Ans: B
Let $a$ and $r$ be the first term and the common ratio of the given sequence respectively.
Let $a$ and $r$ be the first term and the common ratio of the given sequence respectively.
$\left\{\begin{array}{ll}
ar^2 = 21 & \ldots \unicode{x2460} \\
ar^6 = 189 & \ldots \unicode{x2461}
\end{array}\right.$
$\unicode{x2461} \div \unicode{x2460}$, we have
$\begin{array}{rcl}
r^4 & = & 9 \\
r & = & \sqrt{3} \ \text{ or} \ -\sqrt{3}
\end{array}$
Hence, $a=7$.
I may not be true. If $r = \sqrt{3}$, $r > 1$.
II must be true. For example, $a_2 = \pm7\sqrt{3}$, which is irrational number.
III may not be true. If $r=-\sqrt{3}$, then the sum of the first $99$ terms
$\begin{array}{cl}
= & \dfrac{7(1-(-\sqrt{3})^{99})}{1-(-\sqrt{3})} \\
= & 1.06 \times 10^{24} \\
< & 3 \times 10^{24}
\end{array}$
