If sum of second, third and fourth terms of a positive term G.P. is 3 and sum of its sixth, seventh

English

Gujarati

Hindi

8. Sequences and Series

hard

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

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

(JEE MAIN-2020)

Solution

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)

$3 a+9 a+27 a=3$

$a=\frac{1}{13}$

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

Standard 11

Mathematics

Share

Similar Questions

If the range of $f(\theta)=\frac{\sin ^4 \theta+3 \cos ^2 \theta}{\sin ^4 \theta+\cos ^2 \theta}, \theta \in \mathbb{R}$ is $[\alpha, \beta]$, then the sum of the infinite $G.P.$, whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$, is equal to………..

hard

(JEE MAIN-2024)

$x = 1 + a + {a^2} + ….\infty \,(a < 1)$ $y = 1 + b + {b^2}…….\infty \,(b < 1)$ Then the value of $1 + ab + {a^2}{b^2} + ……….\infty $ is

medium

If ${G_1}$ and ${G_2}$ are two geometric means and $A$ the arithmetic mean inserted between two numbers, then the value of $\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}}$is

medium

Consider two G.Ps. $2,2^{2}, 2^{3}, \ldots$ and $4,4^{2}, 4^{3}, \ldots$ of $60$ and $n$ terms respectively. If the geometric mean of all the $60+n$ terms is $(2)^{\frac{225}{8}}$, then $\sum_{ k =1}^{ n } k (n- k )$ is equal to.

hard

(JEE MAIN-2022)

If in a geometric progression $\left\{ {{a_n}} \right\},\;{a_1} = 3,\;{a_n} = 96$ and ${S_n} = 189$ then the value of $n$ is

easy