Question

Let $a_{1},a_{2},a_{3}$ be three positive numbers which are in geometric progression with common ratio $r$ . The inequality $a_{3}>a_{2}+2a_{1}$ holds true if $r$ is equal to

• A. $2$
• B. $1.5$
• C. $0.5$
• D. $2.5$

Answer 1

Answer: (D)

$a_{2}=a_{1}r,a_{3}=a_{1}r^{2}$
$a_{3}>a_{2}+2a_{1}$
$\Rightarrow a_{1}r^{2}>a_{1}r+2a_{1}$
$\Rightarrow r^{2}-r-2>0$
$\Rightarrow \left(r - 2\right)\left(r + 1\right)>0$
$\Rightarrow r < -1$ or $r>2$