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 the G.P. is

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

Answer 1

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

Since x, y, z are in G.P.
$\mathrm{Let}x=a,y=\mathrm{ar},z={\mathrm{ar}}^2,\mathrm{where}\text{'}a\text{'}\mathrm{is}\mathrm{the}\mathrm{first}\mathrm{term}\mathrm{and}r\mathrm{is}\mathrm{the}\mathrm{common}\mathrm{ratio}\mathrm{of}\mathrm{the}G.P.\therefore y^2=\mathrm{xz}$
Also, x, 2y, 3z in A.P.
$2\left(2y\right)=x+3z\Rightarrow 4\mathrm{ar}=a+3{\mathrm{ar}}^2\Rightarrow 4r=1+3r^2\Rightarrow 3r^2-4r+1=0\Rightarrow 3r^2-3r-r+1=0\Rightarrow 3r(r-1)-1(r-1)=0\Rightarrow r=1\left(\mathrm{Not}\mathrm{Possible\; as\; x,\; y,\; z\; are\; distinct}\right)\mathrm{or}\frac{1}{3}$