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 :
Correct Option: , 2
Solution:
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)
$3 a+9 a+27 a=3$
$a=\frac{1}{13}$
$S_{50}=\frac{a\left(r^{50}-1\right)}{(r-1)}=\frac{1}{26}\left(3^{50}-1\right)$
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