Given a,b,c are in AP

then 2b = a+c

   => b = (a+c)/2............1

Again given b,c,d are in GP

then c² = bd.......2

Again 1/c,1/d,1/e are in AP

then

         2/d = 1/c + 1/e

   => 2/d  = (c+e)/c*e

  =>d/2 = c*e/(c+e)

  =>d = 2*c*e/(c+e)............3

Now put the value of b and d in equation 2

c² = {(a+c)/2}*{ 2*c*e/(c+e)}

=>c² = {(a+c)*2*c*e}/{2*(c+e)}

=>c² = {(a+c)*c*e}/{(c+e)}  (2 and 2 is divided by 2)

=>c = {(a+c)*e}/(c+e)  (c is cancelled from left and right side)

=>c*(c+e) = (a+c)*e

=>c² + ce = ae + ce

=>c² = ae

=> a,c,e are in GP