Question

Let $p<q<r$ be three integers such that $p,q,r$ is an arithmetic progression and $p,r,q$ is a geometric progression. The least positive value of $r$ is

• A. $-2$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is C $2$

Since $p,q,r$ is an A.P. ,
$2q=p+r\cdots \left(1\right)$
$p,r,q$ is a G.P. $\Rightarrow r^2=pq\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$ we get
$r^2=(2q-r)q$
$\Rightarrow 2q^2-rq-r^2=0$
$\Rightarrow (2q+r)(q-r)=0$
$\Rightarrow r=-2q[\because q<r]$

$\therefore$ The numbers in A.P. are $4q,q,-2q$
Since $4q<q<-2q,$ we have $q<0$
The least value of $r$ is $2$ when $q=-1$