Are Geometric Progressions Always Less Than or Equal to Arithmetic Progressions?

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Discussion Overview

The discussion revolves around the relationship between geometric progressions (GP) and arithmetic progressions (AP), specifically whether geometric progressions are always less than or equal to arithmetic progressions. The scope includes theoretical considerations and comparisons of sums of these sequences.

Discussion Character

  • Debate/contested

Main Points Raised

  • Some participants propose that geometric progressions are less than or equal to arithmetic progressions, referencing the inequality between arithmetic and geometric means.
  • Others question whether the discussion assumes finiteness in the sequences being considered.
  • A participant seeks clarification on whether the inquiry pertains to the inequality of arithmetic and geometric means.
  • Another participant argues against the idea that the sum of any arithmetic progression is greater than or equal to the sum of any geometric progression, providing a counterexample with specific sequences.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are competing views regarding the relationship between geometric and arithmetic progressions, particularly concerning their sums.

Contextual Notes

The discussion includes assumptions about the nature of the sequences (finite vs. infinite) and the specific context of the inequality being considered, which remain unresolved.

Dustinsfl
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Is it true that geometric progressions are \leq arithmetic?
 
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Doesn't seem a far shot. We know that arithmetic mean is greater or equal than geometric mean, perhaps applying that you could get to your result. Are we assuming finiteness or not?
 
dwsmith said:
Is it true that geometric progressions are \leq arithmetic?

Hi dwsmith, :)

Can you please clarify your question a bit more. Do you mean the inequality of arithmetic and geometric means ?

Kind Regards,
Sudharaka.
 
I am wondering if GP $\leq$ AP
 
dwsmith said:
I am wondering if GP $\leq$ AP

So your question seems to be whether the sum of any arithmetic progression is greater than or equal to the sum of any geometric progression. That is not the case. For example, \(1,\,3,\,5\) is an arithmetic progression and \(2,\,4,\,8\) is a geometric progression. But, \(1+3+5=9\mbox{ and }2+4+8=14\)
 
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