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

If a, b, c are in A.P. and a,b, d are in G.P., prove that a, a-b, d-c are in G.P.

If x^(1/3)+y^(1/3)+z^(1/3)=0 , then prove that (x+y+z)^3=27xyz

If a, b, c are in A.P. and a, b, c + 1 are in G.P. then prove that 4a=(a-c)^2 .

If a,b,c are in G.P., show that 1/(a+b),1/(2b),1/(b+c) are in A.P.

If x,y,z are in G.P then show that log x,log y,log z are in A.P. (x>0,y>0,z>0) .

If x,y,z are in G.P and p,q,r are in A.P then prove that x^(q-r) y^(r-p) z^(p-q)=1 .

If a,b,c,d are in G.P show that each of the following three term are also in G.P 1/(a^2+b^2),1/(b^2+c^2),1/(c^2+d^2) .

If a,b,c,d are in G.P show that each of the following three term are also in G.P 1/(a^3+b^3),1/(b^3+c^3),1/(c^3+d^3) .

If a, b, c are in A.P. then show that the terms given below are also in A.P. a(1/b+1/c),b(1/c+1/a),c(1/a+1/b) .

If a, b, c are in A.P. then show that the terms given below are also in A.P. 1/(bc),1/(ca),1/(ab)