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

[Dustinsfl]

Is it true that geometric progressions are \leq arithmetic?

 

[Fantini]

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?

 

[Sudharaka]

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.

 

[Dustinsfl]

I am wondering if GP $\leq$ AP

 

[Sudharaka]

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, \(1+3+5=9\mbox{ and }2+4+8=14\)