Show that the inequality is true | Geometric Mean

Click For Summary

Homework Help Overview

The discussion revolves around proving an inequality involving strictly positive numbers and their geometric mean. The inequality states that \((1+R_{G})^{n} \leq V\), where \(R_{G}\) is the geometric mean of the numbers and \(V\) is a product involving the numbers increased by one.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods to approach the problem, including taking the logarithm of both sides and expanding terms. There is mention of using the arithmetic mean-geometric mean inequality and a suggestion to relate the variables to exponential functions.

Discussion Status

The discussion is active with multiple approaches being explored. Some participants are considering different mathematical inequalities, while others are looking for additional insights or alternative methods. No consensus has been reached yet.

Contextual Notes

Participants note hints from the professor regarding the use of logarithmic functions and the relationship between the variables and exponential forms, indicating potential constraints or specific directions for the problem-solving process.

michonamona
Messages
120
Reaction score
0

Homework Statement



Let r_{1}, r_{2}, ... , r_{n} be strictly positive numbers. Show that the inequality

(1+R_{G})^{n} \leq V

is true. Where R_{G} = (r_{1}r_{2}...r_{n})^{1/n} and V= \Pi_{k=1}^{n} (1+r_{k})

Homework Equations





The Attempt at a Solution



I've tried taking the log of both sides, as well as expanding out the term. Any insight?

Thanks,
M
 
Physics news on Phys.org
Use the arithmetic mean-geometric mean inequality... several times.
 
Any other insights?

The prof hinted that we should use log(1+e^x) and associate r with e^x.
 
That's an entirely different way to approach it. The approach I was thinking of uses the fact that the terms of the right side are the elementary symmetric functions of the r_j.
 

Similar threads

##\phi(R_{180})##, if ##\phi:D_n\to D_n## is an automorphism
  • · Replies 8 ·
Replies
8
Views
2K
Prove this inequality : Geometric Mean and Arithmetic Mean
  • · Replies 1 ·
Replies
1
Views
2K
Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Two basic results from measure theory -- on volumes of rectangles
  • · Replies 2 ·
Replies
2
Views
3K
High School Attempted proof of the Contracted Bianchi Identity
  • · Replies 1 ·
Replies
1
Views
3K
Show that ## 11\mid R_{n} ## if and only if ## n ## is even
  • · Replies 3 ·
Replies
3
Views
1K
Prove that a matrix can be reduced to RRE and CRE
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10
  • · Replies 10 ·
Replies
10
Views
3K
Computing representation number quad forms
  • · Replies 4 ·
Replies
4
Views
2K
Factor the repunit ## R_{6}=111111 ## into a product of primes
  • · Replies 3 ·
Replies
3
Views
2K