Question

If $a^x=b^y=c^z,x\ne 0,y\ne 0,z\ne 0$ and $b^2=ac$, then show that $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in A.P.

Answer 1

We are given that $a,b$ and $c$ are in G.P.

Let $r$ be common ratio.

$\Rightarrow b=ar$ and $c=a{r}^2$

Now $a^x=b^y$

Taking logarithm on both sides, we get

$x\log a=y\log b$

$x\log a=y\log ar$

$\frac{1}{y}=\frac{1}{x}+\frac{\log r}{\log a}$ ....(i)

Similarly applying logarithm to equation $a^x=c^z$

We obtain

$\frac{1}{z}=\frac{1}{x}+2\frac{\log r}{\log a}$ .....(ii)

Calculate $2$ (i) $-$ (ii)

We obtain $2\frac{1}{y}=\frac{1}{x}+\frac{1}{z}$

Therefore $\frac{1}{x},\frac{1}{y},\frac{1}{z}$ are in A.P.