Compare

**lebanon** · January 6th 2011, 11:36 PM

a and b being two strictly positive numbers , compare (a+b)/2 and √a√b

**melese** · January 6th 2011, 11:59 PM

Originally Posted by lebanon

a and b being two strictly positive numbers , compare (a+b)/2 and √a√b

Consider $(\sqrt{a}-\sqrt{b})^2=(\sqrt{a})^2-2\sqrt{a}\sqrt{b}+(\sqrt{b})^2=a+b-2\sqrt{a}\sqrt{b}$ .

Then divide by 2 to get $\displaystyle\frac{(\sqrt{a}-\sqrt{b})^2}{2}=\frac{a+b}{2}-\sqrt{a}\sqrt{b}$ . But since $\displaystyle\frac{(\sqrt{a}-\sqrt{b})^2}{2}$ is nonnegative for any positive $a$ and $b$ (why?) it follows that $\displaystyle\frac{a+b}{2}-\sqrt{a}\sqrt{b}\geq0$ .

Therefore, $\displaystyle\frac{a+b}{2}\geq\sqrt{a}\sqrt{b}$ for any positive numbers $a$ and $b$ .

**TheCoffeeMachine** · January 7th 2011, 12:08 AM

And it can further be proven by simple induction that:

$\displaystyle \left(\prod_{1\le i \le n}a_{i}\right)^{\frac{1}{n}} \le \frac{1}{n}\sum_{1\le i\le n}a_{i}$ for $a_{i} \in\mathbb{R}^{+}$ and $n\in\mathbb{N}$ .

Thread: Compare

LinkBack

Thread Tools

Display

Compare

Similar Math Help Forum Discussions

Compare A ,B

Compare-numbers

compare variability HELP !

compare 3 alt. w/MARR

How to compare magnitudes

Search Tags