Explanation

(a)
Let the numbers in GP be $\frac{a}{r},a,ar$
Given, $\frac{a}{r} + a + ar = 38$ …(i)
and $\frac{a}{r}.a.ar = 1728$
$\Rightarrow a^{3} = 1728\ \Rightarrow a = 12$ …(ii)
$\therefore$ From Eq. (i)
$a\left( \frac{1}{r} + 1 + r \right) = 38$
$\Rightarrow \ \frac{1}{r} + 1 + r = \frac{38}{12}$ [from Eq. (ii)]
$\Rightarrow \ 6r^{2} - 13r + 6 = 0$
$\Rightarrow \ (3r - 2)(2r - 3) = 0\ \Rightarrow r = \frac{2}{3}$ or $\frac{3}{2}$
Hence, required GP is 18, 12, 8 or 8, 12, 18
$\therefore$ Greatest number is 18.