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There is no simple answer. There are simple formulae for simple sequences such as arithmetic or geometric progressions; there are less simple solutions arising from Taylor or Maclaurin series. But for the majority of sequences there are no solutions.
an arithmetic sequeunce does not have the sum to infinty, and a geometric sequence has.
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a sequential series of geometric shapes
A few examples: Counting numbers are an arithmetic sequence. Radioactive decay, (uncontrolled) bacterial growth follow geometric sequences. The Fibonacci sequence is widespread in nature.