If the sum of the second, third and fourth terms of a positive term G.P.

Let first term $=a>0$

Common ratio $=r>0$

$a r+a r^{2}+a r^{3}=3$  $\ldots(i)$

$a r^{5}+a r^{6}+a r^{7}=243$ $\ldots$ (ii)

$r^{4}\left(a r+a r^{2}+a r^{3}\right)=243$

$r^{4}(3)=243 \Rightarrow r=3$ as $r>0$

from (1)

$3 a+9 a+27 a=3$

$a=\frac{1}{13}$

$S_{50}=\frac{a\left(r^{50}-1\right)}{(r-1)}=\frac{1}{26}\left(3^{50}-1\right)$