Question

If $a_{1}, a_{2}, a_{3}\left(a_{1}>0\right)$ are three successive terms of a $G.P$. with common ratio $r$, the value of $r$ for which $a_{3} > 4 a_{2}$ $3 a_{1}$ holds is given by

• A. $1 < r < 3$
• B. $-3 < r < -1$
• C. $r > 3$ or $r < 1$
• D. None of these

Answer 1

Answer: (C)

Let $a_{1}, a_{2}, a_{3}$ be first three consecutive terms of G.P. with common ratio $r .$
Then, $a_{2}=a_{1} r$ and $a_{3}=a_{1} r^{2}$
Now, $a_{3}>4 a_{2}-3 a_{1}$
$\Rightarrow a_{1} r^{2}>4 a_{1} r-3 a_{1}$
$\Rightarrow r^{2}>4 r-3$
$\Rightarrow r^{2}-4 r+3>0$
$\Rightarrow (r-1)(r-3)>0$
$\Rightarrow r<1$ or $r>3$