## Using Area to Prove the Arithmetic-Geometric Mean Inequality

If we have real numbers and , we call the arithmetic mean (AM) and the geometric mean (GM) of and . In this post, we are going to examine the relationship of these two means.

To start off, let’s have a few examples. If and , then and ; if and , then and ; if and , then and . What do you observe? Try a few example and see if your observations hold.

From a few examples above, and from your trials, you have probably observed that which means that for some positive real numbers and . This is actually true for all positive real numbers and . In the following discussion, we are going to use the concept of area to prove that the statement is true.

To begin the proof, we construct a square with side length made up of four rectangles and a square at the center (technically, a square is also a rectangle). Clearly, the area of each of the four rectangles is , and the square at the center has are (Can you see why?). If we remove the square at the center, the remaining area is represented by the equation . Note that is the total area of the four rectangles. » Read more