The Art and Craft of Problem Solving

182 CHAPTER 5 ALGEBRA

We conclude that

(^10000 1)
2 VlOOO I - 2 <
n�^1 vn
<^2 VlOOOO.
This tells us that l A J is either 198 or 19 9. We can easily refine this estimate, because
the original trick of bracketing 1/ vn between 2 (v'n+T -vn) and 2 ( vn -vn=t)

was pretty crude for small values of n. For example, when n = 1, we ended up using

the "estimate" that

2(v'2-I) < (^1) < 2 V1.
The lower limit is not too bad, but the upper limit is a silly overestimate by exactly 1.

So let's not use it! Start the summation at n = 2 and write

(^10000 1)
A=I+ � tv;.

n= 2 yn

Now we estimate

The lower limit in the above expression is a little bit larger than 19 7, while the upper

limit is 19 8. Thus we conclude that A is between 198 and 19 9, so l A J = 19 8. •

The Cauchy-Schwarz Inequality

Letal ,a 2 , ... ,an and bl ,b 2 , ... ,bn be sequences of real numbers. The Cauchy-Schwarz

inequality states that

with equality holding if and only if aJ/bl = az/b 2 = ... = an/bn. If n = 1, this in­

equality reduces to two-variable AM-GM. If n = 3, the inequality (using friendlier

variables) is

(7)

for all real a, b, c, x, y, z.

5.5.20 Prove (7) by multiplying out efficiently and looking at cross-terms and using

two-variable AM-GM whenever possible. This method generalizes to any value of n.

Another way to prove (7) uses the simple but important tool that

A sum of squares of real numbers is non-negative, and equal to zero if

and only if all the numbers are zero.