# If #a,b,c# are in arithmetic progression; #p,q,r# are in harmonic progression; #ap,bq,cr# are in geometric progression then prove that #a:b:c# is equal to #1/r:1/q:1/p#?

##### 1 Answer

Please see below.

#### Explanation:

As

or **................................(1)**

Also

**................................(2)**

As

or **................................(3)**

Here we assume that

Multiplying (1) and (2), we get

or

or

or

- Observe that
#sqrtk+1/sqrtk=sqrtl+1/sqrtl# is equivalent to #k+1/k=l+1/l# - just square and you get it. This gives us#k-l=1/l-1/k=(k-l)/(kl)# i.e. either#k=l# or#k=1/l#

Now as

but

or

Hence