If a,b,c  are three unequal numbers such that a, b, c  are in A.P and b−a,  c−b,  a  are in G.P, then a  :  b  :  c  is

Given that a, b, c  are in A.P ⇒            2b=a+c and b−a,  c−b,  a  are in G.P.⇒                (c−b)2=(b−a)a ⇒   c2+b2−2bc=ab−a2 ⇒   c2+(a+c2)2−2(a+c2)c  =a(a+c2)−a2        [using  Eq.  (i)] ⇒    c2+a2+c2+2ac4−ac−c2=a22+ac2−a2 ⇒   (c−a)2=2a(c−a) ⇒               c=3a         (∵   c≠a) From Equation.  (i)                2b=a+3a                   b=2a ∴      a  :  b  :  c=a  :  2a  :  3a                        =1  :  2  :3