Choose any two numbers. Call them a and b (b ≤ a). Work out the arithmetic mean (a + b)/2 and the geometric mean

√

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 2 )2.

What does this tell us about the arithmetic mean and the geometric mean of two numbers?