Copyright © University of Cambridge. All rights reserved.

## 'AMGM' printed from http://nrich.maths.org/

Choose any two numbers. Call them $a$ and $b$ ($b < a$). Work
out the arithmetic mean $(a + b)/2$ and the geometric mean
$\sqrt{ab}$. Which is bigger? Repeat for other pairs of numbers.
What do you notice?

In the diagram PQRS is a rectangle measuring $a$ units by $b$
units. The green rectangle measures $(a + b)/2$ by $b$ and the
orange and blue rectangles both measure $(a - b)/2$ by $b$. By
considering the areas of the rectangles explain why this diagram
shows that $$ab < ({{a + b}\over 2})^2. $$ What does this tell
us about the arithmetic mean and the geometric mean of two
numbers?