Mathematics - Sequence & Series Question with Solution | TestHub

Let

$S={\log }_{9^{1/2}}x+{\log }_{9^{1/3}}x+{\log }_{9^{1/4}}x+\dots ..+{\log }_{9^{1/22}}x$

$\Rightarrow S=2{\log }_{9}x+3{\log }_{9}x+\dots \dots +22{\log }_{9}x$

$\Rightarrow S={\log }_{9}x(2+3+\dots ..+22)$

$\Rightarrow S={\log }_{9}x\left\{\frac{21}{2}(2+22)\right\}$

$\Rightarrow S=252{\log }_{9}x$

Given,

$S=504$

$\Rightarrow 252{\log }_{9}x=504$

$\Rightarrow {\log }_{9}x=2$

$\Rightarrow x=81$