If the sum of the second, third and fourth te | vedclass

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Question

Let first term $=a>0$

Common ratio $=r>0$

$ar + ar ^{2}+ ar ^{3}=3$

$ar ^{5}+ ar ^{6}+ ar ^{7}=243$

$r^{4}\left(a r+a r^{2}+a r^{3}\

Vedclass · Accepted Answer

$\frac{1}{26}\left(3^{50}-1\right)$

Vedclass · Answer

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

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

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