# Thread: Arithmetic Mean

1. ## Arithmetic Mean

If a < b, show that a < (a + b)/2 < b. The number (a + b)/2 is called the ARITHMETIC MEAN of a and b.

2. Originally Posted by symmetry
If a < b, show that a < (a + b)/2 < b. The number (a + b)/2 is called the ARITHMETIC MEAN of a and b.
Since $\displaystyle a<b$

Then: $\displaystyle 2a<a+b$

So: $\displaystyle a<\frac{a+b}{2}$

Also since $\displaystyle a<b$

Then: $\displaystyle a+b<2b$

So: $\displaystyle \frac{a+b}{2}<b$

Therefore: $\displaystyle a<\frac{a+b}{2}<b$

3. ## ok

Very interesting notes in this prove question and answer.