If a,b,c,d are in harmonic progression then is ab + bc + cd = 3ad?

Question

I don't know where to start this problem, can anyone please help me out? thanks a lot!

Answer 1

Write a=b/(1-rb), d=c/(1-rc). The question reduces to

b^2/(1-rb) + bc + c^2/(1-rc) =3bc /( 1-rb)(1-rc).

You multiply eveything by ( 1-rb)(1-rc). and conclude using that 1/c - 1/b = r.