Difference between revisions of "User:Superagh"
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====Theorems worth noting==== | ====Theorems worth noting==== | ||
=====Power mean===== | =====Power mean===== | ||
− | Statement: Given that <math>a_1, a_2, a_3, ... a_n > 0</math>, <math>a_{i} \in \ | + | Statement: Given that <math>a_1, a_2, a_3, ... a_n > 0</math>, <math>a_{i} \in \mathbb{R}</math> where <math>1 \leq i \leq n</math>. Define the <math>pm_x(a_1, a_2, \cdots , a_n)</math> as: <cmath>(\frac{a_1^x+a_2^x+\cdots+a_n^x}{n})^{\frac{1}{x}},</cmath> where <math>x\neq0</math>, and: <cmath>sqrt[n]{a_1a_2a_3\cdotsa_n}.</cmath> where <math>x=0</math>. |
If <math>x\geqy</math>, then <cmath>pm_x(a_1, a_2, \cdots , a_n)\geqpm_y(a_1, a_2, \cdots , a_n).</cmath> | If <math>x\geqy</math>, then <cmath>pm_x(a_1, a_2, \cdots , a_n)\geqpm_y(a_1, a_2, \cdots , a_n).</cmath> |
Revision as of 09:04, 24 June 2020
Contents
Introduction
Ok, so inspired by master math solver Lcz, I have decided to take Oly notes (for me) online! I'll probably be yelled at even more for staring at the computer, but I know that this is for my good. (Also this thing is almost the exact same format as Lcz's :P ). (Ok, actually, a LOT of credits to Lcz)
Algebra
Problems worth noting/reviewing
I'll leave this empty for now, I want to start on HARD stuff yeah!
Inequalities
We shall begin with INEQUALITIES! They should be fun enough. I should probably begin with some theorems.
Theorems worth noting
Power mean
Statement: Given that ,
where
. Define the
as:
where
, and:
\[sqrt[n]{a_1a_2a_3\cdotsa_n}.\] (Error compiling LaTeX. Unknown error_msg)
where .
If $x\geqy$ (Error compiling LaTeX. Unknown error_msg), then
\[pm_x(a_1, a_2, \cdots , a_n)\geqpm_y(a_1, a_2, \cdots , a_n).\] (Error compiling LaTeX. Unknown error_msg)