Question

In a $G.P.$ $t_2 + t_5 = 216$ and $t_4 : t_6$ = $1 : 4$ and all terms are integers, then its first term is

• A. 16
• B. 14
• C. 12
• D. None of these

Answer 1

Answer: (C)

12

Answer 2

$G.P.$ is $a, ar,ar^{2}, ar^{3}, ar^{4}, .....$
$t_{2}+t_{5} = ar+ar^{4} = 216$
$\frac{t_{4}}{t_{6}} = \frac{ar^{3}}{ar^{5}} = \frac{1}{4}$
$\Rightarrow r^{2} = 4$
$\Rightarrow r= \pm2$
For $r=2$
$a\left(2+2^{4}\right) =216$
$\Rightarrow a\left(18\right) = 216$
$\Rightarrow a = \frac{216}{18} = 12$
For $r=-2$,
$a\left(-2+2^{4}\right) = 216$
$\Rightarrow a\left(14\right) =216$
$\Rightarrow a=\frac{216}{14} = \frac{108}{7}$
$\therefore a=12$ (since all terms are integers)