If the sum of the second, third and fourth
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:

  1. (1) $\frac{1}{26}\left(3^{49}-1\right)$

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

  3. (3) $\frac{2}{13}\left(3^{50}-1\right)$

  4. (4) $\frac{1}{13}\left(3^{50}-1\right)$


Correct Option: , 2

Solution:

Let the first term be ‘ $a$ ‘ and common ratio be ‘ $r$ ‘.

$\because \operatorname{ar}\left(1+r+r^{2}\right)=3$  …(1)

and $a r^{5}\left(1+r+r^{2}\right)=243$..(2)’

From (1) and (2),

$r^{4}=81 \Rightarrow r=3$ and $a=\frac{1}{13}$

$\therefore S_{50}=\frac{a\left(r^{50}-1\right)}{r-1}=\frac{3^{50}-1}{26} \quad\left[\because S_{n}=\frac{a\left(r^{n}-1\right)}{(r-1)}\right]$

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