Let $S_n(x)=\log _{a^{\frac{1}{2}}} x+\log _{a^{1 / 3}} x+\log _{a^{\frac{1}{6}}} x+$ $\log _{a^{1 / 11}} x+\log _{a^{1 / 18}} x+\log _{a^{1 / 27}} x+\ldots$ up to $n$-terms. Where $a1$. If $S _{24}(x)=1093$ and $S _{12}(2x)=265$, then the value of $a$ is equal to ___ [JEE Main 2021, 16 Mar (Shift 2)]

Question

Let $S_n(x)=\log _{a^{\frac{1}{2}}} x+\log _{a^{1 / 3}} x+\log _{a^{\frac{1}{6}}} x+$ $\log _{a^{1 / 11}} x+\log _{a^{1 / 18}} x+\log _{a^{1 / 27}} x+\ldots$ up to $n$-terms. Where $a>1$. If $S _{24}(x)=1093$ and $S _{12}(2x)=265$, then the value of $a$ is equal to ___ [JEE Main 2021, 16 Mar (Shift 2)]

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Let \(S_n(x)=\log _{a^{\frac{1}{2}}} x+\log _{a^{1 / 3}} x+\log _{a^{\frac{1}{6}}} x+\) \(\log _{a^{1 / 11}} x+\log _{a^…

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