Help with an algebraic proof

**paulrb** · March 29th 2008, 01:20 PM

Prove that if $0 \leq a<b$ , then

$a < \sqrt{ab} < \frac{a+b}{2} < b$

I know how to find $a < \sqrt{ab}$ , but I don't understand how to find $\sqrt{ab} < \frac{a+b}{2}$ , or that that will be less than b.

Can this even be proved? It doesn't appear to be true.

**Peritus** · March 29th 2008, 01:33 PM

$ \begin{gathered} ab < \left( {\frac{{a + b}} {2}} \right)^2 \hfill \\ \Leftrightarrow 4ab < a^2 + 2ab + b^2 \hfill \\ \Leftrightarrow 0 < \left( {a - b} \right)^2 \hfill \\ \end{gathered}$

**Plato** · March 29th 2008, 01:37 PM

The last part first.
$a < b \Rightarrow \quad \frac{a}{2} < \frac{b}{2} \Rightarrow \quad \frac{a}{2} + \frac{a}{2} < \frac{b}{2} + \frac{a}{2}\;\& \quad \frac{a}{2} + \frac{b}{2} < \frac{b}{2} + \frac{b}{2}$ .

Now the first:
Lemma: $\left( {\forall x\,\& \,y} \right)\left[ {x^2 + y^2 \ge 2xy} \right]$ .
Proof: $\left( {x - y} \right)^2 \ge 0 \Rightarrow \quad x^2 - 2xy + y^2 \ge 0 \Rightarrow \quad x^2 + y^2 \ge 2xy$

Now let $x = \sqrt a \;\& \; y = \sqrt b$ and divide by 2.

**galactus** · March 29th 2008, 01:37 PM

Here's a geometric proof.

If we try to construct a square equal in area to a given rectangle and the

dimensions of the rectangle are a and b, then the problem is to determine x

such that $x^{2}=ab$ .

Look at the semi-circle in the diagram. The inscribed triangle is a right triangle, then the two smaller ones are similar. Then, we have:

$\frac{a}{x}=\frac{x}{b}$

Therefore, x is mean proportional between a and b, so there geometric mean is $\sqrt{ab}$ .

Since the radius of the semi-circle is $\frac{a+b}{2}$ , it follows that:

$\sqrt{ab}\leq\frac{a+b}{2}$ .

This holds even when a=b.

The geometric mean of two positive numbers never exceeds their arithmetic mean

Another thing to look at algebraically is since

$(a+b)^{2}-(a-b)^{2}=4ab$ , then

$\frac{a+b}{2}\geq\sqrt{ab}$

**paulrb** · March 29th 2008, 02:14 PM

Thanks for the proofs - they were very helpful.

Originally Posted by Plato

Now the first:
Lemma: $\left( {\forall x\,\& \,y} \right)\left[ {x^2 + y^2 \ge 2xy} \right]$ .

Hmm, I don't recognize that notation.

Does anyone know of a good book that explains/gives practice with proofs? I don't need to for any of my courses (I'm currently in Calc II), but I think proofs are interesting, and I'd like to get better at them. Sadly I have little experience because none of the math courses I've taken required me to do proofs. Even when I took geometry in high school, they dumbed it down and barely required any proofs. It's kind of unfortunate, because I'd have more mathematical discipline if it was for a grade. Anyway, I would be interested to learn about proofs in my spare time.

**Plato** · March 29th 2008, 02:35 PM

Originally Posted by paulrb

Hmm, I don't recognize that notation.

$\left( {\forall x\,\& \,y} \right)$
That is read "For all x & y" what follows is true.
The up-side-down A is read For all.
The backwards E, $\exists$ is read "There exist" or "for some".
Both notations are common in mathematical proofs.

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