Question

If $x,y,z$ are in G.P. and $x+y,y+z,z+x$ are in A.P., where $x\ne y\ne z$, then common ratio of the G.P. is

• A. $4$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

The correct option is D $-2$

Let the common ratio of the G.P., $x,y,z$ be $r$, so
Now,
$x+y,y+z,z+x$ are in A.P., so
$x+y+z+x=2(y+z)\Rightarrow 2x=y+z$
Using G.P., we get
$\Rightarrow 2x=xr+x{r}^2\Rightarrow x(r^2+r-2)=0\Rightarrow x(r+2)(r-1)=0\Rightarrow x=0,r=-2,1$

As $x\ne y\ne z$, so
$x\ne 0,r\ne 1$

Therefore, the only possible value of $r=-2$