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

  • Context: MHB 
  • Thread starter Thread starter Dustinsfl
  • Start date Start date
  • Tags Tags
    Geometric
Click For Summary
SUMMARY

The discussion confirms that geometric progressions (GP) are not always less than or equal to arithmetic progressions (AP). Specifically, the example provided illustrates 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 exceed GP. The conversation also touches on the relationship between arithmetic and geometric means, reinforcing the established mathematical principle that the arithmetic mean is greater than or equal to the geometric mean.

PREREQUISITES
  • Understanding of arithmetic and geometric progressions
  • Familiarity with the concepts of arithmetic mean and geometric mean
  • Basic knowledge of mathematical inequalities
  • Ability to perform summation of sequences
NEXT STEPS
  • Study the properties of arithmetic and geometric means in depth
  • Explore the implications of the Cauchy-Schwarz inequality
  • Investigate the conditions under which GP and AP can be compared
  • Learn about convergence in sequences and series
USEFUL FOR

Mathematicians, educators, students studying sequences and series, and anyone interested in the properties of mathematical inequalities.

Dustinsfl
Messages
2,217
Reaction score
5
Is it true that geometric progressions are \leq arithmetic?
 
Physics news on Phys.org
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, \(1+3+5=9\mbox{ and }2+4+8=14\)
 
Last edited:

Similar threads

High School Geometric and arithmetic means and contraction factor
  • · Replies 6 ·
Replies
6
Views
845
High School Arithmetic progression, Geometric progression and Harmonic progression
  • · Replies 5 ·
Replies
5
Views
1K
MHB Arithmetic Progression: Finding the First Term and Common Difference
  • · Replies 2 ·
Replies
2
Views
4K
MHB Geometric Interpretation of k-Forms from H&H's Vector Calculus
  • · Replies 2 ·
Replies
2
Views
2K
MHB Arithmetic Progression: Expressing d in Terms of x,y,z,n
  • · Replies 1 ·
Replies
1
Views
1K
Arithmetic progression used to determine geometric progression
  • · Replies 3 ·
Replies
3
Views
2K
Triangle determined by arithmetic and geometric sequences
  • · Replies 7 ·
Replies
7
Views
2K
How to Solve Arithmetic Sequence Problems
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Norm of integral less than or equal to integral of norm of function
  • · Replies 3 ·
Replies
3
Views
2K
MHB How Do You Solve This Geometric Progression Problem?
  • · Replies 3 ·
Replies
3
Views
2K