Given,

$S=109+\frac{108}{5}+\frac{107}{5^2}+\dots +\frac{2}{5^{107}}+\frac{1}{5^{108}} ....\left(1\right)$

$\frac{S}{5}=\frac{109}{5}+\frac{108}{5^2}+\dots +\dots +\frac{2}{5^{108}}+\frac{1}{5^{109}} ....\left(2\right)$

Now subtracting above equations we get,
$\frac{-4S}{5}=-109+\left(\frac{1}{5}+\frac{1}{5^2}+\dots +\frac{1}{5^{108}}+\frac{1}{5^{109}}\right)$

$\Rightarrow \frac{-4S}{5}=-109+\frac{1}{5}\frac{\left(1-{\left(\frac{1}{5}\right)}^{109}\right)}{1-\frac{1}{5}}$

$\Rightarrow \frac{4S}{5}=-109+\frac{1}{4}\left(1-\frac{1}{5^{109}}\right)$

$\Rightarrow \frac{16S}{5}=-436+\left(1-\frac{1}{5^{109}}\right)$

$\Rightarrow 16S=-436\times 5+\left(5-5^{-108}\right)$

$\Rightarrow 16S-(25{)}^{-54}=2175$