Arithmetic Mean: Show that a<(a + b)/2<b for a<b

[cherie327]

If anyone could help or give me some pointers on this problem it's much appreciated!

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

Thanks!

 

[pka]

If \(\displaystyle a < b\) then \(\displaystyle \frac{a}{2} < \frac{b}{2}.\)
So add \(\displaystyle \frac{a}{2}\) to both sides: \(\displaystyle \frac{a}{2} + \frac{a}{2} < \frac{b}{2} + \frac{a}{2} = \frac{{b + a}}{2}\).

To finish do it again: add \(\displaystyle \frac{b}{2}\) to both sides.

 

[cherie327]

Thank you very much! that helped a lot!