## LogicalApple 2 years ago Fun Challenge So here is the challenge. Prove that (a + b)/2 (the arithmetic mean) is greater than or equal to sqrt(ab) (the geometric mean) for any real numbers a and b!

1. LogicalApple

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

2. watchmath

Squaring both sides and rearranging we have $$(a-b)^2\geq 0$$ which is always true. BTW you need the condition that $$a,b$$ are non negative though!!

3. Homeworksucks

5+6/2=11 sqrt(30)= 5.47722557505 11 is greater than 5.4 1+1.00001/2= 1.00005 1*1.00001=1.00001 sqrt(1.00001)= 1.00000499999 I haven't studied calculus yet but isn't something like as a+b approaches 2 at the very limit, it should be equal to sqrt(ab)?

4. LogicalApple

I should probably say that no calculus is required :)

5. Homeworksucks

Dammit, that probably wasn't the kind of proof you were looking for wasn't it?

6. watchmath

a=-2 b=-2. Arithmetic mean is -2 , but geometric mean is 2. So AM < GM in this case. You need a and b to be positive

7. LogicalApple

Good point! That's what I get for working out of a textbook, huh? Great counterexample. Ok let's assume a and b are positive then.

8. watchmath

The proof is as I said in the first post

9. LogicalApple

You were not very explicit but you got the same result I did. Magic?