If `x=sum_(n=0)^(oo) a^(n),y=sum_(n=0)^(oo)b^(n),z=sum_(n=0)^(oo)(ab)^(n)`, where `a,blt1`, then

If x = sum_(n=0)^(oo) a^(n) .,y=sum_(n=0)^(oo) b^(n) , z = sum_(n=0)^(oo) (ab)^(n) , where a,blt 1 , then :

If x = sum_(n=0)^(oo) a^(n), y=sum_(n=0)^(oo) b^(n), z = sum_(n=0)^(oo) C^(n) where a,b,c are in A.P. and |a| lt 1, |b| lt 1, |c| lt 1 , then x,y,z are in

If a=sum_(n=0)^(oo)x^(n),b=sum_(n=0)^(oo)y^(n),c=sum_(n=0)^(oo)(xy)^(n) where |x|,|y|<1 then

If a=sum_(n=0)^(oo)x^(n),b=sum_(n=0)^(oo)y^(n),c=sum_(n=0)^(oo)(xy)^(n) where |x|,|y|<1 then

If s= sum_(n=0)^(oo) a^(n) , y = sum_(n=0)^(oo) b^(n) , z= sum _(n=0)^(oo) c^(n) , where a, b c are in A.P. and |a| lt 1, |b|lt 1 , |c| lt 1 then x,y,z are in :

If quad sum_(n=0)^(oo)a^(n),y=sum_(n=0)^(oo)b^(n),z=sum_(n=0)^(oo)c^(n), wherer a,b, and c are in A.P.and |a|<,|b|<1, and |c|<1, then prove that x,y and z are in H.P.

If x = sum _(n =0) ^(oo) a^(n) , y = sum _(n=0) ^(oo) b ^(n), z= sum _(n=0)^(oo) c^(n) where a,b,c are in A.P. and |a| lt 1, |b| lt 1, |c| lt 1, thenx,y,z are in-

If x= sum_(n=0)^(oo) a^n, y=sum_(n=0)^(oo) b^n, z= sum_(n=0)^(oo) c^n , where a ,b, c are in A.P. and |a|<1, |b| <1, |c|<1, then x,y,z are in :