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If $ \$ $1000 is invested at $ 6 \% $ interest, c…

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Problem 64 Hard Difficulty

(a) Determine whether the sequence defined as follows is convergent or divergent:
$ a_1 = 1 $ $ a_{n + 1} = 4 - a_n $ for $ n \ge 1 $

(b) What happens if the first term is $ a_1 = 2 $ ?


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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 11

Infinite Sequences and Series

Section 1

Sequences

Related Topics

Sequences

Series

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Top Calculus 2 / BC Educators
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Oregon State University

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Lectures

Video Thumbnail

01:59

Series - Intro

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

Video Thumbnail

02:28

Sequences - Intro

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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Watch More Solved Questions in Chapter 11

Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
Problem 16
Problem 17
Problem 18
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30
Problem 31
Problem 32
Problem 33
Problem 34
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Problem 36
Problem 37
Problem 38
Problem 39
Problem 40
Problem 41
Problem 42
Problem 43
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Problem 45
Problem 46
Problem 47
Problem 48
Problem 49
Problem 50
Problem 51
Problem 52
Problem 53
Problem 54
Problem 55
Problem 56
Problem 57
Problem 58
Problem 59
Problem 60
Problem 61
Problem 62
Problem 63
Problem 64
Problem 65
Problem 66
Problem 67
Problem 68
Problem 69
Problem 70
Problem 71
Problem 72
Problem 73
Problem 74
Problem 75
Problem 76
Problem 77
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Problem 79
Problem 80
Problem 81
Problem 82
Problem 83
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Problem 85
Problem 86
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Problem 89
Problem 90
Problem 91
Problem 92
Problem 93

Video Transcript

whose problem it helps to just write out the first few terms. So we know a one is one Hey, two is going to be four minus a one. So four minus one, which is three pay three is going to be four minus a too. So four minus three, which is going to be one. And from here, we know that the sequence should just continue in this way. One, three, one. But, you know, if you really want to convince yourself, you keep going. I thought I was going to be safe. Four minus a three is four minus one, which is three. A five is four minus a four four minus three, which is one. So this just switches between one and three, so that's definitely going to diverge. Okay. And once we got Tio Tio here, we knew that it was just going to switch between one and three, because each term is just defined only by the previous term. Okay, once you've gotten to hear, we've already been in this situation once before, right? So we know that if one is the previous term, that next term has to be three. So once we get to this one. We know that the next term has to be three. Okay, And then once we're at three, we know that the next term has to be one, because again, each term is defined only by the previous terms. So if we've been in the situation where we've gotten one before and the next term was three, we know that if one shows up again, the next time has to be three. And likewise, if three shows up in the next term is one, if that ever happens again and the three shows up, the next term is going to be one. But it's always good to, you know, you really convince yourself of it by, you know, right, Not a few more terms and seeing that indeed it is just going to alternate between one and three. So certainly divergence

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Video Thumbnail

01:59

Series - Intro

In mathematics, a series is, informally speaking, the sum of the terms of an infinite sequence. The sum of a finite sequence of real numbers is called a finite series. The sum of an infinite sequence of real numbers may or may not have a well-defined sum, and may or may not be equal to the limit of the sequence, if it exists. The study of the sums of infinite sequences is a major area in mathematics known as analysis.

Video Thumbnail

02:28

Sequences - Intro

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed. Like a set, it contains members (also called elements, or terms). The number of elements (possibly infinite) is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first "n" natural numbers (for a finite sequence). A sequence can be thought of as a list of elements with a particular order. Sequences are useful in a number of mathematical disciplines for studying functions, spaces, and other mathematical structures using the convergence properties of sequences. In particular, sequences are the basis for series, which are important in differential equations and analysis. Sequences are also of interest in their own right and can be studied as patterns or puzzles, such as in the study of prime numbers.

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