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an arithmetic sequeunce does not have the sum to infinty, and a geometric sequence has.
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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.
A few examples: Counting numbers are an arithmetic sequence. Radioactive decay, (uncontrolled) bacterial growth follow geometric sequences. The Fibonacci sequence is widespread in nature.
they look like arithmetic and geometric patterns in math