If in a G.P. of 64 terms, sum of all terms is 7 times sum of odd terms of G.P, then common ratio of

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8. Sequences and Series

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If in a $G.P.$ of $64$ terms, the sum of all the terms is $7$ times the sum of the odd terms of the $G.P,$ then the common ratio of the $G.P$. is equal to

A

$7$

B

$4$

C

$5$

D

$6$

(JEE MAIN-2024)

Solution

$ a+a r+a r^2+a r^3+\ldots+a r^{63} $

$=7\left(a+a r^2+a r^4 \ldots+a r^{62}\right) $

$\Rightarrow \frac{a\left(1-r^{64}\right)}{1-r}=\frac{7 a\left(1-r^{64}\right)}{1-r^2}$

$r=6$

Standard 11

Mathematics

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