Find the sum to indicated number of terms in each of the geometric progressions in $\left.1,-a, a^{2},-a^{3}, \ldots n \text { terms (if } a \neq-1\right)$
Find the sum to indicated number of terms in each of the geometric progressions in $\left.1,-a, a^{2},-a^{3}, \ldots n \text { terms (if } a \neq-1\right)$
The given $G.P.$ is $1,-a, a^{2},-a^{3} \ldots \ldots$
Here, first term $=a_{1}=1$
Common ratio $=r=-a$
$S_{n}=\frac{a_{1}\left(1-r^{n}\right)}{1-r}$
$\therefore S_{n}=\frac{1\left[1-(-a)^{n}\right]}{1-(-a)}=\frac{\left[1-(-a)^{n}\right]}{1+a}$
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Let $A _{1}, A _{2}, A _{3}, \ldots \ldots$ be an increasing geometric progression of positive real numbers. If $A _{1} A _{3} A _{5} A _{7}=\frac{1}{1296}$ and $A _{2}+ A _{4}=\frac{7}{36}$, then, the value of $A _{6}+ A _{8}+ A _{10}$ is equal to
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- [JEE MAIN 2022]
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- [IIT 2008]