If `(a-x)/(p x)=(a-y)/(q y)=(a-z)/ra n dp ,q ,a n dr` are in A.P., then prove that `x ,y ,z` are in H.P.

Let a(a != 0) is a fixed real number and (a-x)/(px)=(a-y)/(qy)=(a-z)/(rz) . If p, q, r are in A.P., show that 1/x,1/y,1/z are in A.P.

If x ,1,a n dz are in A.P. and x ,2,a n dz are in G.P., then prove that x ,a n d4,z are in H.P.

If x ,1,a n dz are in A.P. and x ,2,a n dz are in G.P., then prove that x ,a n d4,z are in H.P.

If y-z,2(y-a),y-x are in H.P. prove that x-a,y-a,z-a are in G.P.

If a^((1)/(x))= b^((1)/(y))= c^((1)/(z)) and a,b,c are in G.P., prove that x,y,z are in A.P.

a, b, x are in A.P., a,b,y are in G.P. and a, b, z are in H.P. then:

If a^x=b^y=c^z and a,b,c are in G.P. show that 1/x,1/y,1/z are in A.P.

If reciprocals of (y-x),2(y-a), (y-z) are in A.P., prove that x-a,y-a,z-a are in G.P.

STATEMENT-1 : If log (x + z) + log (x -2y +z) = 2 log (x -z) then x,y,z are in H.P. STATEMENT-2 : If p , q , r in AP and (a -x)/(px) = (a-y)/(qy) = (a-z)/(rz) , then x, y, z are in A.P. STATEMENT-3 : If (a + b)/(1 - ab), b, (b + c)/(1 - bc) are in A .P. then a, (1)/(b) , c are in H.P.

If a, b, c are in G.P. and a^(1/x)=b^(1/y)=c^(1/z), prove that x, y, z are in A.P.