Difference between revisions of "User:Superagh"
(→Theorems worth noting) |
(→Power mean) |
||
| Line 9: | Line 9: | ||
====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 \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_{1}a_{2}a_{3} \cdots a_{n}}.</cmath> where <math>x=0</math>. | + | 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_{1}a_{2}a_{3} \cdots a_{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 09:20, 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: ![\[(\frac{a_1^x+a_2^x+\cdots+a_n^x}{n})^{\frac{1}{x}},\]](http://latex.artofproblemsolving.com/e/b/c/ebc47ff3b8f1015a599578cca800e555507e382c.png)
where 
, and: ![\[\sqrt[n]{a_{1}a_{2}a_{3} \cdots a_{n}}.\]](http://latex.artofproblemsolving.com/a/2/d/a2dd19fdaf0686ffca569aceaf5f1ec2f02916f3.png)
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)
