Let $\mathrm{S}=109+\frac{108}{5}+\frac{107}{5^2}+....+\frac{2}{5^{107}}+\frac{1}{5^{108}}$
Then the  value of $(16\mathrm{S}-{\left(25\right)}^{-54})$ is equal to _____________

$$\begin{gathered} \mathrm{S}=109+\frac{108}{5}+\frac{107}{5^2}+....\frac{2}{5^{107}}+\frac{1}{5^{108}} ...\left(1\right) \\ \frac{\mathrm{S}}{5}=\frac{109}{5}+\frac{108}{5^2}+\frac{107}{5^3}+....+\frac{2}{5^{108}}+\frac{1}{5^{109}} ....\left(2\right) \left(1\right)-\left(2\right) \mathrm{we} \mathrm{get} \\ \mathrm{S}-\frac{\mathrm{S}}{5}=109-[\frac{1}{5}+\frac{1}{5^2}+...+\frac{2}{5^{108}}+\frac{1}{5^{109}}] \\ \Rightarrow \frac{4}{5}\mathrm{S}=109-\left[\frac{{\displaystyle \frac{{\displaystyle 1}}{{\displaystyle 5}}(1-{\left(\frac{{\displaystyle 1}}{{\displaystyle 5}}\right)}^{109})}}{{\displaystyle 1-\frac{{\displaystyle 1}}{{\displaystyle 5}}}}]\right] \\ \Rightarrow \frac{4}{5}\mathrm{S}=109-\left[\frac{1}{4}(1-{\left(\frac{1}{5}\right)}^{109})\right]\Rightarrow \mathrm{S}=\frac{5}{4}[109-\left(\frac{1}{4}(1-{\left(\frac{1}{5}\right)}^{109})\right)] \\ =\frac{5}{4}\left[109-\frac{{\displaystyle 5}}{{\displaystyle 16}}{(1-\frac{{\displaystyle 1}}{{\displaystyle 5}})}^{109}\right]\Rightarrow \mathrm{S}=\frac{5\left(109\right)}{4}-\frac{5}{16}+\frac{1}{16}{5}^{-108}\Rightarrow \mathrm{S}=\frac{20\left(109\right)}{16}-\frac{5}{16}+\frac{1}{16}{5}^{-108} \\ \Rightarrow 16\mathrm{S}=20\left(109\right)-5+5^{-108}\Rightarrow 16\mathrm{S}=2175+{25}^{-54}\Rightarrow 16\mathrm{S}-{25}^{-54}=2175 \end{gathered}$$