Question

In a GP with alternatively positive and negative terms and any term is the AM of the next two terms. Then, the common ratio of the GP is?

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

Answer 1

Answer: (C)

$-2$

Let the first term and the common ratio of a GP be $a$ and $r$ respectively.
Since, any term is the AM of the next two terms.
Therefore, $a_n=\frac{a_{n+1}+a_{n+2}}{2}$
$\Rightarrow a{r}^{n-1}=\frac{a{r}^n+a{r}^{n+1}}{2}$
$\Rightarrow 2=r+r^2$
$\Rightarrow r^2+r-2=0$
$\Rightarrow (r+2)(r-1)=0$
$\Rightarrow r=-2$ .....$[\because r\ne 1]$