Question

If $d\ne 0$ and $a(a+d),(a+d)(a+2d),(a+2d)a$ are in G.P., then the common ratio is

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

Answer 1

The correct option is B $-2$

Since, $a(a+d),(a+d)(a+2d),(a+2d)a$ are in G.P.
$\Rightarrow (a+d{)}^2(a+2d{)}^2=a^2(a+d)(a+2d)$
Since, $(a+d)(a+2d)\ne 0$
because no term of G.P. should be $0$
$\Rightarrow (a+d)(a+2d)=a^2$
$\Rightarrow a^2+3ad+2d^2=a^2$
$\Rightarrow d(3a+2d)=0$
$\Rightarrow 3a=-2d(\because d\ne 0)$
Common ratio
$=\frac{a+2d}{a}=\frac{a-3a}{a}=-2$