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.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 39E

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