Question

Given $a^x=b^y=c^z=d^u$ and $a,b,c,d$ are in G.P., the $x,y,z,u$ are in

• A. A.P.
• B. G.P.
• C. H.P.
• D. none

Answer 1

Answer: (C)

H.P.

Since, $a,b,c,d$ are in G.P
let, $b=ar,c=a{r}^2,d=a{r}^3$
Given that, $b=arc=a{r}^{2}d=a{r}^3$

Taking logarithm, we get

$b=arc=a{r}^{2}d=a{r}^3$

$\Rightarrow x\left(\log a\right)=y(\log a+\log r)=z(\log a+2\log r)=u(\log a+3\log r)=k$

Therefore, $x=\frac{k}{\log a}$; $y=\frac{k}{\log a+\log r}$; $z=\frac{k}{\log a+2\log r}$; $u=\frac{k}{\log a+3\log r}$
Since, $\log a$, $\log a+\log r$, $\log a+2\log r$, $\log a+3\log r$ are in A.P

Therefore, $x,y,z,u$ are in H.P
Ans: C