$+10(11)^{9}$
$k=1+2\left(\frac{11}{10}\right)+3\left(\frac{11}{10}\right)^{2}+\ldots 10\left(\frac{11}{10}\right)^{9}$ ......$(i)$
$\left(\frac{11}{10}\right) k=1\left(\frac{11}{10}\right)+2\left(\frac{11}{10}\right)^{2}+\ldots+9\left(\frac{11}{10}\right)^{9}$
$+10\left(\frac{11}{10}\right)^{10}$ .......$(ii)$
On subtracting Eq. $(ii)$ from Eq. $(i),$ we get
$k\left(1-\frac{11}{10}\right)=1+\frac{11}{10}+\left(\frac{11}{10}\right)^{2}+\ldots+\left(\frac{11}{10}\right)^{9}$
$ - 10{\left( {\frac{{11}}{{10}}} \right)^{10}}$
$\Rightarrow k\left(\frac{10-11}{10}\right)=\frac{1\left[\left(\frac{11}{10}\right)^{10}-1\right]}{\left(\frac{11}{10}-1\right)}-10\left(\frac{11}{10}\right)^{10}$
$\Rightarrow-k=10\left[10\left(\frac{11}{10}\right)^{10}-10-10\left(\frac{11}{10}\right)^{10}\right]$
$\Rightarrow \quad k=100$
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$(A)$ $\frac{\pi}{4}$ $(B)$ $\frac{\pi}{6}$ $(C)$ $\frac{\pi}{12}$ $(D)$ $\frac{5 \pi}{12}$