Question

In a G.P. of alternating positive and negative terms, any term is the A.M. of the next two terms. Then the common ratio is

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

Answer 1

Answer: (C)

Let the G.P. be $a-ar+a{r}^2-a{r}^3+\dots$ with common ratio $=-r$
By the given condition
$a=\frac{a{r}^2-ar}{2}$
$\Rightarrow 2a=a{r}^2-ar$
$\Rightarrow 2=r^2-r$
$\Rightarrow r^2-r-2=0$
$\Rightarrow (r-2)(r+1)=0$
$\Rightarrow r=2,-1$
$\therefore$ Common ratio $=-2$ or $1$
Hence, common ratio $=-2$
$(\because$ common ratio is $-ve)$