Question

If the sum of the second, third and fourth terms of a positive term $G.P$ . is $3$ and the sum of its sixth, seventh and eighth terms is $243,$ then the sum of the first $50$ terms of this $G.P$. is :

• A. $\frac{2}{13}\left(3^{50}-1\right)$
• B. $\frac{1}{26}\left(3^{50}-1\right)$
• C. $\frac{1}{13}\left(3^{50}-1\right)$
• D. $\frac{1}{26}\left(3^{49}-1\right)$

Answer 1

Answer: (B)

Let first term $= a >0$
Common ratio $= r >0$
$a r+a r^{2}+a r^{3}=3$
$a r^{5}+a r^{6}+a r^{7}=243$
$r^{4}\left(a r+a r^{2}+a r^{3}\right)=243$
$r^{4}(3)=243$
$ \Rightarrow r=3$ as $r>0$
from (1)
$3 a+9 a+27 a=3$
$a =\frac{1}{13}$
$S_{50}=\frac{a\left(r^{50}-1\right)}{(r-1)}$
$=\frac{1}{26}\left(3^{50}-1\right)$