Today we explore inequalities related to mean values of a set of real
numbers. Define:
Quadratic mean as \[\sqrt{\frac{a_1^2 + a_2^2 +
...a_n^2}{n}}\] Arithmetic mean as \[\frac{a_1 + a_2 + ...+a_n}{n}\]
Geometric mean as \[\sqrt[n]{a_1a_2...a_n}\] Harmonic
mean as \[\frac{n}{\frac{1}{a_1} +
\frac{1}{a_2} + ... \frac{1}{a_n}}.\] Then the following
inequality holds: \[\sqrt{\frac{a_1^2 + a_2^2
+ ...a_n^2}{n}} \geq \frac{a_1 + a_2 + ...+a_n}{n} \geq
\sqrt[n]{a_1a_2...a_n} \geq
\frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + ... \frac{1}{a_n}}.\] We
will prove \(QM\geq AM\) and infer the
\(HM \leq GM\) part from the \(AM \geq GM\) in examples. However, the
\(AM\geq GM\) part itself is more
technical. The Mean Inequality is a well known theorem and you can use
it in solutions today or refer to it on olympiads.