Question

if a,b,c, are in ap and a,b,d are in gp ,then prove that a, a-b, d-c are in gp

Answer 1

 if a,b,c are in AP then

  b-a=c-b

   a+c=2b ...........1

now we have to prove that b+c,c+a,a+b are in AP,

     let these are in AP then

                 (a+c)-(b+c)=(a+b)-(a+c)

                      (a+c)=2b

from eq 1 we have a+c=2b ,RHS=LHS

so these terms are also in AP...