Question

If $x,2y,3z$ are in A.P. where the distinct numbers $x,y,z$ are in G.P, then the common ratio of G.P. is

• A. $3$
• B. $\frac{1}{3}$
• C. $2$
• D. $\frac{1}{2}$

Answer 1

The correct option is B $\frac{1}{3}$

It is given that the distinct numbers $x,y,z$ are in G.P.
Let $r$ be the common ratio of this G.P. Then,
$y=xr$ and $z=x{r}^2$
Now, $x,2y,3z$ are in A.P.
$\Rightarrow x,2xr,3x{r}^2$ are in A.P.
$\Rightarrow 2xr-x=3x{r}^2-2xr$
$\Rightarrow 3x{r}^2-4xr+x=0$
$\Rightarrow 3r^2-4r+1=0$
$[\because x\ne 0,\text{as it is a term of G.P}]$
$\Rightarrow (r-1)(3r-1)=0\Rightarrow r=\frac{1}{3}$
$[\because x\ne y\ne z\Rightarrow r\ne 1]$

Hence, the common ratio of the G.P. is $\frac{1}{3}$.