14.2.13: Chapter 13
- Page ID
- 118267
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13.1 Sequences and Their Notations
The first five terms are
The first five terms are
The first six terms are
The first five terms are
13.2 Arithmetic Sequences
The sequence is arithmetic. The common difference is
The sequence is not arithmetic because
There are 11 terms in the sequence.
The formula is and it will take her 42 minutes.
13.3 Geometric Sequences
The sequence is not geometric because .
The sequence is geometric. The common ratio is .
- ⓐ
- ⓑThe number of hits will be about 333.
13.4 Series and Their Notations
38
$2,025
9,840
$275,513.31
The sum is not defined.
The sum of the infinite series is defined.
The sum of the infinite series is defined.
3
The series is not geometric.
$32,775.87
13.1 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.
Yes, both sets go on indefinitely, so they are both infinite sequences.
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
First four terms:
First four terms: .
First four terms: .
First four terms: .
First four terms:
First five terms:
First five terms:
First four terms:
First four terms:
First five terms: , , , ,
First five terms: 2, 3, 5, 17, 65537
First six terms: 0.042, 0.146, 0.875, 2.385, 4.708
First four terms: 5.975, 2.765, 185.743, 1057.25, 6023.521
If is a term in the sequence, then solving the equation for will yield a non-negative integer. However, if then so is not a term in the sequence.
13.2 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.
The common difference is
The sequence is not arithmetic because
First five terms:
There are 10 terms in the sequence.
There are 6 terms in the sequence.
The graph does not represent an arithmetic sequence.
Answers will vary. Examples: and
The sequence begins to have negative values at the 13th term,
Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: First 4 terms:
13.3 Section Exercises
A sequence in which the ratio between any two consecutive terms is constant.
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.
The common ratio is
The sequence is geometric. The common ratio is 2.
The sequence is geometric. The common ratio is
The sequence is geometric. The common ratio is
There are terms in the sequence.
The graph does not represent a geometric sequence.
Answers will vary. Examples: and
The sequence exceeds at the 14th term,
is the first non-integer value
Answers will vary. Example: Explicit formula with a decimal common ratio: First 4 terms:
13.4 Section Exercises
An partial sum is the sum of the first terms of a sequence.
A geometric series is the sum of the terms in a geometric sequence.
An annuity is a series of regular equal payments that earn a constant compounded interest.
The series is defined.
The series is defined.
Sample answer: The graph of seems to be approaching 1. This makes sense because is a defined infinite geometric series with
49
254
$3,705.42
$695,823.97
9 terms
$400 per month
420 feet
12 feet
13.5 Section Exercises
There are ways for either event or event to occur.
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.
A combination;
9
Yes, for the trivial cases and If then If then
13.6 Section Exercises
A binomial coefficient is an alternative way of denoting the combination It is defined as
The Binomial Theorem is defined as and can be used to expand any binomial.
15
35
10
12,376
The expression cannot be expanded using the Binomial Theorem because it cannot be rewritten as a binomial.
13.7 Section Exercises
probability; The probability of an event is restricted to values between and inclusive of and
An experiment is an activity with an observable result.
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
The sequence is arithmetic. The common difference is
4, 16, 64, 256, 1024
$5,617.61
6
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 |