Question

If $x,2y,3z$ are in AP, where the distinct numbers $x,y,z$ are in GP, then the common ratio of the GP is:

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

Answer 1

Answer: (B)

$\frac{1}{3}$

Given $4y=x+3z$ _____(1)
and $y^2=xz$
Let common ratio be $r$.
$\therefore \frac{y}{x}=r$ and $\frac{z}{x}=r^2$
Dividing (1) by x, $4\frac{y}{x}=1+3\frac{z}{x}$
$\Rightarrow 4r=1+3r^2$
$\Rightarrow 3r^2-4r+1=0$
$\Rightarrow (3r-1)(r-1)=0$
$\Rightarrow r=\frac{1}{3},1$
But $r\ne 1(\because x,y,z$ are distinct)
$\therefore r=\frac{1}{3}$