Advanced High-School Mathematics

148 CHAPTER 3 Inequalities

Harmonic Mean:

HM(x 1 ,x 2 ,...,xn) =

n

1

x 1 +

1

x 2 +···+

1

xn

;

Quadratic Mean:^2

QM(x 1 ,x 2 ,...,xn) =

Ã

x^21 +x^22 +···+x^2 n

n

.

Note that if a 1 , a 2 , ... is an arithmetic sequence, then an is the
arithmetic mean ofan− 1 andan+1. Likewise ifa 1 , a 2 , ...is a geometric
sequence (and allan>0), thenanis the geometric mean ofan− 1 and
an+1.
A harmonic sequence is by definition the reciprocal of nonzero
terms in an arithmetic sequence. Thus, the sequences

1 ,

1

2

,

1

3

, ..., and

2

3

,

2

5

,

2

7

, ...

are harmonic sequences. In general, ifa 1 , a 2 , ...is a harmonic sequence,
thenanis the harmonic mean ofan− 1 andan+1.

One of our aims in this section is to prove the classical inequalities

HM≤GM≤AM≤QM.

Before doing this in general (which will require mathematical induc-
tion), it’s instructive first to verify the above in casen= 2.
Indeed, starting with 0≤(

√

x−

√

y)^2 we expand and simplify the
result as
2

√

xy≤x+y⇒GM(x 1 ,x 2 )≤AM(x 1 ,x 2 ).

Having proved this, note next that

HM(x 1 ,x 2 ) =

(

AM

( 1

x 1

+

1

x 2

))− 1

;

(^2) Sometimes called theroot mean square