Let a, b, c, d, e are non-zero and distinct positive real numbers. If a, b, c are in A.P.; b, c, d are in G.P. and c, d, e are in H.P., then a, c, e are in:

Step-by-step solution

Given a, b, c are in A.P. b = a+c 2 c, d, e are in H.P. d = 2ce c+e b, c, d are in G.P. c^2 = bd Substituting b and d in the G.P. condition: c^2 = ( a+c 2 ) ( 2ce c+e )c^2 = ce(a+c) c+e c(c+e) = e(a+c)c^2 + ce = ae + cec^2 = ae This means a, c, e are in G.P.