Question

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

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

Answer 1

The correct option is C HP

Let $a^x=b^y=c^z=d^t=k$

Applying $\log$ on both sides, we get

$x\log a=\log k$

$y\log b=\log k$

$z\log c=\log k$

$t\log d=\log k$

Also $a,b,c,d$ are in GP.

This means that $b^2=ac$ and $c^2=bd$

Substituting for $a,b,c$ and $d$ in terms of $\log$, we have

$2\log b=\log a+\log c$

$\frac{2\left(\log k\right)}{y}=\frac{\log k}{x}+\frac{\log k}{z}$

$\frac{2}{y}+\frac{1}{x}+\frac{1}{z}$ .....(1)

$2\log c=\log b+\log d$

$\frac{2\left(\log k\right)}{z}=\frac{\log k}{y}+\frac{\log k}{t}$

$\frac{2}{z}+\frac{1}{y}=\frac{1}{t}$ ....(2)

From (1) and (2), we can prove that $x,y,z$ and $t$ are in harmonic progression (HP).

Hence option C is correct.