Expert: Paul Klarreich - 6/1/2008

Question
Three Quantities a,b,c are said yo be in harmonical progression when a/c = a-b/b-c

Prove, is a+b/2, b, b+c/2 be in HP, then a, b, c are in GP

thanks

Answer
Questioner:   Brent
Category:  Advanced Math
Private:  No
 
Subject:  Harmonical Progression
Question:  Three Quantities a,b,c are said yo be in harmonical progression when a/c = a-b/b-c

Prove, if (x+y)/2, y, (y+z)/2 are in HP, then x, y, z are in GP (geometric progression)

thanks
............................
Hi, Brent,

[Please, if you send another question, make ALL of your question clear.  I don't like trying to figure out your question before answering it.  Asking the question properly is your job, not mine.]

I changed the notation in the question part, to z,y,z.
If x,y,z are in G.P. then you must prove:

y/x = z/y

or  y^2 = xz
..........................
Now use your three things:

If (x+y)/2, y, (y+z)/2 are in harmonic progression, that means [Please note the parentheses]

(x + y)/2    (x + y)/2 - y
---------- = -------------
(y + z)/2    y - (y + z)/2

(x + y)      x + y - 2y
---------- = -----------
(y + z)      2y - y - z

x + y     x - y
------ = --------
y + z     y - z

xy - xz + y^2 - yz = xy - y^2 + xz - yz

  - xz + y^2      =  - y^2 + xz

       2y^2      =   2xz

       y^2      =   xz

That's it.