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

If a, b, c are in A.P., b, c, d are in G.P. and 1/c,1/d,1/e are in A.P. prove that a,c,e are in GP.

If a^(x)=b, b^(y)=c,c^(z)=a, prove that xyz = 1 where a,b,c are distinct numbers

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

If x, y, z are in A.P. and tan^(-1)x , tan^(-1)y and t a n^(-1)z are also in A.P., then (1) 2x""=""3y""=""6z (2) 6x""=""3y""=""2z (3) 6x""=""4y""=""3z (4) x""=""y""=""z

If a, b, c, d are in G.P, prove that (a^n + b^n), (b^n + c^n), (c^n + d^n) are in G.P.

If a,b,c are three terms in A.P and a^(2), b^(2), c^(2) are in G.P. and a + b + c = (3)/(2) then find a. (where a lt b lt c )

If the 4th, 10th and 16th terms of a G.P. are x, y and z, respectively. Prove that x, y, z are in GP.

Three real numbers a, b, c are inG.P. If a+b + c = xb then prove that x lt -1 or x gt 3

a,b,c,d are in G.P. Prove that a^(2)-b^(2),b^(2)-c^(2), c^(2)-d^(2) are also in G.P.

Suppose a is a fixed real number such that (a - x)/(px) = (a - y)/(qy) = (a - z)/(rz) If p,q,r, are in A.P., then prove that x,y,z are in H.P.