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

Answer 1

The correct option is D $\frac{1}{26}(3^{50}-1)$

First term $=a>0$
Common ratio $=r>0$
$\Rightarrow \frac{ar+a{r}^2+a{r}^3}{a{r}^5+a{r}^6+a{r}^7}=\frac{3}{243}$
$\Rightarrow \frac{ar(1+r+r^2)}{a{r}^5(1+r+r^2)}=\frac{1}{81}$
$\Rightarrow \frac{1}{r^4}=\frac{1}{81}$
$\Rightarrow r=3$

$a(3+9+27)=3$
$\Rightarrow a=\frac{3}{39}=\frac{1}{13}$

$S_{50}=a\left(\frac{r^{50}-1}{r-1}\right)$
$=\frac{1}{13}\left\{\frac{3^{50}-1}{2}\right\}$
$\therefore S_{50}=\frac{1}{26}(3^{50}-1)$