Question

Let three distinct numbers $x,y,z$ be in G.P. where as $x,2y,3z$ are in A.P., then the common ratio of the G.P. is ______.

• A. $3$
• B. $\frac{1}{3}$
• C. $2$
• D. $\frac{1}{2}8$

Answer 1

Answer: (B)

$\frac{1}{3}$

If $x,y,z$ are in G.P.

$\frac{y}{x}=\frac{z}{y}=ry=xr,z=yr=x{r}^2$

$x,3y$ & $3z$ are in A.P.

$\therefore 4y=3z+x4xr=3\left(x{r}^2\right)+x$

$4r={3r}^2+1$

${3r}^2-4r+1=0$

${3r}^2-3r-r+1=0$

$(r-1)(3r-1)=0$

$r=1,\frac{1}{3}$

If $r=1$ $x,y,z$ has to be distinct.

$\therefore r=\frac{1}{3}.$