Question

If a, b, c , d are in GP and $a^x=b^y=c^z=d^v$, then x, y, z, v are in?

• A. AP
• B. GP
• C. HP
• D. None of these

Answer 1

Answer: (C)

HP

$\begin{array}{ccc}a,b,cinG.P& & \\ b^2=ac\to \left(i\right)& & \\ a^x=b^y=x^z=d^v=k& & \\ a=k^{\frac{1}{x}}& b=k^{\frac{1}{y}}& c=k^{\frac{1}{z}}\\ putinequation\left(i\right)& & \\ {\left(k\right)}^{\frac{2}{y}}={\left(k\right)}^{\frac{1}{x}+\frac{1}{z}}& & \\ \frac{d}{y}=\frac{1}{x}+\frac{1}{z}=\frac{x+z}{xz}& & \\ \frac{1}{y}=\frac{x+z}{2x+z}\left(harmonicmeans\right)& & \\ x,y,z,vinH.P& & \end{array}$