Try It
9.1 Sequences and Their Notations
9.2 Arithmetic Sequences
9.3 Geometric Sequences
9.4 Series and Their Notations
9.5 Counting Principles
9.6 Binomial Theorem
9.1 Section Exercises
A sequence is an ordered list of numbers that can be either finite or infinite in number. When a finite sequence is defined by a formula, its domain is a subset of the non-negative integers. When an infinite sequence is defined by a formula, its domain is all positive or all non-negative integers.
A factorial is the product of a positive integer and all the positive integers below it. An exclamation point is used to indicate the operation. Answers may vary. An example of the benefit of using factorial notation is when indicating the product It is much easier to write than it is to write out 13⋅12⋅11⋅10⋅9⋅8⋅7⋅6⋅5⋅4⋅3⋅2⋅1.
If an=−421 is a term in the sequence, then solving the equation −421=−6−8n for n will yield a non-negative integer. However, if −421=−6−8n, then n=51.875 so an=−421 is not a term in the sequence.
9.2 Section Exercises
A sequence where each successive term of the sequence increases (or decreases) by a constant value.
We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.
Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.
Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: a1=3,an=an−1−3. First 4 terms: 3,0,−3,−6a31=−87
9.3 Section Exercises
Divide each term in a sequence by the preceding term. If the resulting quotients are equal, then the sequence is geometric.
Both geometric sequences and exponential functions have a constant ratio. However, their domains are not the same. Exponential functions are defined for all real numbers, and geometric sequences are defined only for positive integers. Another difference is that the base of a geometric sequence (the common ratio) can be negative, but the base of an exponential function must be positive.
Answers will vary. Example: Explicit formula with a decimal common ratio: an=400⋅0.5n−1; First 4 terms: 400,200,100,50;a8=3.125
9.4 Section Exercises
Sample answer: The graph of Sn seems to be approaching 1. This makes sense because ∞∑k=1(12)k is a defined infinite geometric series with S=121–(12)=1.
9.5 Section Exercises
The addition principle is applied when determining the total possible of outcomes of either event occurring. The multiplication principle is applied when determining the total possible outcomes of both events occurring. The word “or” usually implies an addition problem. The word “and” usually implies a multiplication problem.
Yes, for the trivial cases r=0 and r=1. If r=0, then C(n,r)=P(n,r)=1. If r=1, then r=1, C(n,r)=P(n,r)=n.
9.6 Section Exercises
A binomial coefficient is an alternative way of denoting the combination C(n,r). It is defined as
The expression cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.
9.7 Section Exercises
The probability of the union of two events occurring is a number that describes the likelihood that at least one of the events from a probability model occurs. In both a union of sets and a union of events the union includes either or both. The difference is that a union of sets results in another set, while the union of events is a probability, so it is always a numerical value between and
1 | 2 | 3 | 4 | 5 | 6 | |
1 | (1,1) 2 |
(1,2) 3 |
(1,3) 4 |
(1,4) 5 |
(1,5) 6 |
(1,6) 7 |
2 | (2,1) 3 |
(2,2) 4 |
(2,3) 5 |
(2,4) 6 |
(2,5) 7 |
(2,6) 8 |
3 | (3,1) 4 |
(3,2) 5 |
(3,3) 6 |
(3,4) 7 |
(3,5) 8 |
(3,6) 9 |
4 | (4,1) 5 |
(4,2) 6 |
(4,3) 7 |
(4,4) 8 |
(4,5) 9 |
(4,6) 10 |
5 | (5,1) 6 |
(5,2) 7 |
(5,3) 8 |
(5,4) 9 |
(5,5) 10 |
(5,6) 11 |
6 | (6,1) 7 |
(6,2) 8 |
(6,3) 9 |
(6,4) 10 |
(6,5) 11 |
(6,6) 12 |
Review Exercises
1 | 2 | 3 | 4 | 5 | 6 | |
1 | 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 |
2 | 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6 |
3 | 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6 |
4 | 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6 |
5 | 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6 |
6 | 6,1 | 6,2 | 6,3 | 6,4 | 6,5 | 6,6 |