Out of interest, I am trying to proof QM-AM-GM-HM inequality. If you don't know it, it's something like this…
Let there be $n$ numbers $x_1, x_2, x_3…x_n$, where $x_1, x_2, …,x_n>0$.
Proof that $$\sqrt{\frac{x_1^2+x_2^2…+x_n^2}{n}}\geqslant{\frac{x_1+x_2…+x_n}{n}}\geqslant{\sqrt[n]{x_1x_2…x_n}}\geqslant{\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}…+\frac{1}{x_n}}}$$
I thought of using induction (for n). The base case was something that took me about 20 mins to solve. I used n=2 (n=1 was trivial) but I am stuck. Can anyone give me a hint to continue me? To be exact, I need help in apply the induction hypothesis to the induction step. The numbers/fractions are starting to get … uh … ugly…
Update 1: I don't want to see the answer. Just a hint…
Algebra Precalculus – How to Prove QM-AM-GM-HM Inequality
algebra-precalculusinequalitymeans
Best Answer
If you're not restricted to proof by induction, you can try to show that $$ M(p; x_1,x_2,\dotsc,x_n) := \left(\frac{1}{n}\sum _{i=1} ^n x_i^p\right)^{1/p},$$ is an increasing function of $p\in\mathbb{R}$. You only need Jensen's inequality to prove this.
update: For proof without calculus, you only need to prove the AM-GM inequality (e.g., through the Cauchy induction as others suggested). QM-AM is a simple case of the Cauchy-Schwarz inequality (which has an elementary proof). Furthermore, GM-HM is the same as AM-GM for the numbers $y_i = 1/x_i$.