Geometric series i do not understand this i need extraordinary amount of help

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1.How would I take the summation at any point in a series that doesn't increase by a pattern. Oher summations ...

mations like arithmetic and geometric assume there is a pattern or equation, but for a random list how would I do that without just adding up all the numbers before the nth term?

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2.Geometric series I do not understand this. I need extraordinary amount of help.

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