• 4.1: Sequences

    • University of California, San Diego: Edward Bender and S. Williamson's "Arithmetic, Logic and Numbers: Induction, Sequences, and Series" URL

      Read Section 2, "Sequences" on pages IS-12 - IS-19. In addition, read the examples mentioned below. This reading also applies to subunits 4.1.2 and 4.1.3.

      See the definition of alternating sequence, immediately before example 14. Please pay particular attention to examples 7, 14 (optional), and 17 for information specific to the topics in subunits 4.1.2 and 4.1.3. Given a sequence, f(k), f(k + 1), f(k + 2), f(k + 3), ..., where k is a fixed integer, we may be able to determine a general expression for each term of this sequence. The general expression will have the form f(n), where n is a variable that takes on values from the set {n, n > k}. If we can determine the form for f, we write f(n)n >= k. See the exercises for section 2 for examples.

      If one takes the first term of a sequence, then the sum of the first two terms, then the sum of the first three terms, and so on, the result is a new sequence, called a series. Alternating series are defined on page IS-24 of the Bender and Williamson reading above. An alternating sequence is defined just like an alternating series; namely, the signs of the adjacent terms of the sequence alternate in their signs. 

      The problem of finding an explicit formula for a sequence or series is, in general, a hard problem. In some cases, there might not even exist such a formula.

      Finding an explicit formula pertains to three situations:

      1. Given a sequence of numbers, find a general formula for the nth term of the sequence that depends on the n-1 term.
      2. Given a sequence of numbers, find a general formula for the nth term of the sequence that depends on n.
      3. Given a sequence of numbers, find a general formula for the nth term of the sequence that depends on one or more of the preceding terms.

      If the sequence has certain properties - such as the ability of each term to be calculated from the preceding term - as in an arithmetic sequence (where a fixed number is added to the n-1 term to obtain the nth term), or a geometric sequence (where a fixed number is used to multiply the n-1 term, to obtain the nth term), then one could use that information to deduce a formula for each term. Thus, one makes assumptions about the relationship of the n-1 term and the nth term. Or, if one is given a formula for each term that depends on that term, one could deduce a formula for the nth term.

      An example for the third situation above is that of the Fibonacci series - f(n) = n + (n-1) and f(0) = 0, f(1) = 1.