
n 1

21.
3
 4  2 
n 0

n
22.
 1
 10   2 
n 1
n 1
For questions 23-28, solve each problem.
23. Tim wants to buy one of three different albums he sees in a music store. Each is available on
tape and on CD. From how many combinations of album and format does he have to choose?
24. A video store has 8 new releases this week. Each is available on videotape and on DVD. How
many ways can a customer choose a new release and a format to rent?
25. Carlos has homework to do in math, chemistry, and English. How many ways can he choose
the order in which to do his homework?
26. The menu for a banquet has a choice of two types of salad, five main courses, and three
desserts. How many ways can a salad, main course, and dessert be selected to form a meal?
Math Analysis Notes prepared by Mrs. Atkinson
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27. How many ways can six different books be arranged on a shelf if one of the books is a
dictionary and it must be on an end?
28. Each question on a five-question multiple-choice quiz has answer choices labeled A, B, C,
and D. How many different ways can a student answer the five questions?
For questions 29-29, Find the number of possibilities.
29. The winner and first, second, and third runners-up in a contest with 10 finalists.
30. Selecting two of eight employees to attend a business seminar.
31. An arrangement of the letters in the word “algebra”
32. Placing an algebra book, a geometry book, a chemistry book, an English book, and a health book
on a shelf.
33. Selecting nine books to check out of the library from a reading list of twelve.
34. Choosing two CDs to buy from ten that are on sale.
35. Selecting three of fifteen flavors of ice cream at the grocery store.
36. Choosing a hand of five cards from a standard deck of cards consisting of four cards from one
suit and one card from another suit.
37. An airline is hiring 5 flight attendants. If 8 people apply for the job, how many different groups of
5 attendants can the airline hire?
38. A photographer is taking pictures of a bride and groom and their 6 attendants. If she takes
photographs of 3 people in a group, how many different groups can she photograph?
39. A school librarian would like to buy subscriptions to 7 new magazines. His budget, however, will
allow him to buy only 4 new subscriptions. How many different groups of 4 magazines can he
choose?
For questions 40-43, Lisa has 9 rings in her jewelry box. Five are gold and 4 are silver. If she
randomly selects 3 rings to wear to a party, find each probability.
40. P(2 silver or 2 gold)
41. P(all gold or all silver)
42. P(at least 2 gold)
43. P(at least 1 silver)
Math Analysis Notes prepared by Mrs. Atkinson
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For questions 44-49, seven girls and six boys walk into a video store at the same time. There are
five salespeople available to help them. Find the probability that the salespeople will first help
the given number of girls and boys.
44. P(4 girls or 4 boys)
45. P(3 girls or 3 boys)
46. P(all girls or all boys)
47. P(at least 3 girls)
48. P(at least 4 girls or at least 4 boys)
49. P(at least 2 boys)
For questions 50-16, Then find the probability.
50. There are 3 literature books, 4 algebra books, and 2 biology books on a shelf. If a book is
randomly selected, what is the probability of selecting a literature books or an algebra book?
51. A die is rolled. What is the probability of rolling a 5 or a number greater than 3?
52. In the Math Club, 7 of the 20 girls are seniors, and 4 of the 14 boys are seniors. What is the
probability of randomly selecting a boy or a senior to represent the Math Club at a statewide math
contest?
53. A card is drawn from a standard deck of cards. What is the probability of drawing an ace or a
face card? (Hint: A face cards is a jack, queen, or king).
54. One tile with each letter of the alphabet is placed in a bag, and one is drawn at random. What is
the probability of selecting a vowel or a letter from the word equation?
55. Each of the numbers from 1 to 30 is written on a card and placed in a bag. If one card is drawn at
random, what is the probability that the number is a multiple of 2 or a multiple of 3?
In 56-58, Find S3.
56. 1 + 2 + 3 + ….. + n =
n  n  1
57. 1 + 22 + 32 + …. + n2 =
2
n  n  1 2n  1
58. 5 + 25 + 125 +…..+ 5n =
6
5n1  5
4
In 59-61, Use math induction to prove the give statements.
59. 1 + 2 + 3 + ….. + n =
n  n  1
60. 1 + 22 + 32 + …. + n2 =
2
n  n  1 2n  1
61. 5 + 25 + 125 +…..+ 5n =
6
5n1  5
4
Math Analysis Notes prepared by Mrs. Atkinson
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