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

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=Sigma_(n=0)^(oo) a^n,y=Sigma_(n=0)^(oo) b^n,z=Sigma_(n=0)^(oo) c^n where a, b,and c are in A.P and |a|lt 1 ,|b|lt 1 and |c|1 then prove that x,y and z are in H.P

If a,b,c are proper fractiion are in H.P. and x sum_(n=1)^oo a^n, y=sum_(n=1)^oo b^n, z= sum_(n=1)^oo c^n then x,y,z are in (A) A.P. (B) G.P. (C) H.P. (D) none of these

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 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 x=sum a^(n),y=sum b^(n),z=sum c^(n) where a,b,c are in AP then x,y,z are in

If x=sum_(n=0)^oo a^n, y=sum_(n=0)^oo b^n where |a|<1,|b|<1 then prove that sum_(n=0)^oo (ab)^n=(xy)/(x+y-1)

If x= sum_(n=0)^oo (Costheta)^(2n) , y= sum_(n=0)^oo (Sinphi)^(2n) , z= sum_(n=0)^oo (Cosphi)^(2n). (Sintheta)^(2n) Then which of the following is true ??