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

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In summary, the question is whether geometric progressions are always less than or equal to arithmetic progressions in terms of their sums. This is not true, as shown by the example of \(1,\,3,\,5\) and \(2,\,4,\,8\).
  • #1
Dustinsfl
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Is it true that geometric progressions are \leq arithmetic?
 
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  • #2
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?
 
  • #3
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.
 
  • #4
I am wondering if GP $\leq$ AP
 
  • #5
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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FAQ: Are Geometric Progressions Always Less Than or Equal to Arithmetic Progressions?

What is the difference between arithmetic and geometric sequences?

Arithmetic sequences have a constant difference between each term, while geometric sequences have a constant ratio between each term.

How can arithmetic and geometric series be used in real life?

Arithmetic series can be used to calculate the total cost of regularly occurring expenses, such as rent or loan payments. Geometric series can be used to model population growth or investment returns.

What is the formula for finding the sum of an arithmetic series?

The formula for finding the sum of an arithmetic series is Sn = (n/2)(a1 + an), where Sn is the sum, n is the number of terms, a1 is the first term, and an is the last term.

How can you determine if a sequence is arithmetic or geometric?

If the difference between consecutive terms is constant, the sequence is arithmetic. If the ratio between consecutive terms is constant, the sequence is geometric.

Can a sequence be both arithmetic and geometric?

No, a sequence can only be one or the other. However, a sequence can switch from arithmetic to geometric at a certain point.

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