![]() ![]() 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. There is a common difference between each pair of terms. Some of the interesting ones can be found at the online encyclopedia of integer sequences at I like to call them progressions and reserve the word sequence for the definition I proposed in the first 2 paragraphs. #2,8,14,20.# which has first term #2# and common difference #6#. This sequence is made by adding the previous two numbers on the list to form the next one and so on. Let's for example take a Fibonacci sequence. They don't fit right with the definition I gave. However, there can be other ordered arrays of numbers which are sometimes referred at as sequences. Thus if #x_n = 1/n^2#, the sequence may be given as, # where #x_n# is the #n#th element related to the a corresponding natural number. It can be defined in any way you like.įinite 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. ![]() In general an infinite sequence is any mapping from #NN -> S# for any set #S#. The terms of the Fibonacci sequence are expressible by the formula: The ratio of successive pairs of terms tends towards the golden ratio #phi = 1/2 + sqrt(5)/2 ~= 1.618034# An example of this would be the Fibonacci sequence: There are also sequences where the next number is defined iteratively in terms of the previous 2 or more terms. If the sequence is a geometric progression with first term #a_1#, then the terms will be of the form: ![]() If the sequence is an arithmetic progression with first term #a_1#, then the terms will be of the form: ![]()
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