If three unequal numbers p, q, r are in HP and their squares are in AP ,then find the ratio of p:q:r.

p,q,r are in HP.so

q=2pr/(p+r)

Or,2pr=q(p+r)

p²,q²,r² are in AP.so

p²+r²=2q²

or,(p+r)²-2pr=2q²

or,(p+r)²-(p+r)q=2q²

or,(p+r)²-q(p+r)-2q²=0

Solving

Either p+r=2q or p+r=-q.

Take p+r=2q first.So 2pr=2q².

Now (p-r)²=(p+r)²-4pr=4q²-4q²=0

Or,p=r.This cannot happen because p?q,r are distinct.

So p+r=-q.Then 2pr=-q²

(p-r)²=q²+2q²=3q²

Solving this equation with helo of 

p+r=-q you will get

And 

So

This is the answer.Hope it helps.If it does please approve..........................................................................................................................................