Question

Three distinct numbers $x,y,z$ from a geometric progression in that order, and $x+y,y+z,z+x$ form an arithmetic progression in that order. Find the common ratio of the geometric progression.

• A. $-2$
• B. $2$
• C. $0.5$
• D. $-0.5$

Answer 1

Answer: (A)

$-2$

$x,y,z$ are in G.P.
$x+y,y+z,z+x$ are in A.P. .....(i)
$\therefore$ Common ratio of the
G.P $=\frac{y}{x}=\frac{z}{y}=r$ (let)
Also (i)
$\Rightarrow y+z=\frac{(x+y)+(z+x)}{2}$
$\Rightarrow 2x=y+z$
$\Rightarrow r=\frac{y}{x}=\frac{y}{(y+z)/2}=\frac{2y}{y+z}=\frac{2}{1+\frac{z}{y}}$
$\Rightarrow r=\frac{2}{1+r}$
$\Rightarrow r^2+r-2=0$
$\Rightarrow (r-1)(r+2)=0$
$\Rightarrow r=-2$ ....[because $r=1$ would mean $\Rightarrow x=y=z]$