(b) Consider the degree n version of the AM/GM inequality, which states for positive real numbers x₁,x2,...,xn that x₁ + x₂ + + xn n ... >√√xx₂xn (i) Fove that if the AM/GM inequality is true for n=k, then it will be true for n = 2k. (ii) Prove that if the AM/GM inequality is true for n=k, then it will be true for n = k-1 (iii) Hence explain why the AM/GM inequality is true for all positive integers n.
(b) Consider the degree n version of the AM/GM inequality, which states for positive real numbers x₁,x2,...,xn that x₁ + x₂ + + xn n ... >√√xx₂xn (i) Fove that if the AM/GM inequality is true for n=k, then it will be true for n = 2k. (ii) Prove that if the AM/GM inequality is true for n=k, then it will be true for n = k-1 (iii) Hence explain why the AM/GM inequality is true for all positive integers n.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(b) Consider the degree n version of the AM/GM inequality, which states for positive real
numbers x1, X2, .., Xn that
X1 + X2 + •…·+ xn
Mx1x2 • Xn
n
(i) F ve that if the AM/GM inequality is true for n = k, then it will be true for n = 2k.
(ii) Prove that if the AM/GM inequality is true for n = k, then it will be true for n = k -1.
(iii) Hence explain why the AM/GM inequality is true for all positive integers n.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7a736a9-b3d2-4a38-a3ba-aa09cfe77d1b%2Ff98e1757-d358-4228-a3f8-9433c985300c%2Fu9k65iq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) Consider the degree n version of the AM/GM inequality, which states for positive real
numbers x1, X2, .., Xn that
X1 + X2 + •…·+ xn
Mx1x2 • Xn
n
(i) F ve that if the AM/GM inequality is true for n = k, then it will be true for n = 2k.
(ii) Prove that if the AM/GM inequality is true for n = k, then it will be true for n = k -1.
(iii) Hence explain why the AM/GM inequality is true for all positive integers n.
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