if you want hints for the exercises (credits to Evan Chen) please ask me through PM (@LLLhk)
Math Trick of the Week: AM-GM
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Jul 22, 2020
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#3
btw i think my paper has a small mistake, please correct me through PM if there is a mistake and you found it
Jul 22, 2020
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#6
we learnt the GM and AM formula (rlly simple one) ...I did not understand anything else o-o
Jul 22, 2020
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#8
hmm...ya ..it looks pretty interesting...ill ask my dad to teach me this ><
Jul 22, 2020
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#9
Hobi_Mochi wrote:
hmm...ya ..it looks pretty interesting...ill ask my dad to teach me this ><
reading this article is better xD
What is AM-GM?
AM-GM is an inequality. It states that AM≥GM for positive variables and equality holds if and only if they are all equal.
What is AM?
AM is the "mean" you are used to, the Arithmetic Mean. This is just adding the n variables and divide by n. It is called that because when there are 2 variables, the variables and the AM form an Arithmetic sequence.
What is GM?
GM is not the "mean" you are used to. This, the Geometric Mean, is just multiplying the n variables and taking the whole thing to the nth root. It is called that because when there are 2 variables, the variables and the GM form an Geometric sequence.
Example:
(15)^(1/3)≤3
This is true since 3=(27)^(1/3).
But, another way to look at it is that (1*3*5)^(1/3)≥(1+3+5)/3=3
How do we prove AM-GM?
See this article I wrote (a few months ago for no reason):
I cannot attach files, so why not see screenshots I made of it? xD
Example
:
a^7+b^7+c^7 ≥ a^4*b^3+b^4*c^3+c^4*a^3
(a^7+a^7+a^7+a^7+b^7+b^7+b^7)/7≥(a^28*b^21)^(1/7)=a^4*b^3.
So, (4a^7+3b^7)≥a^4*b^3, and similarly we can write two other inequalities with b and c, c and a.
Summing them up gives the desired result.
Exercises:
Chapter 1.3:https://web.evanchen.cc/handouts/Ineq/en.pdf
THE END
Next time, we will be looking at some geometry!