Choose any two numbers. Call them $a$ and $b$ ( $b\leqq a$). Work out the arithmetic mean $(a+b)/2$ and the geometric mean $\sqrt{\mathrm{ab}}$. 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

${\displaystyle \mathrm{ab}<(\frac{a+b}{2}){}^2.}$

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