The Art and Craft of Problem Solving

180 CHAPTER 5 ALGEBRA

The AM-GM inequality is the starting point for many interesting inequalities.

Here is one example (see the problems for several more).

Example 5.5.18 Let al ,a 2 , ... ,an be a sequence of positive numbers. Show that

(al +a 2 +···+an) (�+ � + ... +�) 2: n^2 ,

al a 2 an

with equality holding if and only if the ai are equal.

Solution: First, make it easier by examining a simpler case. Let's try to prove

(

1 I)?

(a+b) �+b (^24).
Multiplying out, we get
or

a b?

1+-+-+1> 4

b a -

,

a b?

- + -> 2.

b a-

This inequality is true because of AM-GM:

It is worth remembering this result in the following form:

1

If x > 0, then x + -2: 2, with equality if and only if x = 1.

x

Returning to the general case, we proceed in exactly the same way. When we

multiply out the product

n n 1

�I

aj

�I ak'

we get n^2 terms, namely all the terms of the form

aj

, 1 k '5. j, '5. n.

ak

For n of these, j = k, and the term equals 1. The remaining n^2 - n terms can be paired

up in the form

aj ak

-+-, 1 '5.j<k'5.n.

ak aj

(Note that the expression "1 '5. j < k '5. n" ensures that we get every pair with no

duplications.) Applying AM-GM to each of these pairs yields