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In this exercise we consider sequences defined over the positive natural numbers 1, 2, 3, ... The n-th element in the sequence is denoted as an and therefore the elements in the sequence are a1, a2, a3, . .. Each of the following sequences is defined using a closed formula that directly gives an for any positive natural number n. For each sequence, give an equivalent recursive definition, i.e., a basis step and an inductive step defining the n-th element in the sequence as a function of elements already in the sequence (either the previous one or some other element preceding an.) a) an = 4n – 2 b) an = 1+ (-1)" c) an = n(n – 1) d) an = n2 Suggestion: it may be convenient to first tabulate the values of the sequence for a few values of n, observe the pattern, and then guess the basis and inductive steps. Then, make sure that the basis and inductive steps give the same elements you tabulated. Note: to be fully correct, one should formally prove that the inductive definition of the sequences generate all and only the elements in the sequence. This would require some additional steps, but we omit them for brevity.

In this exercise we consider sequences defined over the positive natural numbers 1, 2, 3, ... The n-th element in the sequence is denoted as an and therefore the elements in the sequence are a1, a2, a3, . .. Each of the following sequences is defined using a closed formula that directly gives an for any positive natural number n. For each sequence, give an equivalent recursive definition, i.e., a basis step and an inductive step defining the n-th element in the sequence as a function of elements already in the sequence (either the previous one or some other element preceding an.) a) an = 4n – 2 b) an = 1+ (-1)" c) an = n(n – 1) d) an = n2 Suggestion: it may be convenient to first tabulate the values of the sequence for a few values of n, observe the pattern, and then guess the basis and inductive steps. Then, make sure that the basis and inductive steps give the same elements you tabulated. Note: to be fully correct, one should formally prove that the inductive definition of the sequences generate all and only the elements in the sequence. This would require some additional steps, but we omit them for brevity.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In this exercise we consider sequences defined over the positive natural numbers 1, 2, 3, ... The n-th element in the sequence is denoted as an and therefore the elements in the sequence are a1, a2, a3, ... Each of the following sequences is defined using a closed formula that directly gives an for any positive natural number n. For each sequence, give an equivalent recursive definition, i.e., a basis step and an inductive step defining the n-th element in the sequence as a function of elements already in the sequence (either the previous one or some other element preceding an.) a) an = 4n – 2 b) an = 1+ (-1)" с) аn 3 п(п — 1) d) аn — п? Suggestion: it may be convenient to first tabulate the values of the sequence for a few values of n, observe the pattern, and then guess the basis and inductive steps. Then, make sure that the basis and inductive steps give the same elements Note: to be fully correct, one should formally prove that the inductive definition of the sequences generate all and only the elements in the sequence. This would require some additional steps, but we omit them for brevity. you tabulated.
Transcribed Image Text:In this exercise we consider sequences defined over the positive natural numbers 1, 2, 3, ... The n-th element in the sequence is denoted as an and therefore the elements in the sequence are a1, a2, a3, ... Each of the following sequences is defined using a closed formula that directly gives an for any positive natural number n. For each sequence, give an equivalent recursive definition, i.e., a basis step and an inductive step defining the n-th element in the sequence as a function of elements already in the sequence (either the previous one or some other element preceding an.) a) an = 4n – 2 b) an = 1+ (-1)" с) аn 3 п(п — 1) d) аn — п? Suggestion: it may be convenient to first tabulate the values of the sequence for a few values of n, observe the pattern, and then guess the basis and inductive steps. Then, make sure that the basis and inductive steps give the same elements Note: to be fully correct, one should formally prove that the inductive definition of the sequences generate all and only the elements in the sequence. This would require some additional steps, but we omit them for brevity. you tabulated.
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