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

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Geometric progressions (GP) are not always less than or equal to arithmetic progressions (AP). The discussion highlights that while the arithmetic mean is greater than or equal to the geometric mean, this does not imply that the sum of any GP will be less than or equal to the sum of any AP. An example provided shows that the sum of the arithmetic progression \(1, 3, 5\) equals 9, while the sum of the geometric progression \(2, 4, 8\) equals 14, demonstrating that AP can be less than GP. The conversation also raises questions about the assumptions of finiteness in the context of these progressions. Overall, the relationship between GP and AP is not universally applicable.
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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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