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Infinite Sequences

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Key Questions

  • Answer:

    It depends.

    Explanation:

    There are many types of sequences. Some of the interesting ones can be found at the online encyclopedia of integer sequences at https://oeis.org/

    Let's look at some simple types:

    Arithmetic Sequences

    #a_n = a_0 + dn#

    e.g. #2, 4, 6, 8,...#

    There is a common difference between each pair of terms.

    If you find a common difference between each pair of terms, then you can determine #a_0# and #d#, then use the general formula for arithmetic sequences.

    Geometric Sequences

    #a_n = a_0 * r^n#

    e.g. #2, 4, 8, 16,...#

    There is a common ratio between each pair of terms.

    If you find a common ratio between pairs of terms, then you have a geometric sequence and you should be able to determine #a_0# and #r# so that you can use the general formula for terms of a geometric sequence.

    Iterative Sequences

    After the initial term or two, the following terms are defined in terms of the preceding ones.

    e.g. Fibonacci

    #a_0 = 0#
    #a_1 = 1#
    #a_(n+2) = a_n + a_(n+1)#

    For this sequence we find: #a_n = (phi^n - (-phi)^(-n))/sqrt(5)# where #phi = (1+sqrt(5))/2#

    There are many ways to make these iterative rules, so there is no universal method to provide an expression for #a_n#

    Polynomial Sequences

    If the terms of a sequence are given by a polynomial, then given the first few terms of the sequence you can find the polynomial.

    e.g.

    #color(red)(1), 2, 4, 7, 11,...#

    Form the sequence of differences of these values:

    #color(red)(1), 2, 3, 4,...#

    Form the sequence of differences of these values:

    #color(red)(1), 1, 1,...#

    Once you reach a constant sequence like this, pick out the initial terms from each sequence. In this case #1#, #1# and #1#.

    These form the coefficients of a polynomial expression:

    #a_n = color(red)(1)/(0!) + (color(red)(1)*n)/(1!) + (color(red)(1)*n(n-1))/(2!)#

    #=n^2/2+n/2+1#

  • It depends on the type of sequence.

    If the sequence is an arithmetic progression with first term #a_1#, then the terms will be of the form:

    #a_n = a_1 + (n-1)b#
    for some constant b.

    If the sequence is a geometric progression with first term #a_1#, then the terms will be of the form:

    #a_n = a_1 * r^(n-1)#
    for some constant #r#.

    There are also sequences where the next number is defined iteratively in terms of the previous 2 or more terms. An example of this would be the Fibonacci sequence:

    #0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,...#

    Each term is the sum of the two previous terms.

    The ratio of successive pairs of terms tends towards the golden ratio #phi = 1/2 + sqrt(5)/2 ~= 1.618034#

    The terms of the Fibonacci sequence are expressible by the formula:

    #F_n = (phi^n-(-phi)^-n)/sqrt(5)# (starting with #F_0 = 0#, #F_1 = 1#)

    In general an infinite sequence is any mapping from #NN -> S# for any set #S#. It can be defined in any way you like.

    Finite sequences are the same, except that they are mappings from a finite subset of #NN# consisting of those numbers less than some fixed limit, e.g. #{n in NN: n <= 10}#

  • A sequence is an infinite set of points which are ordered and a mapping can be associated with with that set and the set of natural numbers.

    The general notion of a sequence is that it is an infinite set with every element associated with a natural number even though all infinite sets may not be sequences.

    A sequence may be represented by, #{x_n}# where #x_n# is the #n#th element related to the a corresponding natural number.

    Thus if #x_n = 1/n^2#, the sequence may be given as,

    #{1,1/4,1/9,1/16,....}#

    For #x_n = n^3# we shall have,

    #{1, 8, 27, 64,....}#

    Now for #x_n = n# we can have,

    #{1,2,3,.....}#

    This is indeed the set of naturals.

    However, there can be other ordered arrays of numbers which are sometimes referred at as sequences. They don't fit right with the definition I gave.

    Let's for example take a Fibonacci sequence.

    #1,1,2,3,5,8,13,....#

    This sequence is made by adding the previous two numbers on the list to form the next one and so on.

    There can be arithmetic sequences, like

    #2,8,14,20,....# which has first term #2# and common difference #6#.

    I like to call them progressions and reserve the word sequence for the definition I proposed in the first 2 paragraphs.

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