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