This can be solved as under:-
p/q=q/r, which implies that q2=pr. -(1)
(p+x), (q+x), (r+x) are in HP,
Thus,
2(p+x)(r+x)=(q+x)(p+r+2x)
or (2pr+2px+2rx+2x2)=(pq+rq+2qx+px+rx+2x2)
which can be written as:-
x(2p+2r-p-r-2q)=(pq+rq-2pr) -(2)
LHS1=x(p+r-2q) -(2L)
RHS1=[(p+r)q-2pr]=[(p+r)q-2q2] [From eq. 1]
RHS1=(p+r-2q)q=q(LHS1)/x -(2R) [From eq. 2L]
This implies that:-
LHS1=q(LHS1)/x [From eqs. 2, 2L, 2R]
x=q
